Topology of closure systems in algebraic lattices

Abstract

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system $\langle X \rangle _P$. The closure system $\langle Y \rangle _P$ generated by the patch closure Y of X is the patch closure of $\langle X \rangle _P$. If X is contained in the set of nontrivial prime elements of P then $\langle X \rangle _P$ is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.

1 Introduction

Algebraic lattices, [4, Section I.4], [8, Section I-4], [9, p. 106, Definition 12], and closure systems, [5, section II.1], [4, Section I.5], [8, p. 26 ff], are classical topics in lattice theory and poset theory. Algebraic lattices are a class of complete lattices and abound in algebra, cf. [6, Examples 7.2.13], [8, Theorem I$-$4.16]. By [6, Theorem 7.2.8] every algebraic lattice is a spectral space for its coarse lower topology (see [6, p. 589] or 2.9 for the definition). Closure systems in algebraic lattices are studied as subspaces.

Every topological space T carries a binary relation ${\,\rightsquigarrow \,}$, which is called specialization and is defined by: $x {\,\rightsquigarrow \,}y$ if $y \in \overline{\{x\}}$, cf. [6, 1.1.3]. It is well known that the specialization relation is a partial order if and only if X is a $T_0$-space, and then specialization establishes connections between topological spaces and posets. Moreover, a $T_0$-topology on a poset is a lower topology, resp. an upper topology, if the specialization order is the given partial order, resp. the inverse partial order. Every poset P has at least one, frequently many, upper and lower topologies. Some particularly important examples are described in 2.9, e.g., the coarse lower topology, $\tau ^{\ell }(P)$, and the Scott topology, $\sigma (P)$ (which is an upper topology).

Let P be a poset. See [8, Definition I$-$1.1] for the notion of compact elements in P. The set of compact elements is denoted by ${{\mathbb {K}}}(P)$. In particular, if T is a topological space and ${\mathcal {O}}(T)$ is the frame of open sets then ${{\mathbb {K}}}({\mathcal {O}}(T)) = \overset{\circ }{{\mathcal {K}}}(T)$ is the set of quasi-compact open subsets of T. For a bounded lattice P the bottom element is compact and the join of two compact elements is compact, but the top element and the meet of two compact elements need not be compact. If these are also compact then P is called coherent. A complete lattice P is algebraic if every element is the supremum of the compact elements below it. Thus an algebraic lattice has many compact elements, and ${{\mathbb {K}}}(P)$ is a decisive part of its structure. If P is an algebraic lattice then $\tau ^{\ell }(P)$ is a spectral topology and $\sigma (P)$ is its inverse topology, [6, Theorem 7.2.8]. Recall that a topological space T is spectral if it is $T_0$ and sober and $\overset{\circ }{{\mathcal {K}}}(T)$ is a basis of open sets and a bounded sublattice of the frame of open sets, [6, Definition 1.1.5]. An important part of the structure of a spectral space is the patch topology (or constructible topology), which is the Boolean topology generated by the quasi-compact open sets and their complements, [6, Section 1.3]. The coarse lower topology and the Scott topology are defined for arbitrary posets, but usually are not spectral topologies. Although the focus of the paper is on algebraic lattices, various results can and will be proved for closure systems in larger classes of posets.

The first two sections are of a preparatory nature. They fix notation, recall terminology and exhibit various elementary facts about posets and closure systems. Section 2 deals with posets and topologies on posets. Morphisms between posets are the monotonic maps. Order-theoretic properties of posets and poset maps are related to topological properties, several typical results are contained in Theorem 2.11. Closure operators and closure systems in posets are discussed in Section 3. Closure systems in a poset P correspond bijectively to closure operators. The closure operator $P \rightarrow P$ belonging to a closure system $Q \subseteq P$ is denoted by $\eta _{Q,P}$. Conversely, the image of a closure operator $\eta :P \rightarrow P$ is a closure system. The set of closure systems is denoted by ${\mathscr {C}}(P)$. The algebraic closure systems are a particularly important subset of ${\mathscr {C}}(P)$, which is denoted by ${\mathscr {A}}(P)$. If P is a complete lattice then ${\mathscr {C}}(P)$ and ${\mathscr {A}}(P)$ are both closure systems in the power set ${\mathfrak {P}}(P)$. Thus, every subset $X \subseteq P$ generates a closure system, which is denoted by $\langle X \rangle _P$. First topological properties of closure systems appear in Remark 3.6 and Proposition 3.7. For example, if P is a poset and Q is a subset then both P and Q carry their respective coarse lower topologies and Scott topologies. It is always true that $\tau ^{\ell }(Q) \subseteq \tau ^{\ell }(P)|_Q$, and equality holds if Q is a closure system. The relation $\sigma (Q) \subseteq \sigma (P)|_Q$ holds for all closure systems, and equality holds if and only if Q is an algebraic closure system.

The analysis of closure systems in algebraic lattices starts with Section 4. If $Q \subseteq P$ is a closure system in an algebraic lattice then it is always true that ${{\mathbb {K}}}(Q) \subseteq \eta _{Q,P}({{\mathbb {K}}}(P))$, Proposition 4.2. It is algebraic if and only if it is patch closed in P, if and only if ${{\mathbb {K}}}(Q) = \eta _{Q,P}({{\mathbb {K}}}(P))$, Theorem 4.5.

Generating sets of closure systems enter the picture in Section 5. Let X be any subset in an algebraic lattice P and Y the patch closure of X. Then $\langle Y \rangle _P$ is the patch closure of $\langle X \rangle _P$ in P, and the compact elements of $\langle Y \rangle _P$ are contained in $\langle X \rangle _P$ (but need not be compact in $\langle X \rangle _P$), Theorem 5.2. Section 6 is devoted to complete lattices and closure systems generated by sets of prime elements. An element $p \in P$ is prime if $a \wedge b \le p$ implies $a \le p$ or $b \le p$, [8, Proposition I$-$3.12]. The set of prime elements is denoted by ${{\mathbb {P}}}(P)$. The top element is always prime, the trivial prime element. The set of nontrivial prime elements is denoted by ${{\mathbb {P}}}(P)^{\flat }$ and may be empty. The inclusion ${{\mathbb {P}}}(P) \cap Q \subseteq {{\mathbb {P}}}(Q)$ holds for any closure system $Q \subseteq P$ and may be proper. But if $Q = \langle {{\mathbb {P}}}(Q) \rangle _P$ then Q is isomorphic to the frame ${\mathcal {O}}({{\mathbb {P}}}(Q)^{\flat })$, Theorem 6.13, and the equality ${{\mathbb {P}}}(P) \cap Q = {{\mathbb {P}}}(Q)$ holds if and only if the closure operator $\eta _{Q,P}$ is a $\wedge$-homomorphism, Corollary 6.9. The set of closure systems satisfying these conditions is denoted by ${\mathscr {N}}(P)$. The poset ${\mathscr {N}}(P)$ (with inclusion) is isomorphic to the set of closed sets for the b-topology associated with $\tau ^{\ell }(P)$, Remark 6.15. Coherent algebraic lattices are particularly important in applications. It is shown in Theorem 6.20 that ${{\mathbb {P}}}(P)^{\flat } \subseteq P$ is patch closed if P is a coherent algebraic frame. On the other hand, if P is an algebraic lattice, X is patch closed in P and is contained in ${{\mathbb {P}}}(P)^{\flat }$ then $\langle X \rangle _P$ is a coherent algebraic frame.

In Section 7 results of the previous sections are specialized to frames. If P is a frame and a closure system Q belongs to ${\mathscr {N}}(P)$ then the closure operator $\eta _{Q,P}$ is a nucleus of P. If P is a coherent algebraic frame then ${\mathscr {A}}(P) \cap {\mathscr {N}}(P)$, the set of algebraic closure systems in ${\mathscr {N}}(P)$, is isomorphic (as a poset) to the set of patch closed subsets of ${{\mathbb {P}}}(P)^{\flat }$ and is a closure system in ${\mathfrak {P}}(P)$, Theorem 7.1.

The literature about posets, complete and algebraic lattices, frames, and closure systems is huge. We mention [4, 5, 8, 9, 11, 15] as general references. The terminology and notation for posets is mostly the same as in [6, Appendix]. Everything needed about spectral spaces can be found in [6].

2 Topological spaces and posets

To study connections between posets and topological spaces it is first necessary to fix the notation and terminology and note some elementary facts. In particular we mention the b-topology associated with any topological space and its connections with irreducible sets, generic points and soberness, cf. 2.2 and 2.3. We recall the specialization order in $T_0$-spaces, cf. 2.8, as well as intrinsic topologies, cf. [14], on posets, most importantly the coarse lower topology, the Scott topology and the Lawson topology, 2.9. The morphisms between posets are the monotonic maps, cf. 2.5. A continuous map between $T_0$-spaces is monotonic for the specialization order. Various order-theoretic properties of a poset map are related to continuity properties with respect to suitable upper and lower topologies, cf. Theorem 2.11 and its corollaries. We pay particular attention to left adjoint and right adjoint maps of poset maps. They are used to define closure systems and are decisive tools for their study.

2.1 Notation for topological spaces

For a topological space $T = (T,\tau )$ the frame of open sets is denoted by ${\mathcal {O}}(T)$ or ${\mathcal {O}}(\tau )$, and ${\mathcal {A}}(T) = {\mathcal {A}}(\tau )$ is the set of closed sets. A continuous map $f: S \rightarrow T$ yields the inverse image maps ${\mathcal {O}}(f): {\mathcal {O}}(S) \rightarrow {\mathcal {O}}(T)$ and ${\mathcal {A}}(f): {\mathcal {A}}(S) \rightarrow {\mathcal {A}}(T)$.

2.2 The b-topology associated with a topological space

The locally closed sets in a topological space $T = (T,\tau )$ are a basis of open sets for the b-topology, denoted by $\beta = \beta (\tau )$, cf. [19, 2.2] or [6, 4.5.20]. We call $\beta (T) = (T,\beta (\tau ))$ the b-space of T. The b-topology has also been studied under the name Goldman topology or G-topology, see [3, section 4], in particular loc.cit., Proposition 4.2. If ${\mathcal {B}}$ is a basis of open sets for $\tau$ then the sets $\overline{\{x\}} \cap O$ with $O \in {\mathcal {B}}$ and $x \in T$ are a basis of open sets for $\beta (\tau )$. The closure of $A \subseteq T$ for the b-topology is denoted by $\overline{A}^{\beta } = \overline{A}^{\beta (\tau )}$, and A is said to be very dense in a $\beta$-closed set $B \subseteq T$ if $\overline{A}^{\beta } = B$. A continuous map $f:S \rightarrow T$ is also continuous as a map of the associated b-spaces. In particular, if S is a subspace of T then $\beta (S) \subseteq \beta (T)$ is a subspace.

2.3 Irreducible sets, sober spaces, and sobrification

A $T_0$-space T is sober if every nonempty closed and irreducible subset C is equal to $\overline{\{c\}}$ for a unique $c \in C$, the generic point of C. Every $T_0$-space is contained in a smallest sober space, its sobrification. The sobrification of T is denoted by $\textbf{Sob}(T)$, and $\textbf{Sob}_T:T \rightarrow \textbf{Sob}(T)$ is the inclusion map. The sobrification was first studied in [10, Section 0.2.9] where it is presented as the set of nonempty closed and irreducible subsets of T with a suitable topology. Another presentation is given in [6, Section 11.2], which also contains a collection of basic facts of the sobrification. In particular, the subspace $T \subseteq \textbf{Sob}(T)$ is very dense, [6, Corollary 11.2.4]. The paper [3] presents various results about the sobrification and connections with the b-topology. In particular, loc.cit., Theorem 4.16 and Theorem 4.20 show that for any sober space T

• (a) the sober subspaces are the b-closed subsets, and

• (b) for any subspace S the inclusion $S \rightarrow \overline{S}^{\beta }$ is isomorphic to the sobrification.

By [2], a continuous map $f:S \rightarrow T$ of $T_0$-spaces is an epimorphism in the category of $T_0$-spaces if and only if $f(S) \subseteq T$ is very dense. Thus, $\textbf{Sob}(S)$ is (up to isomorphism) the largest epimorphic extension of S and is contained in any sober space containing S.

2.4 Notation and terminology for posets

Let $P = (P,\le )$ be a poset. Top and bottom elements in P, if they exist, are denoted by $\top = \top _P$ and $\bot = \bot _P$. If both exist then the poset is bounded. For a subset $X \subseteq P$ we define $X^{\flat } = X {\setminus } \{\top _P\}$ and $X^{\sharp } = X \cup \{\top _P\}$. (Nothing happens if $\top$ does not exist.) Every $a \in P$ generates a principal upset, $a^{\uparrow }$, and a principal downset, $a^{\downarrow }$. For $S \subseteq P$ we set $S^{\uparrow } = \bigcup _{a \in S} a^{\uparrow }$ and $S^{\downarrow } = \bigcup _{a \in S} a^{\downarrow }$. The sets of upper, resp. lower, bounds of S are $S^{\Uparrow } = \bigcap _{a \in S}a^{\uparrow }$, resp. $S^{\Downarrow } = \bigcap _{a \in S} a^{\downarrow }$. For $Q \subseteq P$ we define $a^{\uparrow Q} = a^{\uparrow } \cap Q$, $S^{\uparrow Q} = S^{\uparrow } \cap Q$, $S^{\Uparrow Q} = S^{\Uparrow } \cap Q$, and so on. The set of all upsets, resp. principal upsets, are denoted by ${\Uparrow }(P)$, resp. ${\uparrow }(P)$.

We call P a dcpo, resp. an fcpo, if $\bigvee _P D$ exists for every up-directed $D \subseteq P$, resp. $\bigwedge _P D$ exists for every down-directed $D \subseteq P$, cf. [8, 0$-$2.1]. Note that up-directed and down-directed sets are non-empty by definition.

For the way-below relation on P and the definition of compact elements we refer to [8, I$-$1.1] or [6, 7.1.1]. If $a \in P$ then $a^{\ll }= a^{\ll }_P$ is the set of elements that are way below a and $a^{\gg } = a^{\gg }_P = \{x \in P \mid a \ll x\}$. The set of compact elements is denoted by ${{\mathbb {K}}}(P)$. Bottom elements, if they exist, are trivially compact. There need not be any other compact elements. Now assume that P is a bounded lattice. Then ${{\mathbb {K}}}(P)$ is always a $\vee$-subsemilattice of P. If the meet of two compact elements is compact (i.e., ${{\mathbb {K}}}(P) \subseteq P$ is a sublattice) then P is arithmetic, cf. [8, I$-$4.7]. Moreover, P is coherent if ${{\mathbb {K}}}(P)$ is a bounded sublattice, cf. [1, p. 2]. The way below relation and compact elements are used in [8, Definitions I$-$1.6 and I$-$4.2] to define continuous lattices (a complete lattice P such that $a^{\ll }$ is up-directed for each $a \in P$ and $a = \bigvee a^{\ll }$) and algebraic lattices (see the introduction).

2.5 Poset morphisms

A map $\varphi :P \rightarrow Q$ between posets is a poset morphism if it is monotonic. A poset map may have additional properties, which will always be announced explicitly.

• $\varphi$ is a dcpo-homomorphism if $\bigvee _Q \varphi (D)$ exists and is equal to $\varphi (\bigvee _P D)$ for all up-directed $D \subseteq P$ such that $\bigvee _P D$ exists. The notion of fcpo-homomorphisms is defined similarly.

• $\varphi$ is a $\wedge$-homomorphism if $\varphi (a) \wedge _Q \varphi (b)$ exists and is equal to $\varphi (a \wedge _P b)$ for all $a,b \in P$ such that $a \wedge _P b$ exists. Homomorphisms for $\bigwedge$, $\vee$ and $\bigvee$ are defined accordingly. Clearly, a $\bigwedge$-map, resp. a $\bigvee$-map, is also a $\wedge$-map and an fcpo-map, resp. a $\vee$-map and a dcpo-map.

• $\varphi$ is a poset-embedding if $x \le y$ is equivalent to $\varphi (x) \le \varphi (y)$ for $x,y \in P$, resp. a $\wedge$-embedding if it is a poset-embedding, a $\wedge$-morphism and $x \wedge _P y$ exists if $\varphi (x) \wedge _Q \varphi (y)$ exists. Embeddings for $\bigwedge$, $\vee$, $\bigvee$, as well as dcpo-embeddings and fcpo-embeddings are defined similarly.

• $\varphi :P \rightarrow Q$ is coherent if $\varphi ({{\mathbb {K}}}(P)) \subseteq {{\mathbb {K}}}(Q)$, cf. [1, p. 2].

If $\varphi$ is a poset map then the inverse image map ${\mathfrak {P}}(\varphi ):{\mathfrak {P}}(Q) \rightarrow {\mathfrak {P}}(P)$ of the power sets restricts to a poset map ${\Uparrow }(\varphi ): {\Uparrow }(Q) \rightarrow {\Uparrow }(P)$. The inclusion ${\mathfrak {P}}(\varphi )({\uparrow }(Q)) \subseteq {\uparrow }(P)$ need not hold, but if it holds then ${\uparrow }(\varphi ):{\uparrow }(Q) \rightarrow {\uparrow }(P)$ denotes the restriction of ${\mathfrak {P}}(\varphi )$.

2.6 Adjoint pairs of poset morphisms

See [8, p. 22 ff] and [7, p. 155 ff] for the following notions and facts. Consider poset morphisms $\varphi :P \rightarrow Q$ and $\psi :Q \rightarrow P$. We say that $(\varphi ,\psi )$ is an adjoint pair with $\psi$ right adjoint to $\varphi$ and $\varphi$ left adjoint to $\psi$ if the equivalence $\varphi (a) \le b \Leftrightarrow a \le \psi (b)$ holds for all $a \in P$ and $b \in Q$, if and only if $\varphi ^{-1}(b^{\downarrow }) = \psi (b)^{\downarrow }$ for all $b \in Q$, if and only if $\psi ^{-1}(a^{\uparrow }) = \varphi (a)^{\uparrow }$ or all $a \in P$. Each member of the pair $(\varphi ,\psi )$ determines the other one uniquely. The map $\varphi$ has a right adjoint $\varphi _*$ if and only if for all $b \in Q$ the set $\varphi ^{-1}(b^{\downarrow })$ has a largest element (which then is equal to $\varphi _*(b)$). Similarly, $\psi$ has a left adjoint $\psi ^*$ if and only if for all $a \in P$ the set $\psi ^{-1}(a^{\uparrow })$ has a smallest element (which then is equal to $\psi ^*(a)$), if and only if the map ${\uparrow }(\psi )$ is defined, cf. 2.5. Let $(\varphi ,\psi )$ be an adjoint pair.

• (a) If $a \in P$ and $b \in Q$ then $a \le \psi \circ \varphi (a)$ and $\varphi \circ \psi (b) \le b$.

• (b) $\varphi \circ \psi \circ \varphi = \varphi$ and $\psi \circ \varphi \circ \psi = \psi$, [7, p. 159, Lemma 7.26].

• (c) $\psi$ is injective if and only if $\varphi \circ \psi = {{\,\textrm{id}\,}}_Q$, if and only if $\varphi$ is surjective, whereas $\psi$ is surjective if and only if $\psi \circ \varphi = {{\,\textrm{id}\,}}_P$, if and only if $\varphi$ is injective, [8, Proposition 0$-$3.7]. If $\psi$ is injective then it is a poset-embedding. For, $x,y \in Q$ and $\psi (x) \le \psi (y)$ implies $x = \varphi \circ \psi (x) \le \varphi \circ \psi (y) = y$. The same holds for $\varphi$.

If P is complete then $\varphi :P \rightarrow Q$ has a right adjoint, resp. left adjoint, if and only if it is a $\bigvee$-morphism, resp. a $\bigwedge$-morphism, [8, Corollary 0$-$3.5].

Proposition 2.7

Let $(\varphi :P \rightarrow Q,\psi :Q \rightarrow P)$ be an adjoint pair.

• (a) $\psi$ is a $\bigwedge$-morphism and $\varphi$ is a $\bigvee$-morphism.

• (b) If $\psi$ is injective then it is a $\bigwedge$-embedding, hence a $\wedge$-embedding and an fcpo-embedding. If $\varphi$ is injective then it is a $\bigvee$-embedding, hence a $\vee$-embedding and a dcpo-embedding.

• (c) If $\psi$ is injective and a $\bigvee$-homomorphism, resp. a $\vee$-homomorphism, resp. a dcpo-morphism, then it is a $\bigvee$-embedding, resp. a $\vee$-embedding, resp. a dcpo-embedding. If $\varphi$ is injective and a $\bigwedge$-homomorphism, resp. a $\wedge$-homomorphism, resp. an fcpo-morphism, then it is a $\bigwedge$-embedding, resp. a $\wedge$-embedding, resp. an fcpo-embedding.

Proof

It suffices to prove the assertions about $\psi$; the proofs for $\varphi$ are similar.

(a). Pick $S \subseteq Q$ such that $\bigwedge _Q S$ exists. Then $\psi (\bigwedge _Q S) \le \psi (s)$ for all $s \in S$. Pick $a \in P$ with $a \le \psi (s)$ for all $s \in S$. Then $\varphi (a) \le s$ for all $s \in S$, hence $\varphi (a) \le \bigwedge _Q S$. It follows that $a \le \psi (\bigwedge _Q S)$, i.e., $\psi (\bigwedge _Q S) = \bigwedge _P \psi (S)$.

(b). Note that $\psi$ is a poset-embedding by 2.6(c) and is a $\bigwedge$-morphism by (a). Assume $S \subseteq Q$ and $a = \bigwedge _P \psi (S)$ exists. We show that $\varphi (a) = \bigwedge _Q S$. For all $s \in S$ we have $\varphi (a) \le s$. Pick $b \in Q$ with $b \le s$ for all $s \in S$. Then $\psi (b) \le \psi (s)$ for all s, hence $\psi (b) \le a$, and 2.6(c) implies $b = \varphi \circ \psi (b) \le \varphi (a)$. It follows that $\bigwedge _Q S$ exists and is equal to $\varphi (a)$. Clearly, $\psi$ is a $\wedge$-embedding and an fcpo-embedding.

(c). Again, $\psi$ is a poset-embedding by 2.6(c) and is a $\bigvee$-map by hypothesis. Now suppose $S \subseteq Q$ and $a = \bigvee _P \psi (S)$ exists. By 2.6(a) we have $\psi (s) \le a \le \psi \circ \varphi (a)$ for all $s \in S$. Since $\psi$ is a poset-embedding it follows that $s \le \varphi (a)$ for all $s \in S$. Now let $b \in Q$ be an upper bound for S. Then $\psi (b)$ is an upper bound for $\psi (S)$, and we see that $a \le \psi (b)$. It follows that $\varphi (a) \le b$.

If $\psi$ is a $\vee$-map, resp. a dcpo-map, then use subsets $S \subseteq Q$ with $|S| = 2$, resp. up-directed sets $S \subseteq Q$. $\square$

2.7 Specialization in topological spaces

See the Introduction for the definition of the specialization relation of a topological space. Every $T_0$-space is considered as a poset via specialization, and $\overline{\{x\}} = x^{\uparrow }$. Continuous maps between $T_0$-spaces are monotonic for specialization.

Consider a $T_0$-space T and a nonempty irreducible subset C. If $\overline{C}$ has the generic point c, cf. 2.3, then $\overline{C} = c^{\uparrow }$ and $\bigwedge _T C$ exists and is equal to c, [6, Proposition 4.2.1(ii)]. Every down-directed set in T is nonempty and irreducible, [6, Proposition 4.2.1(i)], but irreducible sets need not be down-directed, [6, Example 4.2.3]. We say that T is

• (a) sober for down-directed sets if $\overline{C}$ has a generic point whenever $C \subseteq T$ is down-directed;

• (b) sober for irreducible sets with infimum if $\overline{C}$ has a generic point for all nonempty irreducible C such that $\bigwedge _T C$ exists;

• (c) sober for down-directed sets with infimum if $\overline{C}$ has a generic point for all down-directed sets C such that $\bigwedge _T C$ exists.

Note the following simple facts.

• (d) $C \subseteq T$ has an infimum if and only if $\overline{C}$ has an infimum. Then the infima are equal.

• (e) Let $f:S \rightarrow T$ be continuous, $C \subseteq S$ a nonempty irreducible set and assume that $\overline{C}^S$ has a generic point, namely $\bigwedge _S C$. Then f(C) is nonempty irreducible and $\overline{f(C)}^T = f(\bigwedge _S C)^{\uparrow }$, hence $f(\bigwedge _S C) = \bigwedge _T f(C)$.

• (f) Let $S \subseteq T$ a subspace, $C \subseteq S$ nonempty and irreducible. Then $\overline{C}^S$ has a generic point if and only if $\overline{C}^T$ has a generic point and $\bigwedge _T C \in S$.

• (g) Let T be sober and $S \subseteq T$ a subspace. Then S is sober if and only if $\bigwedge _T C \in S$ for all nonempty irreducible $C \subseteq S$.

2.8 Lower and upper topologies on posets

A topology on the poset P is said to be intrinsic, cf. [14], if it can be defined in terms of the partial order. A $T_0$-topology on P is a lower topology, or an upper topology if its specialization order is $\le$, resp. the inverse partial order $\le _{{{\,\textrm{inv}\,}}}$, cf. [6, p. 589]. Every open set of a lower topology is a downset for $\le$, every open set of an upper topology is an upset. If $(T,\tau )$ is a $T_0$-space then $\tau$ is a lower topology for ${\,\rightsquigarrow \,}$. The coarse lower topology $\tau ^{\ell }(P)$ and the coarse upper topology $\tau ^u(P)$ (for the definition see [6, p. 589] ) are intrinsic topologies. The set ${\uparrow }(P)$ (see 2.5 for the notation) is called the canonical subbasis of closed sets for the coarse lower topology. If $\tau$ is any lower topology then ${\uparrow }(P)$ is the set of nonempty closed and irreducible sets with generic point. We need two more intrinsic topologies:

• (a) A set $U \subseteq P$ is open for the lower Scott topology $\sigma ^{\ell }(P)$ if $U = U^{\downarrow }$ and $U \cap D \ne {\emptyset }$ for each down-directed $D \subseteq P$ such that $\bigwedge _P D$ exists and belongs to U.

• (b) A set $U \subseteq P$ is open for the Scott topology $\sigma (P)$ if $U = U^{\uparrow }$ and $U \cap D \ne {\emptyset }$ for each up-directed set $D \subseteq P$ such that $\bigvee _P D$ exists and belongs to U. (This definition is slightly more general than the usual one, where it is assumed that P is a dcpo, [8, Definition II$-$1.3], [14, p. 82], [6, Definition 7.1.6].) In particular, a principal upset $a^{\uparrow }$ is Scott open if and only if $a \in {{\mathbb {K}}}(P)$.

For the case of a dcpo basic properties of $\tau ^{\ell }(P)$, resp. $\sigma (P)$, are presented in [6, 7.1.4], resp. [8, section II-1] or [6, 7.1.6 and 7.1.8]. The coarse upper topology (resp. the lower Scott topology) is the coarse lower topology (resp. the Scott topology) for the inverse partial order $\le _{{{\,\textrm{inv}\,}}}$. Thus properties of $\tau ^{u}(P)$ and $\sigma ^{\ell }(P)$ follow from properties of $\tau ^{\ell }(P)$ and $\sigma (P)$, and vice versa. Note the following useful fact.

• (c) A lower topology is coarser than the lower Scott topology if and only if it is sober for down-directed sets with infimum.

Let $\tau$ be an upper topology on P. Then item (c) says that $\tau \subseteq \sigma (P)$ if and only if $\overline{D}^{\tau }$ has a generic point for every up-directed set D with supremum. This equivalence strengthens [6, 7.1.8(x)], where it is shown that every spectral upper topology on P is coarser than the Scott topology.

The join of $\tau ^{\ell }(P)$ and $\sigma (P)$ in the lattice of topologies on P is the Lawson topology, [8, Definition III$-$1.5], which is denoted by $\lambda (P)$. A detailed discussion of the Lawson topology is contained in [8, Chapter III], always under the assumption that P is a dcpo. However, various results and arguments are true with the more general definition used here. The sets $U {\setminus } F^{\uparrow }$ with $U \in {\mathcal {O}}(\sigma (P))$ and $F \subseteq P$ finite are a basis of open sets for the Lawson topology, [8, p. 211 f.]. The Lawson topology is coarser than both b-topologies $\beta (\tau ^{\ell }(P))$ and $\beta (\sigma (P))$. An upset is Lawson open if and only if it is Scott open, [8, Proposition III$-$1.6(a)].

Let $\varphi :P \rightarrow Q$ be a poset map. If P and Q are both equipped with their coarse lower topologies, resp. their Scott topologies, and so on, and $\varphi$ is continuous then we say that $\varphi$ is coarse lower continuous, resp. Scott continuous, and so on.

2.9 Algebraic lattices as spectral spaces

An algebraic lattice P is a spectral space for the coarse lower topology, and the Scott topology is the inverse topology, [6, Theorem 7.2.8]. A subset $U \subseteq P$ is quasi-compact open if and only if $U = P {\setminus } F^{\uparrow }$ with $F \subseteq {{\mathbb {K}}}(P)$ finite, loc. cit.. Thus the sets $a^{\uparrow }$ with $a \in {{\mathbb {K}}}(P)$ are a subbasis of closed sets for $\tau ^{\ell }(P)$. Usually this is a proper subset of the canonical subbasis ${\uparrow }(P)$, cf. 2.9.

The patch topology of the spectral space P is the join of the spectral topology and the inverse topology, hence is equal to $\lambda (P)$. For any subset $X \subseteq P$ the patch closure of X is denoted by $\overline{X}^{{{\,\textrm{con}\,}}}$. The basic constructible sets $F^{\uparrow } \setminus G^{\uparrow }$, with $F,G \subseteq {{\mathbb {K}}}(P)$ finite, are a basis of open sets for the patch topology. Since $F^{\uparrow } {\setminus } G^{\uparrow } = \bigcup _{a \in F} a^{\uparrow } {\setminus } G^{\uparrow }$ the sets $a^{\uparrow } \setminus G^{\uparrow }$ (with $a \in {{\mathbb {K}}}(P)$ and $G \subseteq {{\mathbb {K}}}(P)$ finite) are a basis as well, and the sets $a^{\uparrow } {\setminus } b^{\uparrow }$ (where $a,b \in {{\mathbb {K}}}(P)$) are a subbasis. Note that ${{\mathbb {K}}}(P)$ is dense in P for the patch topology since a basic open set $a^{\uparrow } \setminus G^{\uparrow }$ is nonempty if and only if $a \in a^{\uparrow } {\setminus } G^{\uparrow }$.

Theorem 2.11

Let $\varphi :P \rightarrow Q$ be a poset map. Then:

• (a) Let $\varphi :(P,\tau _P) \rightarrow (Q,\tau _Q)$ be continuous, where $\tau _P$ and $\tau _Q$ are lower topologies on P and Q. If $\tau _P \subseteq \sigma ^{\ell }(P)$ then $\varphi$ is an fcpo map.

• (b) $\varphi$ is an fcpo map if and only if it is lower Scott continuous.

• (c) If $\varphi$ has a left adjoint then it is coarse lower continuous.

• (d) Let $\varphi :(P,\tau _P) \rightarrow (Q,\tau _Q)$ be continuous, where $\tau _P$ and $\tau _Q$ be upper topologies on P and Q. If $\tau _P \subseteq \sigma (P)$ then $\varphi$ is a dcpo map.

• (e) (cf. [8, Proposition II$-$2.1]) $\varphi$ is a dcpo map if and only if it is Scott continuous.

• (f) If $\varphi$ has a right adjoint then it is coarse upper continuous.

• (g) If $\varphi$ is Lawson continuous then it is Scott continuous (cf. [8, Theorem III$-$1.8]) and lower Scott continuous.

Proof

(a). If $D \subseteq P$ is down-directed and $\bigwedge _P D$ exists, then $\overline{D} = (\bigwedge _P D)^{\uparrow }$, cf. 2.9(c), and $\varphi (\bigwedge _P D) = \bigwedge _Q \varphi (D)$ holds by 2.8(e).

(b). Assume that $\varphi$ is an fcpo-map and let $V \subseteq Q$ be $\sigma ^{\ell }(Q)$-open. It is claimed that the downset $U = \varphi ^{-1}(V) \subseteq P$ is $\sigma ^{\ell }(P)$-open. If $D \subseteq P$ is down-directed such that $\bigwedge _P D$ exists and belongs to U then $\bigwedge _Q \varphi (D) = \varphi (\bigwedge _P D) \in V$, hence $\varphi (D) \cap V \ne {\emptyset }$, i.e., $D \cap U \ne {\emptyset }$. – The converse holds by (a).

(c) follows from 2.6, and (d), (e), (f) are proved exactly as (a), (b), (c), mutatis mutandis (or one applies (a), b), (c) to the inversely ordered sets $P_{{{\,\textrm{inv}\,}}}$ and $Q_{{{\,\textrm{inv}\,}}}$).

(g). For Scott continuity, let $V \subseteq Q$ be Scott open. Then $U = \varphi ^{-1}(V)$ is an upset in P and is Lawson open, hence is Scott open, cf. 2.9. To prove lower Scott continuity we show that $\varphi$ is an fcpo-map, cf. item (b). So, assume $D \subseteq P$ is down-directed and $\bigwedge _P D$ exists. It is claimed that $\varphi (\bigwedge _P D)$ is the infimum of $\varphi (D)$. If this is false there is some $b \in \varphi (D)^{\Downarrow }$ with $b \not \le \varphi (\bigwedge _P D)$. The set $V = Q {\setminus } b^{\uparrow Q}$ is open for $\tau ^{\ell }(Q)$, hence is Lawson open, and $U = \varphi ^{-1}(V)$ is Lawson open in P, say $U = \bigcup _{i \in I} U_i {\setminus } F_i^{\uparrow }$ (with $U_i$ Scott open and $F_i \subseteq P$ finite). Since $\varphi (D) \subseteq b^{\uparrow Q}$ and $b \not \le \varphi (\bigwedge _P D)$ we see that $\bigwedge _P D \in U$ and $D \cap U = {\emptyset }$. Pick $i \in I$ such that $\bigwedge _P D \in U_i {\setminus } F_i^{\uparrow }$ and let $F_i = \{x_1,\ldots ,x_l\}$. For each $j = 1,\ldots ,l$ there is some $d_j \in D \cap P {\setminus } x_j^{\uparrow }$. For, otherwise $x_j \in D^{\Downarrow }$ and $x_j \le \bigwedge _P D$, which is false. If $d \in D$ is a lower bound for $\{d_1,\ldots ,d_l\}$ then $d \in P \setminus F_i^{\uparrow }$. Thus $U_i = U_i^{\uparrow }$ and $\bigwedge _P D \in U_i$ imply $d \in D \cap (U_i {\setminus } F_i^{\uparrow }) \subseteq D \cap U$, a contradiction. $\square$

Remark 2.12

Let X and Y be spectral spaces and $f:X \rightarrow Y$ a spectral map. Then f is monotonic for the specialization order and is both a dcpo map and an fcpo map, cf. 2.8(e). The spectral topologies of X and Y are coarser than the lower Scott topologies and the inverse topologies are coarser then the Scott topologies, [6, 7.1.8(x)]. It follows from Theorem 2.11(b) and (e) that f is continuous both for the lower Scott topology and the Scott topology.

Corollary 2.13

Let $\varphi : P \rightarrow Q$ be a monotonic map of complete lattices.

• (a) The following conditions are equivalent.

  • (i) $\varphi$ preserves all infima.

  • (ii) $\varphi$ preserves finite infima and is coarse lower continuous.

  • (iii) $\varphi$ preserves finite infima and is lower Scott continuous.

• (b) The following conditions are equivalent.

  • (i) $\varphi$ preserves all suprema.

  • (ii) $\varphi$ preserves finite suprema and is coarse upper continuous.

  • (iii) $\varphi$ preserves finite suprema and is Scott continuous.

• (c) The following conditions are equivalent.

  • (i) $\varphi$ preserves all infima and is a dcpo map.

  • (ii) $\varphi$ preserves finite infima and is both coarse lower continuous and Scott continuous.

  • (iii) $\varphi$ preserves finite infima and is continuous for the Lawson topology.

  If the equivalent conditions hold then the left adjoint map $\varphi ^*$ exists and is coherent.

Proof

(a). (i)$\Rightarrow$(ii). The first statement in (ii) is trivial. Coarse lower continuity follows from the existence of a left adjoint, cf. 2.6 and Theorem 2.11(c). The implications (ii)$\Rightarrow$(iii) and (iii)$\Rightarrow$(i) follow from Theorem 2.11(a) and (b).

(b). Apply item (a) to $\varphi :P_{{{\,\textrm{inv}\,}}} \rightarrow Q_{{{\,\textrm{inv}\,}}}$.

(c). (i)$\Rightarrow$(ii) follows from item (a) and Theorem 2.11(e). (ii)$\Rightarrow$(iii) is trivial. Finally, item (a) and Theorem 2.11(e) and (g) yield (iii)$\Rightarrow$(i). If (i)–(iii) hold then $\varphi ^*$ exists by 2.6 and condition (i). An element x in a poset is compact if and only if $x^{\uparrow }$ is Scott open, 2.9(b). Thus, $\varphi ^*(b)^{\uparrow } = \varphi ^{-1}(b^{\uparrow })$ is Scott open if $b \in {{\mathbb {K}}}(Q)$, and then $\varphi ^*(b)$ is compact. $\square$

Corollary 2.14

Let $\varphi : P \rightarrow Q$ be a bounded lattice homomorphism of algebraic lattices.

• (a) $\varphi$ is spectral if and only if it preserves all infima and all suprema.

• (b) If $\varphi _*$ exists then it is spectral if and only if $\varphi$ is coherent.

Proof

(a). Use Corollary 2.13.

(b). If $a \in P$ then $\varphi _*^{-1}(a^{\uparrow }) = \varphi (a)^{\uparrow } \subseteq Q$, see 2.6. Assume $\varphi _*$ is spectral and $a \in {{\mathbb {K}}}(P)$. Then $\varphi (a)^{\uparrow } \subseteq Q$ is closed and constructible, hence $\varphi (a) \in {{\mathbb {K}}}(Q)$. Now let $\varphi$ be coherent. If $a \in {{\mathbb {K}}}(P)$ then $\varphi (a) \in {{\mathbb {K}}}(Q)$, hence $a^{\uparrow }$ and $\varphi (a)^{\uparrow }$ are closed and constructible. Thus $\varphi _*^{-1}(C)$ is closed and constructible if $C \subseteq P$ is closed and constructible. $\square$

2.10 Products of posets

Let P and Q be bounded posets. By [8, Lemma III$-$1.3] the coarse lower topology on $P \times Q$ is the product of the coarse lower topologies on P and Q. The projection $\pi _P:P \times Q \rightarrow P$ (similarly the projection $\pi _Q$) has both right and left adjoints and $(\pi _P)_*(a) = (a,\top _Q)$ and $\pi _P^*(a) = (a,\bot _Q)$. For $x \in P \times Q$ let $x_P = \pi _P(x)$ and $x_Q = \pi _Q(x)$. A subset $D \subseteq P \times Q$ is up-directed if and only if $D_P = \{d_P \mid d \in D\}$ and $D_Q = \{d_Q \mid d \in D\}$ are up-directed, and $x = \bigvee _{P \times Q} D$ if and only if $x_P = \bigvee _P D_P$, $x_Q = \bigvee _Q D_Q$. Thus ${{\mathbb {K}}}(P \times Q) = {{\mathbb {K}}}(P) \times {{\mathbb {K}}}(Q)$ and $\pi _P$, $\pi _P^*$, $\pi _Q$, $\pi _Q^*$ are all coherent. It follows that $P \times Q$ is an algebraic lattice if so are P and Q.

If $Q = P$ then the diagonal map $\Delta :P \rightarrow P \times P, x \mapsto (x,x)$ is both a $\bigwedge$-homomorphism and a $\bigvee$-homomorphism and is coherent.

• (a) The right adjoint $\Delta _*$ exists if and only if P is a $\wedge$-semilattice. If this is the case then $\Delta _*(x,y) = x \wedge y$ and $\Delta _*$ is coarse lower continuous (also see [8, Lemma III$-$1.4]), hence is an fcpo-map and is lower Scott continuous, Theorem 2.11(a), (b), (c).

• (b) The left adjoint $\Delta ^*$ exists if and only if P is a $\vee$-semilattice. Then $\Delta ^*(x,y) = x \vee y$ and $\Delta ^*$ is continuous for the coarse upper topology, hence is a dcpo map and is Scott continuous, Theorem 2.11(d), (e), (f). Moreover, $\Delta ^*$ is coherent.

The Scott topology of products is more complicated. Usually $\sigma (P \times Q)$ is finer than the product of $\sigma (P)$ and $\sigma (Q)$, [8, p. 197].

Corollary 2.16

Let P be an algebraic lattice. Then $\Delta$ and $\wedge = \Delta _*$ are spectral maps.

Proposition 2.17

Let P be a frame. Then $\wedge :P \times P \rightarrow P$ is Scott continuous, hence also Lawson continuous.

Proof

Scott continuity is equivalent to Lawson continuity since $\wedge$ is coarse lower continuous, 2.15(a), Theorem 2.11(g). For Scott continuity pick a Scott open set $U \subseteq P$ and let $D \subseteq P \times P$ be up-directed with $\bigvee _{P \times P} D \in \wedge ^{-1}(U)$. Define $D_1$ and $D_2$ to be the sets of first and second components of elements of D and let $C = \{x_1 \wedge y_2 \mid x,y \in D\} \subseteq P$, which is up-directed as well. Then $\bigvee _{P \times P} D =(\bigvee _P D_1,\bigvee _P D_2)$ and $\wedge (\bigvee _{P \times P} D) = \bigvee _P D_1 \wedge \bigvee _P D_2 = \bigvee _P C$ (as P is a frame). Thus, $\bigvee _P C \in U$ implies the existence of $x,y \in D$ with $x_1 \wedge y_2 \in U$. For any $z \in x^{\uparrow } \cap y^{\uparrow } \cap D$ we have $x_1 \wedge y_2 \le z_1 \wedge z_2 = \wedge (z)$, and it follows that $z \in D \cap \wedge ^{-1}(U)$. $\square$

3 Closure operators and closure systems

In this section we fix the notation and terminology concerning closure systems and closure operators, exhibit some examples and present first topological properties of closure systems, cf. Remark 3.6 and Proposition 3.7.

Notation and Terminology 3.1

A poset map $\eta :P \rightarrow P$ is a closure operator if it is idempotent and inflationary (i.e., $a \le \eta (a)$), resp. a kernel operator if it is idempotent and deflationary (i.e., $\eta (a) \le a$), [8, Definition 0$-$3.8]. It suffices to discuss closure operators since a poset map $\eta :P \rightarrow P$ is a closure operator if and only if $\eta$ viewed as a map $P_{{{\,\textrm{inv}\,}}} \rightarrow P_{{{\,\textrm{inv}\,}}}$ (the inversely ordered poset) is a kernel operator.

Let $Q \subseteq P$ and $\iota = \iota _{P,Q}:Q \rightarrow P$ the inclusion map. Then Q is a closure system if $\iota$ has a left adjoint, equivalently if, for all $a \in P$, the set $a^{\uparrow Q}$ has a smallest element, cf. 2.6. The left adjoint $\iota ^*:P \rightarrow Q$ is denoted by $\vartheta = \vartheta _{Q,P}$ and is called the closure map. The composition $\eta _{Q,P} = \iota _{P,Q} \circ \vartheta _{Q,P}$ is a closure operator. Conversely, if $\eta :P \rightarrow P$ is a closure operator then $Q = \eta (P)$ is a closure system with $\eta = \eta _{Q,P}$, and $\vartheta _{Q,P}$ is the corestriction of $\eta$. The correspondence between closure systems and closure operators of P is bijective. Clearly, $\eta (a)^{\uparrow Q} = a^{\uparrow Q}$ and $\eta (a) = \bigwedge _P a^{\uparrow Q} = \bigwedge _Q a^{\uparrow Q}$ where $a \in P$. The inclusion $\iota$ is a $\bigwedge$-embedding, Proposition 2.7(b), and is coarse lower continuous, Theorem 2.11(c). The closure map is a $\bigvee$-map, hence a dcpo map, Proposition 2.7(a), hence is Scott continuous, Theorem 2.11(e). A closure system $Q \subseteq P$, and the corresponding closure operator, are algebraic, or inductive, if $\iota _{P,Q}$ is a dcpo map (equivalently a dcpo embedding, Proposition 2.7(b)). The set of all closure systems, resp. algebraic closure systems, is denoted by ${\mathscr {C}}(P)$, resp. ${\mathscr {A}}(P)$.

Consider a subset $X \subseteq P$ such that $\bigwedge _P a^{\uparrow X}$ exists for all $a \in P$. Then $\eta :P \rightarrow P, a \mapsto \bigwedge _P a^{\uparrow X}$ is a closure operator, $\langle X \rangle _P:= \eta (P)$ is the smallest closure system containing X and is called the closure system generated by X. In arbitrary posets not every subset generates a closure system. Here are two examples. First, let $P = \{a,b\}$ with a, b incomparable, $X = \{a\}$. Then $b^{\uparrow X} = {\emptyset }$, hence $\bigwedge _P b^{\uparrow X}$ does not exist since there is no top element in P. For the second example, let ${{\mathbb {N}}}^*$ be the inversely ordered set of natural numbers; its elements are denoted by $0^*,1^*, \ldots$. Let $P = {{\mathbb {N}}}\cup {{\mathbb {N}}}^*$ with ${{\mathbb {N}}}$ naturally ordered and $k \le l^*$ for all $k,l \in {{\mathbb {N}}}$. If $X = {{\mathbb {N}}}^*$ then $k^{\uparrow X} = {{\mathbb {N}}}^*$ for all $k \in {{\mathbb {N}}}$, which does not have an infimum in P.

We say that X is dense in P if $\langle X \rangle _P$ exists and is equal to P. If P is a complete lattice and $Q \subseteq P$ then one shows easily that

Q is a closure system if and only if $\bigwedge _P S \in Q$ for all $S \subseteq Q$.

In particular, Q is a complete lattice as well. It follows that ${\mathscr {C}}(P)$ and ${\mathscr {A}}(P)$ are closure systems (not necessarily algebraic) in the complete lattice ${\mathfrak {P}}(P)$. If $X \subseteq P$ then $\eta _{{\mathscr {C}}(P),{\mathfrak {P}}(P)}(X) = \langle X \rangle _P$, and its elements are the infima of subsets $S \subseteq X$.

Example 3.2

Let $\psi ^*:P \rightarrow Q$ be the left adjoint of a poset map $\psi :Q \rightarrow P$. It follows from 2.6 that $\psi \circ \psi ^*:P \rightarrow P$ is a closure operator with corresponding closure system $\psi (Q)$, and $\psi ^* \circ \psi$ is a kernel operator. Let $C \subseteq P$ and $D \subseteq Q$ be closure systems with inclusion maps $\iota _C$, $\iota _D$ and closure maps $\vartheta _C$, $\vartheta _D$. Assume that $\psi (D) \subseteq C$ and let $\varphi :D \rightarrow C$ be the restriction of $\psi$. Then $\varphi$ has a left adjoint and $\varphi ^* \circ \vartheta _C = \vartheta _D \circ \psi ^*$. Moreover, $\psi (D)$ is a closure system in P, and $\psi (D) = \langle \psi (X) \rangle _P$ if $D = \langle X \rangle _Q$.

Left adjoint maps are $\bigvee$-homomorphisms, Proposition 2.7(a). Now assume that $\psi$ is a dcpo map and D is an algebraic closure system, i.e., $\iota _D$ is a dcpo map. Then $\varphi = \vartheta _C \circ \iota _C \circ \varphi = \vartheta _C \circ \psi \circ \iota _D$ is a dcpo map. Moreover, $\psi (D)$ is also algebraic in P. To show this one may assume $C = \psi (D)$, i.e., $\varphi$ is surjective, and then $\iota _C = \iota _C \circ \varphi \circ \varphi ^* = \psi \circ \iota _D \circ \varphi ^*$ is a dcpo map, 2.6(c).

Example 3.3

If P is a poset then the set ${\uparrow }(P)$ generates a closure system ${\mathcal {A}}(P,{\uparrow }(P)) \subseteq {\mathfrak {P}}(P)$, the set of ${\uparrow }(P)$-closed sets. The corresponding closure map sends $A \in {\mathfrak {P}}(P)$ to its ${\uparrow }(P)$-closure $\overline{A}^{{\uparrow }(P)} = \bigcap _{c \in A^{\Downarrow }} c^{\uparrow P}$. The top and bottom elements of ${\mathcal {A}}(P,{{\uparrow }(P)})$ are P, resp. $P \setminus P^{\flat }$. The complements of ${{\uparrow }(P)}$-closed sets in P are the ${\uparrow }(P)$-open sets, and the set of these is denoted by ${\mathcal {O}}(P,{{\uparrow }(P)})$.

Consider a poset map $\psi :Q \rightarrow P$ with left adjoint. The inverse image map ${\mathfrak {P}}(\psi ):{\mathfrak {P}}(P) \rightarrow {\mathfrak {P}}(Q)$ restricts to ${\uparrow }(\psi ):{\uparrow }(P) \rightarrow {\uparrow }(Q)$, 2.6, hence yields the maps

$$\begin{aligned} {\mathcal {A}}(\psi ,{\uparrow }):{\mathcal {A}}(P,{\uparrow }(P)) \rightarrow {\mathcal {A}}(Q,{\uparrow }(Q)), \\ {\mathcal {O}}(\psi ,{\uparrow }):{\mathcal {O}}(P,{\uparrow }(P)) \rightarrow {\mathcal {O}}(Q,{\uparrow }(Q)). \end{aligned}$$

The left adjoints ${\mathfrak {P}}(\psi )^*$ and ${\mathcal {A}}(\psi ,{\uparrow })^*$, as well as the right adjoint ${\mathcal {O}}(\psi ,{\uparrow })_*$ exist and ${\mathfrak {P}}(\psi )^*(B) = \bigcap \{A \in {\mathfrak {P}}(P) \mid \psi (B) \subseteq A\}$, ${\mathcal {A}}(\psi ,{\uparrow })^*(B) = \overline{\psi (B)}^{{\uparrow }(P)}$, ${\mathcal {O}}(\psi ,{\uparrow })_*(O) = Q {\setminus } \overline{\psi (Q {\setminus } O)}^{{\uparrow }(P)}$.

Let $X \subseteq P$ be a subset with inclusion map $e:X \rightarrow P$. We define

$$\begin{aligned} {\uparrow }(X,P) = {\mathfrak {P}}(e)({\uparrow }(P)) = \{a^{\uparrow X} \mid a \in P\} \subseteq {\mathfrak {P}}(X) \end{aligned}$$

and call ${\mathcal {A}}(X,{\uparrow }(X,P)) = \langle {\uparrow }(X,P) \rangle _{{\mathfrak {P}}(X)}$ the set of ${\uparrow }(X,P)$-closed subsets of X. The complements in X are the set ${\mathcal {O}}(X,{\uparrow }(X,P))$ of ${\uparrow }(X,P)$-open subsets. The inclusion e yields the surjective maps

$$\begin{aligned}&{\mathcal {A}}(e,{\uparrow }):{\mathcal {A}}(P,{\uparrow }(P)) \rightarrow {\mathcal {A}}(X,{\uparrow }(X,P)), \\&{\mathcal {O}}(e,{\uparrow }):{\mathcal {O}}(P,{\uparrow }(P)) \rightarrow {\mathcal {O}}(X,{\uparrow }(X,P)). \end{aligned}$$

The adjoints ${\mathcal {A}}(e,{\uparrow })^*$ and ${\mathcal {O}}(e,{\uparrow })_*$ exist and are both injective. The map $\gamma _{X,P}:P \rightarrow {\mathcal {O}}(X,{\uparrow }(X,P)), a \mapsto X {\setminus } a^{\uparrow X}$ is a poset morphism.

The following lemmas record elementary facts about closure systems, generating sets and the maps $\gamma _{X,P}$ (without proof).

Lemma 3.4

Let P be a poset, $S,X \subseteq P$ and assume that $Q:= \langle X \rangle _P$ exists. Then:

• (a) $\bigwedge _P a^{\uparrow X} = \bigwedge _P a^{\uparrow Q}$ for all $a \in P$.

• (b) $Q = \{a \in P \mid \forall b \in P: b \not \le a \Rightarrow a^{\uparrow X} \setminus b^{\uparrow X} \ne {\emptyset }\}$.

• (c) If $\bigvee _P S$ exists then $(\bigvee _P S)^{\uparrow X} = \bigcap _{s \in S} s^{\uparrow X}$.

• (d) $\bigwedge _P S$ exists if and only if $\overline{S}^{{\uparrow }(P)} \in {\uparrow }(P)$, and then $\overline{S}^{{\uparrow }(P)} = (\bigwedge _P S)^{\uparrow }$.

• (e) P is complete if and only if ${\mathcal {A}}(P,{\uparrow }(P)) = {\uparrow }(P)$.

Lemma 3.5

Let P be a poset, $X \subseteq P$.

• (a) The poset map $\gamma _{X,P}$ is an embedding if and only if X is dense in P.

• (b) If P is complete then $\gamma _{X,P}$ is surjective. The converse holds if X is dense in P.

Remark 3.6

Let P be a poset and $Q \subseteq P$ a subset. Recall from [6, 7.1.4(ii)] that the inclusion $\tau ^{\ell }(Q) \subseteq \tau ^{\ell }(P)|_Q$ is always true and may be proper. Now assume that Q is a closure system in P with inclusion map $\iota$, closure map $\vartheta$ and closure operator $\eta$. Since $\iota$ is coarse lower continuous (Theorem 2.11(c)) it follows that $\tau ^{\ell }(Q) = \tau ^{\ell }(P)|_Q$. The closure map $\vartheta$ is a dcpo map and is Scott continuous, Theorem 2.11(d), (e), (f).

• (a) $\eta$ is a dcpo map if $\iota$ is a dcpo map. The converse is true if P is a dcpo.

• (b) $\eta$ is a $\wedge$-homomorphism if and only if $\vartheta$ is a $\wedge$-homomorphism.

Thus, the closure system is algebraic (by definition) if and only if $\iota$ is a dcpo map, if and only if $\iota$ is Scott continuous, and then $\eta$ is Scott continuous (since the left adjoint $\vartheta = \iota ^*$ is a dcpo map, Proposition 2.7(a)). If P is a dcpo then Scott continuity of $\eta$ implies that $\iota$ is Scott continuous, Theorem 2.11(e).

Proposition 3.7

Consider a poset P with a closure system $Q \subseteq P$. Then:

• (a) Q is closed for $\beta = \beta (\tau ^{\ell }(P))$.

• (b) $\sigma (Q) \subseteq \sigma (P)|_Q$.

• (c) $\sigma (P)|_Q = \sigma (Q)$, i.e., $\iota$ is Scott continuous, if and only if $\iota$ is Lawson continuous, if and only if Q is algebraic in P.

• (d) If Q is quasi-compact for $\lambda (P)$ then Q is an algebraic closure system.

• (e) If Q is a complete lattice and is an algebraic closure system then Q is quasi-compact for $\lambda (P)$.

• (f) If P is a complete lattice then Q is an algebraic closure system if and only if $Q \subseteq P$ is quasi-compact for the Lawson topology.

Proof

(a). If $a \in P {\setminus } Q$ then $a < \eta (a)$, hence a belongs to $a^{\uparrow P} \setminus \eta (a)^{\uparrow P}$, which is $\beta$-open and is disjoint from Q.

(b). Let B be $\sigma (Q)$-closed in Q and set $A = B^{\downarrow P}$. As $B = A \cap Q$ it suffices to show that A is $\sigma (P)$-closed. So, pick $D \subseteq A$ up-directed such that $\bigvee _P D$ exists. For $d \in D$ there exists $b \in B$ with $d \le b$. Thus $\vartheta (d) = \eta (d) \le \eta (b) = b$ and $\vartheta (D) \subseteq B$ is an up-directed set in a Scott closed set. As $\vartheta$ is a dcpo-map we have $\bigvee _P D \le \vartheta (\bigvee _P D) = \bigvee _Q \vartheta (D) \in B$, which implies $\bigvee _P D \in A$.

(c). The closure system is algebraic if and only if $\iota$ is a dcpo map, if and only if $\iota$ is Scott continuous, if and only if $\sigma (P)|_Q = \sigma (Q)$ (by (b)). Scott continuity of $\iota$ is equivalent to Lawson continuity (note that $\iota$ is coarse lower continuous and use Theorem 2.11(g)).

(d). Let $D \subseteq Q$ be up-directed such that $\bigvee _Q D$ exists. We have to show that $\bigvee _P D$ exists and is equal to $\bigvee _Q D$. If this is false then there is some $a \in P$ with $D \subseteq a^{\downarrow P}$, but $\bigvee _Q D \not \le a$. The set $a^{\downarrow P}$ is closed for $\sigma (P)$, hence for $\lambda (P)$. Thus $a^{\downarrow Q}$ is closed in Q for $\lambda (P)|_Q$ and is quasi-compact (as Q is quasi-compact). The sets $P {\setminus } d^{\uparrow P}$ with $d \in D$ are open for $\lambda (P)$ and cover $a^{\downarrow Q}$. (For, if $x \in a^{\downarrow Q} \cap \bigcap _{d \in D} d^{\uparrow P}$ then $\bigvee _Q D \le x \le a$, a contradiction.) The cover $a^{\downarrow Q} \subseteq \bigcup _{d \in D} P {\setminus } d^{\uparrow P}$ has a finite subcover, say $a^{\downarrow Q} \subseteq \bigcup _{i=1}^k P {\setminus } d_i^{\uparrow P}$. As D is up-directed there is some $d \in D$ with $d_1,\ldots ,d_k \le d$. Hence $a^{\downarrow Q} \subseteq P {\setminus } d^{\uparrow P}$ and $D \subseteq a^{\downarrow Q}$ implies $d \in P \setminus d^{\uparrow P}$, a contradiction.

(e). Item (c) and Remark 3.6 show that $\lambda (P)|_Q = \lambda (Q)$. Thus it suffices to note that $(Q,\lambda (Q))$ is quasi-compact by [8, Theorem III$-$1.9].

(f) follows from (d) and (e). $\square$

Proposition 3.8

Let P be a poset, $C \subseteq P$ a subset and $\tau$ a lower topology.

• (a) C is irreducible for $\tau ^{\ell }(P)$ if and only if, for all finite $F \subseteq P$ with $C \subseteq F^{\uparrow }$, there is some $x \in F$ with $C \subseteq x^{\uparrow }$.

• (b) If $C \ne {\emptyset }$ is irreducible for $\tau ^{\ell }(P)$ then $\overline{C}^{\tau ^{\ell }(P)} = \overline{C}^{{\uparrow }(P)}$ (for the notation see Example 3.3), hence $\tau ^{\ell }(P)$ is sober for irreducible sets with infimum. In particular, $\tau ^{\ell }(P)$ is sober if P is complete.

Proof

(a). The sets $F^{\uparrow }$ (where $F \subseteq P$ is finite) are a basis of closed sets for $\tau ^{\ell }(P)$. If C is irreducible and $C \subseteq F^{\uparrow } = \bigcup _{x \in F} x^{\uparrow }$ then $C \subseteq x^{\uparrow }$ for some $x \in F$. Conversely, assume that $C \subseteq x^{\uparrow }$ for some $x \in F$ whenever $F \subseteq P$ is finite and $C \subseteq F^{\uparrow }$. Pick finite subsets $G,H \subseteq P$ with $C \subseteq G^{\uparrow } \cup H^{\uparrow } = (G \cup H)^{\uparrow }$. Then $C \subseteq x^{\uparrow }$ for some $x \in G \cup H$, hence $C \subseteq G^{\uparrow }$ or $C \subseteq H^{\uparrow }$ according as $x \in G$ or $x \in H$, and C is irreducible.

(b). The equality $\overline{C}^{\tau ^{\ell }(P)} = \bigcap _{y \in C^{\Downarrow }} y^{\uparrow } = \overline{C}^{{\uparrow }(P)}$ follows from (a). If $\bigwedge _P C$ exists then the set is equal to $(\bigwedge _P C)^{\uparrow }$, Lemma 3.4(d). $\square$

Corollary 3.9

Let P be a complete lattice. A subset Q is a closure system if and only if it is closed in P under finite infima and is sober for $\tau ^{\ell }(P)|_Q$.

Proof

Assume Q is a closure system. Then it is closed in P under all infima. Proposition 3.7(a) shows that Q is $\beta (\tau ^{\ell }(P))$-closed in P. As $\tau ^{\ell }(P)$ is sober, Proposition 3.8(b), it follows that Q is a sober subspace of P, see 2.3(a). Conversely, assume Q is sober for $\tau ^{\ell }(P)|_Q$ and is closed in P under finite infima. It is claimed that $\bigwedge _P S \in Q$ for any subset $S \subseteq Q$. If $S = {\emptyset }$ then $\top _P = \bigwedge _P S \in Q$. For $S \ne {\emptyset }$ let $T \subseteq Q$ be the set of all finite infima of elements of S. Then T is down-directed, hence is irreducible for any lower topology, [6, Proposition 4.2.1(i)], and $S^{\Downarrow } = T^{\Downarrow }$. As Q is sober it follows from 2.8(g) that $\bigwedge _P S = \bigvee _P S^{\Downarrow } = \bigvee _P T^{\Downarrow } = \bigwedge _PT \in Q$. $\square$

4 Algebraic closure systems in algebraic lattices

Every algebraic lattice is a spectral space, 2.10. The main result in this section is Theorem 4.5, which contains several topological conditions characterizing algebraic closure systems in algebraic lattices.

Proposition 4.1

Let P be a continuous lattice and $Q \subseteq P$ an algebraic closure system. Then Q is a continuous lattice as well.

Proof

By [8, Proposition I$-$1.5(ii)] we have to show that for each $b \in Q$ there is an up-directed set $D \subseteq b_Q^{\ll }$ with $\bigvee _Q D = b$. By hypothesis, the set $b_P^{\ll }$ is up-directed and $b = \bigvee _P b_P^{\ll }$. Thus, $\vartheta (b_P^{\ll })$ is up-directed and, since $\vartheta$ is a dcpo-map, $b = \vartheta (b) = \vartheta (\bigvee _P b_P^{\ll }) = \bigvee _Q \vartheta (b_P^{\ll })$. We show that $\vartheta (c) \ll _Q b$ for each $c \in b_P^{\ll }$: Let $D \subseteq Q$ be up-directed with $b \le \bigvee _Q D = \bigvee _P D$. Then $c \ll _P b$ implies $c \le d$ for some $d \in D$, hence $\vartheta (c) \le \vartheta (d) = d$. $\square$

Proposition 4.2

Let P be an algebraic lattice and $Q \subseteq P$ a closure system. Then ${{\mathbb {K}}}(Q) \subseteq \vartheta ({{\mathbb {K}}}(P))$.

Proof

Pick $a \in {{\mathbb {K}}}(Q)$ and write $a = \bigvee _P a^{\downarrow {{\mathbb {K}}}(P)}$. Then

$$\begin{aligned} a = \vartheta (a) = \vartheta \left( \bigvee _P a^{\downarrow {{\mathbb {K}}}(P)}\right) = \bigvee _Q \left\{ \vartheta (c) \mid c \in a^{\downarrow {{\mathbb {K}}}(P)}\right\} \end{aligned}$$

since $\vartheta$ is a dcpo map. As the supremum is up-directed and $a \in {{\mathbb {K}}}(Q)$ there is some $c \in a^{\downarrow {{\mathbb {K}}}(P)}$ with $a \le \vartheta (c) \le \vartheta (a) = a$. $\square$

Definition 4.3

Let P be an algebraic lattice and Q a closure system. An element $a \in P$ is absorbing for Q, or for $\eta$, if $\eta (a^{\downarrow {{\mathbb {K}}}(P)}) \subseteq a^{\downarrow P}$. (This generalizes the notion of absorbing elements in an algebraic frame with respect to a nucleus as defined in [13, Definition and Remarks 4.1]).

Proposition 4.4

Let $Q \subseteq P$ be a closure system in an algebraic lattice and A the set of absorbing elements for Q. Then A is the patch closure of Q.

Proof

The inclusion $Q \subseteq A$ is obvious. First consider any $a \in P {\setminus } A$. There is some $c \in a^{\downarrow {{\mathbb {K}}}(P)}$ with $\eta (c) \not \le a$. The set $c^{\uparrow } {\setminus } \eta (c)^{\uparrow }$ is patch open (by 2.10), contains a and is disjoint from Q (by 3.1). It follows that $\overline{Q}^{{{\,\textrm{con}\,}}} \subseteq A$. For the reverse inclusion pick $a \in A$ and let C be a patch open set containing a. We may assume that $C = c^{\uparrow } {\setminus } F^{\uparrow }$ with $c \in {{\mathbb {K}}}(P)$ and $F \subseteq P$ finite, cf. 2.10. It suffices to show that $\eta (c) \in C$. Note that $c \le \eta (c) \le a$ (as a is absorbing) and $\eta (c) \notin F^{\uparrow }$ (since otherwise $x \le \eta (c) \le a$ for some $x \in F$, hence $a \notin C$, a contradiction). $\square$

Theorem 4.5

(cf. [12, 1.1(9)]). Let P be an algebraic lattice and $Q \subseteq P$ a closure system. The following conditions are equivalent.

• (a) Q is an algebraic closure system.

• (b) Q is the set of Q-absorbing elements.

• (c) Q is patch closed in P.

• (d) The closure map is coherent, hence ${{\mathbb {K}}}(Q) = \vartheta ({{\mathbb {K}}}(P))$.

• (e) Q is sober for $\sigma (P)|_Q$.

If the equivalent conditions hold then Q is an algebraic lattice.

Proof

For the last assertion recall that every algebraic closure system in an algebraic lattice is an algebraic lattice, cf. [8, Proposition I$-$4.13].

(a)$\Rightarrow$(b). Let $a = \bigvee _P a^{\downarrow {{\mathbb {K}}}(P)} \in P$ be absorbing for Q. Then

$$\begin{aligned} a \le \eta (a) = \vartheta \left( \bigvee _P a^{\downarrow {{\mathbb {K}}}(P)}\right) = \bigvee _Q \vartheta (a^{\downarrow {{\mathbb {K}}}(P)}) \le a \end{aligned}$$

(the last inequality since a is absorbing). Thus, $a = \eta (a) \in Q$.

(b)$\Rightarrow$(a). Let $D \subseteq Q$ be up-directed and set $a = \bigvee _P D$. It suffices to show that a is absorbing, i.e., $a \in Q$. If $c \in a^{\downarrow {{\mathbb {K}}}(P)}$ then $c \le \bigvee _P D$ implies $c \le d$ for some $d \in D$, and it follows that $\eta (c) \le \eta (d) = d \le a$, i.e., a is absorbing.

(b)$\Leftrightarrow$(c) is clear by Proposition 4.4.

(a)$\Rightarrow$(d). The inclusion map $\iota :Q \rightarrow P$ is a dcpo map and preserves all infima. Thus $\vartheta = \iota ^*$ is coherent, Corollary 2.13(c), and the equality follows from Proposition 4.2.

(d)$\Rightarrow$(c). For $z \in P \setminus Q$ we must find a patch open set $C \subseteq P$ with $z \in C$ and $C \cap Q = {\emptyset }$. Since $z \notin Q$ it follows that $z < \eta (z) = \bigvee _P \eta (z)^{\downarrow {{\mathbb {K}}}(P)}$ and $u \not \le z$ for some $u \in \eta (z)^{\downarrow {{\mathbb {K}}}(P)}$. As $z = \bigvee _P z^{\downarrow {{\mathbb {K}}}(P)}$, $\vartheta$ preserves suprema and $u \le \eta (z) = \vartheta (z)$ it follows that $\vartheta (u) \le \vartheta (z) = \bigvee _Q \vartheta (z^{\downarrow {{\mathbb {K}}}(P)})$. By hypothesis $\vartheta (u) \in {{\mathbb {K}}}(Q)$, hence there is some $v \in z^{\downarrow {{\mathbb {K}}}(P)}$ with $\vartheta (u) \le \vartheta (v)$. The set $v^{\uparrow P} {\setminus } \eta (v)^{\uparrow P}$ is patch-open in P, contains z (since $v \in z^{\downarrow {{\mathbb {K}}}(P)}$, $u \not \le z$, and $u \le \eta (u) \le \eta (v)$) and is disjoint from Q (as $v^{\uparrow Q} = \eta (v)^{\uparrow Q}$).

(c)$\Rightarrow$(e). According to [6, Theorem 2.1.3] the patch closed subset $Q \subseteq P$ is a spectral subspace, hence is sober for its inverse topology, which is equal to $\sigma (P)|_Q$, [6, Theorem 7.2.8].

(e)$\Rightarrow$(a). Since $\sigma (P)$ is an upper topology its specialization poset is $P_{{{\,\textrm{inv}\,}}}$. Pick $D \subseteq Q$ up-directed, hence down-directed in $P_{{{\,\textrm{inv}\,}}}$. Then $\bigvee _P D$ is the generic point of $\overline{D}^{\sigma (P)}$, [6, Proposition 4.2.1(ii)]. Soberness of $(Q,\sigma (P)|_Q)$ implies $\bigvee _P D \in Q$ (cf. 2.8(g)), i.e., $\bigvee _P D = \bigvee _Q D$. $\square$

Example 4.6

Let P be an algebraic lattice, $Q \subseteq P$ an algebraic closure system. Theorem 4.5 shows that the inclusion $\iota :Q \rightarrow P$ is a spectral map, hence $\eta$ is spectral if and only if $\vartheta$ is spectral. (Clearly, $\vartheta$ spectral implies $\eta = \iota \circ \vartheta$ spectral. If $\eta$ is spectral and $Q \subseteq P$ is a spectral subspace then the corestriction $\vartheta :P \rightarrow Q$ is spectral.) Both maps are dcpo maps, cf. Remark 3.6(a), hence are continuous for the Scott topology, Theorem 2.11(e). If one of them is also continuous for the coarse lower topology then both are spectral, [6, Theorem 1.4.6]. We show that this need not be the case:

Let P be the inverse of the set ${{\mathbb {N}}}\cup \{\omega \}$, and define $Q = \{\omega ,0\} = \langle \{\omega \}\rangle _P$. Note that P is an algebraic lattice and Q is an algebraic closure system. The closure operator $\eta$ is given by $\omega \mapsto \omega$ and $a \mapsto 0$ otherwise. It is a homomorphism of bounded lattices and preserves all suprema, but is not an fcpo-map since $\bigwedge {{\mathbb {N}}}= \omega$ and $\eta (\bigwedge {{\mathbb {N}}}) = \omega < 0 = \bigwedge \eta ({{\mathbb {N}}})$. It follows from Corollary 2.14(a) that the closure operator is not a spectral map. In fact, $\{0\} \subseteq P$ is closed for the coarse lower topology, but $\eta ^{-1}(\{0\}) = {{\mathbb {N}}}$ is not closed.

Example 4.7

Let P be an algebraic lattice and $Q \subseteq P$ a closure system. Then ${{\mathbb {K}}}(Q) \subseteq \vartheta ({{\mathbb {K}}}(P))$, Proposition 4.2. We exhibit a closure system that shows how equality can fail.

Let P be the totally ordered set $2 \cdot \omega +1$, which is an algebraic lattice since every non-limit ordinal is compact. The subset $Q = {{\mathbb {N}}}\cup \{2 \cdot \omega \}$ is a closure system in P, but is not algebraic since $\bigvee _P {{\mathbb {N}}}= \omega < \bigvee _Q {{\mathbb {N}}}= 2 \cdot \omega$. Note that Q is isomorphic to $\omega +1$, hence is an algebraic lattice. The only element that is not compact is $2 \cdot \omega$. The closure map is given by $a \mapsto a$ if $a \in {{\mathbb {N}}}$ and $a \mapsto 2 \cdot \omega$ otherwise. The element $\omega +1 \in P$ is not a limit ordinal, hence is compact. But $\vartheta (\omega +1) = 2 \cdot \omega$ is not compact.

5 Closure systems and generating subsets

An algebraic lattice P is complete, hence every subset $X \subseteq P$ generates a closure system. We explore connections between properties of the generating set and the closure system. The main results are contained in Theorem 5.2. They imply, in particular, that the operators ${\mathfrak {P}}(P) \rightarrow {\mathfrak {P}}(P), X \mapsto \overline{X}^{{{\,\textrm{con}\,}}}$ and ${\mathfrak {P}}(P) \rightarrow {\mathfrak {P}}(P), X \mapsto \langle X \rangle _P$ commute with each other, Remark 5.4.

Notation 5.1

The following notation will be used frequently in the rest of the paper. Consider a poset P and a subset X. By default P is equipped with the coarse lower topology. If P is a complete lattice then we set $L = \langle X \rangle _P$, and if P is an algebraic lattice then we also define $Y = \overline{X}^{{{\,\textrm{con}\,}}}$ and $M = \langle Y \rangle _P$.

Theorem 5.2

Notation as in 5.1. Let P be an algebraic lattice.

• (a) M is patch closed in P.

• (b) $M = \overline{L}^{{{\,\textrm{con}\,}}}$, i.e., M is the set of L-absorbing elements (see Proposition 4.4 and compare with [13, Theorem 4.12]).

• (c) M is the smallest algebraic closure system in P containing X.

• (d) ${{\mathbb {K}}}(M) \subseteq L$, hence $\eta _{M,P}(a) = \eta _{L,P}(a)$ for $a \in {{\mathbb {K}}}(P)$.

Proof

(a). To show that $P \setminus M$ is patch open in P, pick $z \in P {\setminus } M$. Then $z < \eta _{M,P}(z) = \bigvee _P \eta _{M,P}(z)^{\downarrow {{\mathbb {K}}}(P)}$, hence there is some $a \in \eta _{M,P}(z)^{\downarrow {{\mathbb {K}}}(P)}$ with $a \not \le z$. Note that $z^{\uparrow Y} = \eta _{M,P}(z)^{\uparrow Y} \subseteq a^{\uparrow Y} = \eta _{M,P}(a)^{\uparrow Y}$, cf. 3.1. The set $a^{\uparrow P}$ is closed and constructible in P and $Y \subseteq P$ is patch closed, hence $a^{\uparrow Y}$ is closed and constructible in Y. Since $z = \bigvee _P z^{\downarrow {{\mathbb {K}}}(P)}$ and the supremum is up-directed it follows that $z^{\uparrow Y} = \bigcap _{b \in z^{\downarrow {{\mathbb {K}}}(P)}} b^{\uparrow Y}$. The sets $b^{\uparrow Y}$ are closed and constructible in Y, the intersection is down-directed for inclusion and is disjoint from the quasi-compact open set $Y \setminus a^{\uparrow Y}$ (since $z^{\uparrow Y} \subseteq a^{\uparrow Y}$). The patch topology of Y is compact, hence there is some $b \in z^{\downarrow {{\mathbb {K}}}(P)}$ with $b^{\uparrow Y} \subseteq a^{\uparrow Y}$. One concludes that $\eta _{M,P}(a) = \bigwedge _P a^{\uparrow Y} \le \bigwedge _P b^{\uparrow Y} = \eta _{M,P}(b)$. Note that $\eta _{M,P}(b) \not \le z$. For, otherwise $a \le \eta _{M,P}(a) \le \eta _{M,P}(b) \le z$, contradicting the choice of a. Thus $b^{\uparrow P} {\setminus } \eta _{M,P}(b)^{\uparrow P}$ is patch open in P and contains z. Finally, if $x \in b^{\uparrow M}$ then $\eta _{M,P}(b) \le \eta _{M,P}(x) = x$, and $b^{\uparrow P} {\setminus } \eta _{M,P}(b)^{\uparrow P}$ is a patch open neighborhood of z disjoint from M.

(b). First we claim that $\overline{L}^{{{\,\textrm{con}\,}}} \subseteq P$ is closed under finite meets. As $\top _P \in L$ it suffices to prove $a \wedge b \in \overline{L}^{{{\,\textrm{con}\,}}}$ if $a,b \in \overline{L}^{{{\,\textrm{con}\,}}}$. The map $\wedge :P \times P \rightarrow P$ is spectral, Corollary 2.16, hence sends patch closed sets to patch closed sets. As $L \subseteq P$ is closed under infima it follows that $\wedge$ restricts to a map $L \times L \rightarrow L$. The equality $\overline{L \times L}^{{{\,\textrm{con}\,}}} = \overline{L}^{{{\,\textrm{con}\,}}} \times \overline{L}^{{{\,\textrm{con}\,}}}$, cf. [6, Theorem 2.2.1], implies $\wedge (\overline{L}^{{{\,\textrm{con}\,}}} \times \overline{L}^{{{\,\textrm{con}\,}}}) = \overline{L}^{{{\,\textrm{con}\,}}}$. Next we claim that $\overline{L}^{{{\,\textrm{con}\,}}}$ is a closure system in P. For, consider any nonempty subset $S \subseteq \overline{L}^{{{\,\textrm{con}\,}}}$ and let $T \subseteq P$ be the set of infima $\bigwedge F$ with $F \subseteq S$ finite. Then $T \subseteq \overline{L}^{{{\,\textrm{con}\,}}}$ and $\bigwedge _P S = \bigwedge _P T$. The down-directed set T is irreducible, hence $\overline{T}$ is nonempty closed and irreducible in P and has the generic point $\bigwedge _P T$, [6, Proposition 4.2.1]. Moreover, $\overline{T}^{{{\,\textrm{con}\,}}} \subseteq \overline{L}^{{{\,\textrm{con}\,}}}$ is down-directed and contains the generic point of $\overline{T}$, [6, Proposition 4.2.6]. To finish the proof, note that $X \subseteq L$ implies $Y \subseteq \overline{L}^{{{\,\textrm{con}\,}}}$, hence $M \subseteq \overline{L}^{{{\,\textrm{con}\,}}}$, and $\overline{L}^{{{\,\textrm{con}\,}}} = M$ follows from item (a).

(c) follows from item (b) and Theorem 4.5.

(d). Pick $c \in {{\mathbb {K}}}(M)$ and note that the set $C = c^{\uparrow M} {\setminus } \eta _{L,M}(c)^{\uparrow M}$ is patch open in M. Assume $c < \eta _{L,M}(c)$. Then $c \in C$, i.e., $C \ne {\emptyset }$, and (b) implies $c^{\uparrow L} \setminus \eta _{L,M}(c)^{\uparrow L} = C \cap L \ne {\emptyset }$, a contradiction, cf. 3.1. The final claim follows from Theorem 4.5. $\square$

Corollary 5.3

Let P be an algebraic lattice and $Q \subseteq P$ a closure system.

• (a) $\overline{Q}^{{{\,\textrm{con}\,}}}$ is an algebraic closure system and an algebraic lattice.

• (b) ${{\mathbb {K}}}(Q) \subseteq {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})$.

• (c) If ${{\mathbb {K}}}(Q) = {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})$ then $Q = \overline{Q}^{{{\,\textrm{con}\,}}}$.

• (d) Q is dense in $\overline{Q}^{{{\,\textrm{con}\,}}}$ for $\beta (\sigma (P))$. Thus, $\overline{Q}^{{{\,\textrm{con}\,}}}$ with the Scott topology is the sobrification of $(Q,\sigma (P)|_Q)$.

• (e) The elements of $\overline{Q}^{{{\,\textrm{con}\,}}}$ are the suprema in P of up-directed sets in Q.

Proof

(a) follows from Theorem 5.2 and Theorem 4.5.

(b). Pick $a \in {{\mathbb {K}}}(Q)$ and write $a = \bigvee _{\overline{Q}^{{{\,\textrm{con}\,}}}} a^{\downarrow {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})}$. Then ${{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}}) \subseteq Q$, cf. Theorem 5.2(d), implies $a = \bigvee _Q a^{\downarrow {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})}$. Since $a^{\downarrow {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})}$ is up-directed there is some $b \in a^{\downarrow {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})}$ with $a \le b$, hence $a = b \in {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})$.

(c). Pick $b \in \overline{Q}^{{{\,\textrm{con}\,}}}$ and write

$$\begin{aligned} b = \bigvee _{\overline{Q}^{{{\,\textrm{con}\,}}}} b^{\downarrow {{\mathbb {K}}}(Q)} \le a:= \bigvee _Q b^{\downarrow {{\mathbb {K}}}(Q)} = \bigvee _{\overline{Q}^{{{\,\textrm{con}\,}}}} a^{\downarrow {{\mathbb {K}}}(Q)}. \end{aligned}$$

If $a^{\downarrow {{\mathbb {K}}}(Q)} = b^{\downarrow {{\mathbb {K}}}(Q)}$ then $b = a \in Q$. The inclusion $b^{\downarrow {{\mathbb {K}}}(Q)} \subseteq a^{\downarrow {{\mathbb {K}}}(Q)}$ holds trivially. If $c \in a^{\downarrow {{\mathbb {K}}}(Q)}$ then there is some $d \in b^{\downarrow {{\mathbb {K}}}(Q)}$ with $c \le d \le b$, i.e., $c \in b^{\downarrow {{\mathbb {K}}}(Q)}$.

(d). We may assume that $\overline{Q}^{{{\,\textrm{con}\,}}} = P$. The sets $x^{\uparrow }$ with $x \in {{\mathbb {K}}}(P)$ are a basis of open sets for the Scott topology of P, [8, Corollary II$-$1.15] or [6, Theorem 7.2.8]. Hence the sets $a^{\downarrow } \cap x^{\uparrow }$, with $a \in P$, $x \in {{\mathbb {K}}}(P) \subseteq Q$ (Theorem 5.2(d)), are a basis for $\beta (\sigma (P))$, see 2.2. The set $a^{\downarrow } \cap x^{\uparrow }$ is nonempty if and only if it contains x, if and only if $(a^{\downarrow } \cap x^{\uparrow }) \cap Q \ne {\emptyset }$. – The second assertion follows from  2.3.

(e). If $D \subseteq Q$ is up-directed then $\bigvee _P D \in \overline{Q}^{{{\,\textrm{con}\,}}}$ by (a). Conversely, every element $a \in \overline{Q}^{{{\,\textrm{con}\,}}}$ is equal to $\bigvee _{\overline{Q}^{{{\,\textrm{con}\,}}}} a^{\downarrow {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})} = \bigvee _P a^{\downarrow {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})}$, where $a^{\downarrow {{\mathbb {K}}}(\overline{Q}^{{{\,\textrm{con}\,}}})}$ is up-directed and is contained in Q, Theorem 5.2(d). $\square$

Remark 5.4

Let P be an algebraic lattice. As noted in 3.1 the set ${\mathscr {C}}(P)$ of closure systems and the set ${\mathscr {A}}(P)$ of algebraic closure systems in P are closure systems in ${\mathfrak {P}}(P)$. The subset ${\mathcal {A}}(P_{{{\,\textrm{con}\,}}}) \subseteq {\mathfrak {P}}(P)$ of patch closed subsets of P is a closure system as well. Theorem 4.5 shows that ${\mathscr {A}}(P) = {\mathcal {A}}(P_{{{\,\textrm{con}\,}}}) \cap {\mathscr {C}}(P)$, and it follows that

$$\begin{aligned} \eta _{{\mathscr {A}}(P),{\mathfrak {P}}(P)} \circ \eta _{{\mathscr {C}}(P),{\mathfrak {P}}(P)} = \eta _{{\mathscr {A}}(P),{\mathfrak {P}}(P)} = \eta _{{\mathscr {A}}(P),{\mathfrak {P}}(P)} \circ \eta _{{\mathcal {A}}(P_{{{\,\textrm{con}\,}}}),{\mathfrak {P}}(P)}, \end{aligned}$$

i.e., $\overline{\langle X \rangle _P}^{{{\,\textrm{con}\,}}} = \langle \overline{X}^{{{\,\textrm{con}\,}}} \rangle _P$ for $X \in {\mathfrak {P}}(P)$.

6 Prime generated closure systems

Continuing the study of closure systems and generating sets we consider closure systems generated by sets of prime elements. The notion of prime elements in a poset can be found in [8, I$-$3.11]. If P be a $\wedge$-semilattice, which will always be the case in our considerations, then $p \in P$ is prime if $a \wedge b \le p$ implies $a \le p$ or $b \le p$ for all $a,b \in P$, cf. [8, Proposition I$-$3.12]. The set of prime elements is denoted by ${{\mathbb {P}}}(P)$. A top element, if it exists, is always prime and is called the trivial prime element and ${{\mathbb {P}}}(P)^{\flat }$ (which may be empty) is the set of nontrivial prime elements. We show that ${{\mathbb {P}}}(P)$ and ${{\mathbb {P}}}(P)^{\flat }$ are both closed for the b-topology $\beta (\tau ^{\ell }(P))$, Proposition 6.3. A closure system $Q \subseteq P$ is called prime generated if $Q = \langle X \rangle _P$, where $X \subseteq {{\mathbb {P}}}(Q)$.

Let P be a complete lattice and $Q \subseteq P$ a closure system. It is always true that ${{\mathbb {P}}}(P) \cap Q \subseteq {{\mathbb {P}}}(Q)$, and the inclusion may be proper. If Q is prime generated then ${{\mathbb {P}}}(P) \cap Q = {{\mathbb {P}}}(Q)$ if and only if the closure operator is a $\wedge$-homomorphism, Corollary 6.9. Moreover, if Q is prime generated by X then it is isomorphic to the frame ${\mathcal {O}}(X^{\flat })$, Theorem 6.13. If P is even an algebraic lattice then it follows from Theorem 6.20 that a prime generated algebraic closure system is coherent if and only if its set of nontrivial prime elements is patch closed in P.

Lemma 6.1

Let P be an algebraic lattice. An element $p \in P$ is prime if and only if $a \wedge b \le p$ implies $a \le p$ or $b \le p$ for all $a,b \in {{\mathbb {K}}}(P)$.

Proof

If p is prime then the claim holds trivially. Now assume that p is not prime, i.e., there are $x,y \in P$ with $x \wedge y \le p$, but $x \not \le p$ and $y \not \le p$. Since $x = \bigvee x^{\downarrow {{\mathbb {K}}}(P)}$ and $y = \bigvee y^{\downarrow {{\mathbb {K}}}(P)}$ there are $a \in x^{\downarrow {{\mathbb {K}}}(P)}$ and $b \in y^{\downarrow {{\mathbb {K}}}(P)}$ with $a,b \not \le p$, but $a \wedge b \le x \wedge y \le p$. $\square$

Example 6.2

(a). For any set S the power set ${\mathfrak {P}}(S)$ is an algebraic frame, and the set of nontrivial prime elements is equal to $\{T \subseteq S \mid |S {\setminus } T| = 1\}$.

(b). Let K be a field with at least 3 elements and let V be a vector space with $\dim V \ge 2$. Let ${\mathscr {U}}(V)$ be the set of subspaces, which is an algebraic closure system in ${\mathfrak {P}}(V)$, hence is an algebraic lattice. The compact elements are the finite dimensional subspaces. Thus ${\mathscr {U}}(V)$ is always an arithmetic algebraic lattice. Coherence holds if and only if $\dim V$ is finite. There are no nontrivial prime elements in ${\mathscr {U}}(V)$.

(c). Let P be a $\wedge$-semilattice and $X \subseteq P$ a dense subset. Then X is totally ordered if and only if P is totally ordered, if and only if $P = {{\mathbb {P}}}(P)$.

(d). Consider $\wedge$-semilattices P and Q with top elements. One checks that ${{\mathbb {P}}}(P \times Q) = {{\mathbb {P}}}(P) \times \{\top _Q\} \cup \{\top _P\} \times {{\mathbb {P}}}(Q)$. Thus the projection maps $\pi _P$ and $\pi _Q$, as well as their right adjoint maps, send prime elements to prime elements, 2.15. Moreover, $\wedge : P \times P \rightarrow P, (a,b) \mapsto a \wedge b$ maps prime elements to prime elements.

Proposition 6.3

Let P be a $\wedge$-semilattice with the coarse lower topology. Then ${{\mathbb {P}}}(P)$ and ${{\mathbb {P}}}(P)^{\flat }$ are closed in P for $\beta = \beta (\tau ^{\ell }(P))$. Thus, ${{\mathbb {P}}}(P)$ and ${{\mathbb {P}}}(P)^{\flat }$ are sober if P is complete.

Proof

Pick $x \in P \setminus {{\mathbb {P}}}(P)$. There are $a,b \in P$ with $a,b \not \le x$ and $a \wedge b \le x$. The set $(a \wedge b)^{\uparrow } {\setminus } (a^{\uparrow } \cup b^{\uparrow })$ is a $\beta$-neighborhood of x and is disjoint from ${{\mathbb {P}}}(P)$. – The subset $P^{\flat } \subseteq P$ is $\tau ^{\ell }(P)$-open, hence is $\beta$-closed. Thus ${{\mathbb {P}}}(P)^{\flat } = {{\mathbb {P}}}(P) \cap P^{\flat }$ is $\beta$-closed. – If P is complete then $\tau ^{\ell }(P)$ is sober, Proposition 3.8(b). Hence ${{\mathbb {P}}}(P)$ and ${{\mathbb {P}}}(P)^{\flat }$ are sober by 2.3. $\square$

Example 6.4

Let P be a complete lattice. Then the infimum of a down-directed set of prime elements is prime by Proposition 6.3. We show that the supremum of an up-directed set of prime elements need not be prime. Let $P = {{\mathbb {N}}}\cup \{\omega \} \cup \{a,b,c\}$ with the following partial order: The set ${{\mathbb {N}}}\cup \{\omega \}$ of ordinals carries the natural total order and all its elements are smaller than a, b, c. Moreover we define $a,b < c$ and assume that a, b are incomparable. Then P is an algebraic lattice, ${{\mathbb {N}}}\subseteq {{\mathbb {P}}}(P)$ and $\bigvee {{\mathbb {N}}}= \omega \notin {{\mathbb {P}}}(P)$.

Lemma 6.5

Let P and Q be $\wedge$-semilattices, $\varphi :P \rightarrow Q$ a poset map with left adjoint $\varphi ^*$. If $\varphi ^*$ is a $\wedge$-homomorphism then $\varphi$ maps prime elements to prime elements.

Proof

If $p \in {{\mathbb {P}}}(P)$ and $b \wedge b' \le \varphi (p)$ then $\varphi ^*(b) \wedge \varphi ^*(b') = \varphi ^*(b \wedge b') \le p$, hence $\varphi ^*(b) \le p$ or $\varphi ^*(b') \le p$, hence $b \le \varphi (p)$ or $b' \le \varphi (p)$. $\square$

Remark 6.6

Let P be a $\wedge$-semilattice with top element and $Q \subseteq P$ a $\wedge$-subsemilattice, e.g., a closure system. The inclusion ${{\mathbb {P}}}(P) \cap Q \subseteq {{\mathbb {P}}}(Q)$ is obvious and may be proper. For example, consider a power set ${\mathfrak {P}}(S)$ where $|S| \ge 3$, cf. Example 6.2(a). There are distinct elements $A,B \in {\mathfrak {P}}(S) \setminus {{\mathbb {P}}}({\mathfrak {P}}(S))$. We define $Q = \langle \{A,B\} \rangle _{{\mathfrak {P}}(S)} = \{A \cap B,A,B,S\}$. Then $A,B \in {{\mathbb {P}}}(Q) {\setminus } {{\mathbb {P}}}({\mathfrak {P}}(S))$.

Now assume that Q is a closure system. If the closure map $\vartheta = \iota ^*$ (equivalently, the closure operator, cf. Remark 3.6) is a $\wedge$-homomorphism then ${{\mathbb {P}}}(P) \cap Q = {{\mathbb {P}}}(Q)$ by Lemma 6.5. But note that the equality ${{\mathbb {P}}}(P) \cap Q = {{\mathbb {P}}}(Q)$ may be true without $\eta$ being a $\wedge$-homomorphism. For an example let K be a field and V a vector space, ${\mathscr {U}}(V) \subseteq {\mathfrak {P}}(V)$ the algebraic closure system of subspaces, Example 6.2(b). Then ${{\mathbb {P}}}({\mathfrak {P}}(V)) \cap {\mathscr {U}}(V) = \{V\} = {{\mathbb {P}}}({\mathscr {U}}(V))$, but the closure operator is not a $\wedge$-homomorphism. For, consider two disjoint generating sets A and B of a nontrivial subspace $U \subseteq V$. Then $\eta (A) = U = \eta (B)$, but $\eta (A \cap B) = \{0\}$.

Proposition 6.7

Let P be a complete lattice, X a subset and $Q = \langle X \rangle _P$. Then the inclusions $a^{\uparrow X} \cup b^{\uparrow X} \subseteq (\eta _{Q,P}(a) \wedge \eta _{Q,P}(b))^{\uparrow X} \subseteq (a \wedge b)^{\uparrow X}$ hold for all $a,b \in P$, and

• (a) both inclusions are equalities if and only if $X \subseteq {{\mathbb {P}}}(P)$;

• (b) the first inclusion is an equality if and only if $X \subseteq {{\mathbb {P}}}(Q)$;

• (c) the second inclusion is an equality if and only if $\eta _{Q,P}$ is a $\wedge$-homomorphism.

Proof

The inequalities $\eta _{Q,P}(a) \wedge \eta _{Q,P}(b) \le \eta _{Q,P}(a),\eta _{Q,P}(b)$ imply

$$\begin{aligned} a^{\uparrow X} \cup b^{\uparrow X} = \eta _{Q,P}(a)^{\uparrow X} \cup \eta _{Q,P}(b)^{\uparrow X} \subseteq (\eta _{Q,P}(a) \wedge \eta _{Q,P}(b))^{\uparrow X}. \end{aligned}$$

The second inclusion follows from $a \wedge b \le \eta _{Q,P}(a) \wedge \eta _{Q,P}(b)$.

(a) is clear by the definition of prime elements.

(b). The equality $a^{\uparrow X} \cup b^{\uparrow X} = (\eta _{Q,P}(a) \wedge \eta _{Q,P}(b))^{\uparrow X}$ holds for all $a,b \in P$ if and only if the equality $c^{\uparrow X} \cup d^{\uparrow X} = (c \wedge d)^{\uparrow X}$ holds for all $c,d \in Q$, if and only if $X \subseteq {{\mathbb {P}}}(Q)$, cf. item (a).

(c). The second inclusion is an equality if and only if

$$\begin{aligned} \eta _{Q,P}(a \wedge b)^{\uparrow X} = (a \wedge b)^{\uparrow X} = (\eta _{Q,P}(a) \wedge \eta _{Q,P}(b))^{\uparrow X}, \end{aligned}$$

and the claim follows from 3.1. $\square$

Example 6.8

Notation as in Proposition 6.7. We show that each inclusion in Proposition 6.7 can be proper while the other one is an equality. In the first example of Remark 6.6 the second inclusion is proper and the first one is an equality. For the other example let P be a complete lattice such that ${{\mathbb {P}}}(P)$ is not totally ordered and let $Q = \langle {{\mathbb {P}}}(P) \rangle _P$. Both inclusions are equalities, and $\eta _{Q,P}$ is a $\wedge$-homomorphism. Now consider the trivial generating set Q of Q. Then ${{\mathbb {P}}}(Q)$ is a proper subset of Q, cf. Example 6.2(c), hence the first inclusion is proper, whereas the second one is an equality since $\eta _{Q,P}$ does not depend on the particular generating set.

Corollary 6.9

(cf. [12, 1.1(9)]). Let P be a complete lattice, Q a closure system, $Q = \langle {{\mathbb {P}}}(Q) \rangle _P$. Then the following conditions are equivalent:

• (a) $\eta$ is a $\wedge$-homomorphism.

• (b) ${{\mathbb {P}}}(P) \cap Q = {{\mathbb {P}}}(Q)$.

• (c) ${{\mathbb {P}}}(Q) \subseteq {{\mathbb {P}}}(P)$.

Proof

(a)$\Rightarrow$(b). See Remark 6.6. (b)$\Rightarrow$(c) is trivial. (c)$\Rightarrow$(a) follows from Proposition 6.7. $\square$

Corollary 6.10

Let P be an algebraic lattice and use the notation of 5.1. Then

• (a) ${{\mathbb {P}}}(M) \cap L = {{\mathbb {P}}}(L)$.

• (b) If $X \subseteq {{\mathbb {P}}}(L)$ then $\eta _{L,M}$ is a $\wedge$-homomorphism and ${{\mathbb {P}}}(M) \subseteq Y^{\sharp }$.

Proof

(a). The inclusion ${{\mathbb {P}}}(M) \cap L \subseteq {{\mathbb {P}}}(L)$ holds by Remark 6.6. For the other inclusion pick $c \in L {\setminus } {{\mathbb {P}}}(M)$ and $a,b \in {{\mathbb {K}}}(M)$ with $a,b \not \le c$, $a \wedge b \le c$, Lemma 6.1. Since ${{\mathbb {K}}}(M) \subseteq L$, Theorem 5.2(d), we see that $c \notin {{\mathbb {P}}}(L)$.

(b). Proposition 6.7 and item (a) show that $\eta _{L,M}$ is a $\wedge$-homomorphism. For the second assertion pick $b \in M \setminus Y^{\sharp }$. There is a constructible set $c^{\uparrow M} {\setminus } F^{\uparrow M}$ in M (with $c \in {{\mathbb {K}}}(M)$ and $F \subseteq {{\mathbb {K}}}(M)$ finite) containing b and disjoint from X, i.e., $c^{\uparrow X} \subseteq F^{\uparrow M}$. As ${{\mathbb {K}}}(M) \subseteq L$ (Theorem 5.2(d)) it follows that $c = \bigwedge _M c^{\uparrow X}$. Thus $\bigwedge _M F \le c \le b$, hence $b \in (\bigwedge _M F)^{\uparrow M} {\setminus } F^{\uparrow M}$, i.e., $b \notin {{\mathbb {P}}}(M)$. $\square$

Proposition 6.11

Assume P be a complete lattice and consider $X,Y \subseteq {{\mathbb {P}}}(P)^{\flat }$. Then $\langle X \rangle _P = \langle Y \rangle _P$ if and only if $\overline{X}^{\beta } = \overline{Y}^{\beta }$ where $\beta = \beta (\tau ^{\ell }(P))$.

Proof

If $\overline{X}^{\beta } = \overline{Y}^{\beta }$ then $\langle X \rangle _P = \langle \overline{X}^{\beta } \rangle _P = \langle \overline{Y}^{\beta } \rangle _P = \langle Y \rangle _P$, Proposition 3.7(a). Conversely, suppose $\langle X \rangle _P = \langle Y \rangle _P$. We may assume that X and Y are $\beta$-closed, cf. loc.cit., and claim that $X = Y$. If $y \in Y {\setminus } X$ then $y \in \langle X \rangle _P$ implies $y = \bigwedge _P y^{\uparrow X}$, where $y^{\uparrow X} \ne {\emptyset }$. Since X is $\beta$-closed there is a finite nonempty set $F \subseteq P$ such that $y^{\uparrow P} \setminus F^{\uparrow P}$ is a $\beta$-open neighborhood of y and is disjoint from X. But then $y^{\uparrow X} \subseteq F^{\uparrow X} = (\bigwedge _P F)^{\uparrow X}$ (where we use $X \subseteq {{\mathbb {P}}}(P)$). It follows that $\bigwedge _P F \le y$, and $y \in {{\mathbb {P}}}(P)$ yields $y \in F^{\uparrow P}$, a contradiction. $\square$

Corollary 6.12

Let P be a complete lattice, $X \subseteq {{\mathbb {P}}}(P)^{\flat }$ and $Q = \langle X \rangle _P$. Then ${{\mathbb {P}}}(Q)^{\flat }$ is the closure of X for $\beta = \beta (\tau ^{\ell }(P))$, i.e., ${{\mathbb {P}}}(Q)^{\flat }$ is the sobrification of X (cf. 2.3).

Proof

Proposition 6.7 and Corollary 6.9 imply ${{\mathbb {P}}}(Q)^{\flat } = {{\mathbb {P}}}(P)^{\flat } \cap Q$. Both ${{\mathbb {P}}}(P)^{\flat } \subseteq P$ and $Q \subseteq P$ are closed for $\beta$, cf. Proposition 6.3 and Proposition 3.7(a), hence ${{\mathbb {P}}}(Q)^{\flat } \subseteq P$ is closed as well. Thus ${{\mathbb {P}}}(Q)^{\flat } = \overline{X}^{\beta }$ follows from Proposition 6.11 and $Q = \langle X \rangle _P = \langle {{\mathbb {P}}}(Q)^{\flat } \rangle _P$. $\square$

Theorem 6.13

Let P be a complete lattice generated by $X \subseteq {{\mathbb {P}}}(P)^{\flat }$.

• (a) If $A \in {\mathcal {A}}(X)$ then $A = a^{\uparrow X}$ where $a = \bigwedge _P A$.

• (b) P is isomorphic to the spatial frame ${\mathcal {O}}(X^{\flat })$.

Proof

(a). Trivially, $A \subseteq a^{\uparrow X}$. Now pick $p \in X {\setminus } A$. Since A is closed for $\tau ^{\ell }(P)|_X$ there is a finite set $F \subseteq P$ with $A \subseteq F^{\uparrow P} \subseteq (\bigwedge _P F)^{\uparrow P}$ and $p \notin F^{\uparrow P}$. It follows that $\bigwedge _P F \le a$, hence $\bigwedge _P F \not \le p$ (since $p \in {{\mathbb {P}}}(P)$), and we see that $a \not \le p$.

(b). The map $\gamma _{X,P}:P \rightarrow {\mathcal {O}}(X,{\uparrow }(X,P)), a \mapsto X {\setminus } a^{\uparrow X}$ is an isomorphism of posets, Lemma 3.5. Thus it suffices to show ${\mathcal {O}}(X,{\uparrow }(X,P))) = {\mathcal {O}}(X)$. The inclusion ${\mathcal {O}}(X,{\uparrow }(X,P)) \subseteq {\mathcal {O}}(X)$ holds trivially, Example 3.3, and equality follows from item (a). $\square$

Remark 6.14

Theorem 6.13 is a more precise version of [8, I$-$3.15] where it is shown that a complete lattice is a frame if it is prime generated and that continuous (in particular: algebraic) lattices are frames if and only they are prime generated. Prime generated frames are exactly the spatial frames, [11, p. 43]. If $X \subseteq {{\mathbb {P}}}(P)^{\flat }$ is generating then ${{\mathbb {P}}}(P)^{\flat }$ is the sobrification of X, Corollary 6.12, and the canonical map ${\mathcal {O}}({{\mathbb {P}}}(P)^{\flat }) \rightarrow {\mathcal {O}}(X)$ is an isomorphism.

Remark 6.15

Let P be a complete lattice and define ${\mathscr {N}}(P)$ to be the set of prime generated closure systems $Q \subseteq P$ such that $\eta _{Q,P}$ is a $\wedge$-homomorphism. If $Q \in {\mathscr {N}}(P)$ then $Q = \langle {{\mathbb {P}}}(Q)^{\flat } \rangle _P$ and ${{\mathbb {P}}}(Q)^{\flat } \subseteq {{\mathbb {P}}}(P)^{\flat }$, Corollary 6.9. Thus Proposition 6.11 shows that ${\mathscr {N}}(P) \rightarrow {\mathcal {A}}({{\mathbb {P}}}(P)^{\flat },\beta ), Q \mapsto {{\mathbb {P}}}(Q)^{\flat }$ and ${\mathcal {A}}({{\mathbb {P}}}(P)^{\flat },\beta ) \rightarrow {\mathscr {N}}(P), X \mapsto \langle X \rangle _P$ are mutually inverse isomorphisms of posets. Each element of ${\mathscr {N}}(P)$ is a spatial frame by Theorem 6.13.

Example 6.16

Consider a spatial (= prime generated) frame P. For each $Q \in {\mathscr {N}}(P)$ the closure operator $\eta _{Q,P}:P \rightarrow P$ is a nucleus and Q is a spatial frame. Conversely, if $\nu :P \rightarrow P$ is a nucleus such that $\nu (P)$ is a spatial frame then $\nu (P) \in {\mathscr {N}}(P)$, Corollary 6.9. Identifying $Q \in {\mathscr {N}}(P)$ with the nucleus $\eta _{Q,P}$ we consider ${\mathscr {N}}(P)$ as a subset of the assembly (= the set of all nuclei), [11, p. 51 ff], [18], namely the set of nuclei with spatial image. In general this is a proper subset. We show how one can construct spatial frames P and nuclei $\nu :P \rightarrow P$ such that $\nu (P)$ is not spatial, hence is not in ${\mathscr {N}}(P)$.

Let X be a localic space, cf. [17, p. 1163] and [16, Section 3] (where localic spaces were introduced under the name locales), such that the frame $\overset{\circ }{{\mathcal {K}}}(X)$ does not have enough localic points, cf. [11, p. 43], [16, p. 27], i.e., is not spatial and not prime generated. For example, X could be an infinite extremally disconnected Boolean space without isolated points, [6, Examples 9.5.4(ii)]. (Then $\overset{\circ }{{\mathcal {K}}}(X)$ does not have any localic points at all.) The frame $P = {\mathcal {O}}(X)$ is algebraic and coherent. The map $\nu _X:P \rightarrow P, O \mapsto \overline{O}^{{{\,\textrm{con}\,}}}$ is a nucleus with $\nu _X(P) = \overset{\circ }{{\mathcal {K}}}(X)$ and is called the natural nucleus for X, [16, p. 13], [17, p. 1163]. It follows that ${{\mathbb {P}}}(P) \cap \overset{\circ }{{\mathcal {K}}}(X) = {{\mathbb {P}}}(\overset{\circ }{{\mathcal {K}}}(X))$, Remark 6.6. As $\overset{\circ }{{\mathcal {K}}}(X)$ is not prime generated it does not belong to ${\mathscr {N}}(P)$.

Proposition 6.17

Let P and Q be complete lattices generated by $X \subseteq {{\mathbb {P}}}(P)^{\flat }$ and $Y \subseteq {{\mathbb {P}}}(Q)^{\flat }$. Let $\varphi : P \rightarrow Q$ be a $\bigwedge$-homomorphism with $\varphi (X) \subseteq Y$ and let $f:X \rightarrow Y$ be the restriction of $\varphi$. Then the following diagrams are commutative:

Proof

Note that the left adjoint $\varphi ^*$ exists by 2.6. It suffices to prove commutativity for the diagram on the left. Uniqueness of right adjoints implies the claim for the other diagram. Theorem 2.11(c) shows that $\varphi$ is continuous for the coarse lower topology, hence f is continuous as well. For each $b \in Q$ the set $\gamma _{Y,Q}(b) = Y {\setminus } b^{\uparrow Y} \subseteq Y$ is open, hence

$$\begin{aligned} O:= {\mathcal {O}}(f)(Y \setminus b^{\uparrow Y}) = X \setminus f^{-1}(b^{\uparrow Y}) \subseteq X \end{aligned}$$

is open and equals $X \setminus z^{\uparrow X} = \gamma _{X,P}(z)$, where

$$\begin{aligned} z = \bigwedge _P X \setminus O = \bigwedge _P \{x \in X \mid b \le f(x)\}, \end{aligned}$$

Theorem 6.13(a). It remains to show that $\varphi ^*(b) = z$. Recall from  2.6 that $\varphi ^*(b) = \bigwedge _P \{a \in P \mid b \le \varphi (a)\}$, hence the inclusion

$$\begin{aligned} \{x \in X \mid b \le \varphi (x) = f(x)\} \subseteq \{a \in P \mid b \le \varphi (a)\} \end{aligned}$$

yields $\varphi ^*(b) \le z$. To prove equality we show that $b \le \varphi (a)$ implies $z \le a$. As $a = \bigwedge _P a^{\uparrow X}$ we have $\varphi (a) = \bigwedge _Q \varphi (a^{\uparrow X}) = \bigwedge _Q f(a^{\uparrow X})$. Thus $b \le \varphi (a)$ implies $b \le f(x)$ for all $x \in a^{\uparrow X}$, i.e., $a^{\uparrow X} \subseteq \{ x \in X \mid b \le f(x)\}$, hence $z \le \bigwedge _P a^{\uparrow X} = a$. $\square$

We continue with the notation and hypotheses of Proposition 6.17. It follows from Theorem 6.13 that P and Q are frames and ${\mathcal {O}}(f)_*(U)$ is the largest $V \in {\mathcal {O}}(Y)$ with $f^{-1}(V) \subseteq U$. Accordingly $\varphi (a)$ is the largest element $b \in Q$ with $\varphi ^*(b) \le a$. The composition $\varphi \circ \varphi ^*$ is a closure operator, 2.6(a). In fact, it is a nucleus of the frame Q since $\varphi ^*$ is a frame homomorphism and $\varphi$ preserves all infima.

If P and Q are algebraic lattices then they are spectral spaces and $\varphi$ and $\varphi ^*$ are maps between spectral spaces. We ask whether they are spectral maps. We know from 2.6 and Theorem 2.11(c), (e) that $\varphi$ is coarse lower continuous and $\varphi ^*$ is Scott continuous.

Proposition 6.18

• (a) Let $X \subseteq P$ and $Y \subseteq Q$ be patch closed. Then $\varphi$ is spectral if and only if f is spectral.

• (b) $\varphi ^*$ is spectral if and only if for each $a \in {{\mathbb {K}}}(P)$ there is a smallest element $b \in Q$ with $a \le \varphi ^*(b)$. If this is the case then $b \in {{\mathbb {K}}}(Q)$.

Proof

(a). If $\varphi$ is a spectral map then its restriction f is trivially spectral. Conversely assume that f is spectral. By Corollary 2.13(c) it suffices to show that $\varphi$, or ${\mathcal {O}}(f)_*$, is a dcpo map. Recall from Example 3.2 that ${\mathcal {O}}(f)_* \circ {\mathcal {O}}(f)$ is a closure operator and ${\mathcal {O}}(f) \circ {\mathcal {O}}(f)_*$ is a kernel operator. For any up-directed set ${\mathcal {U}}\subseteq {\mathcal {O}}(X)$ the inclusion $\bigcup {\mathcal {O}}(f)_*({\mathcal {U}}) \subseteq {\mathcal {O}}(f)_*(\bigcup {\mathcal {U}})$ holds trivially, and we must prove equality. Assume the inclusion is proper. Then there is some $W \in \overset{\circ }{{\mathcal {K}}}(Y)$ with $W \not \subseteq \bigcup {\mathcal {O}}(f)_*({\mathcal {U}})$, but $W \subseteq {\mathcal {O}}(f)_*(\bigcup {\mathcal {U}})$. It follows that $f^{-1}(W) = {\mathcal {O}}(f)(W) \in \overset{\circ }{{\mathcal {K}}}(X)$ and ${\mathcal {O}}(f)(W) \subseteq {\mathcal {O}}(f) \circ {\mathcal {O}}(f)_*(\bigcup {\mathcal {U}}) \subseteq \bigcup {\mathcal {U}}$. Since ${\mathcal {U}}$ is up-directed there is some $U \in {\mathcal {U}}$ with ${\mathcal {O}}(f)(W) \subseteq U$. But then $W \subseteq {\mathcal {O}}(f)_* \circ {\mathcal {O}}(f)(W) \subseteq {\mathcal {O}}(f)_*(U)$, a contradiction.

(b). Pick $a \in {{\mathbb {K}}}(P)$, i.e., $a^{\uparrow P}$ is closed and constructible, cf. 2.10, and let $b \in Q$ be the smallest element with $a \le \varphi ^*(b)$. It suffices to show that $b \in {{\mathbb {K}}}(Q)$ since then $(\varphi ^*)^{-1}(a^{\uparrow P}) = b^{\uparrow Q}$ is closed and constructible. As $b = \bigvee _Q b^{\downarrow {{\mathbb {K}}}(Q)}$ and $\varphi ^*$ preserves all suprema it follows that $a \le \varphi ^*(b) = \bigvee _P \varphi ^*(b^{\downarrow {{\mathbb {K}}}(Q)})$. The supremum is up-directed and $a \in {{\mathbb {K}}}(P)$, hence there is some $c \in b^{\downarrow {{\mathbb {K}}}(Q)}$ with $a \le \varphi ^*(c)$. Minimality of b implies $c = b$.

Now assume $\varphi ^*$ is spectral. For $a \in {{\mathbb {K}}}(P)$ the set $(\varphi ^*)^{-1}(a^{\uparrow P})$ is closed and constructible in Q, hence is equal to $G^{\uparrow Q}$ with $G \subseteq {{\mathbb {K}}}(Q)$ finite, cf. 2.10. It follows that $a \le \varphi ^*(y)$ for all $y \in G$, thus $a \le \bigwedge _P \varphi ^*(G) = \varphi ^*(\bigwedge _Q G)$ (note that $\varphi ^*$ is a frame homomorphism, Proposition 6.17). We define $b = \bigwedge _Q G$ and claim that this is the smallest $c \in Q$ such that $a \le \varphi ^*(c)$. So, pick any $z \in Q$ with $a \le \varphi ^*(z)$. Then $z \in (\varphi ^*)^{-1}(a^{\uparrow P}) = G^{\uparrow Q} \subseteq b^{\uparrow Q}$, i.e., $b \le z$. $\square$

Example 6.19

We continue with Proposition 6.18(b) and assume $X \subseteq P$ and $Y \subseteq Q$ are patch closed. Since $P = \langle X \rangle _P = \langle {{\mathbb {P}}}(P)^{\flat } \rangle _P$ and $X = \overline{X}^{\tau ^{\ell }(P)}$ it follows from Proposition 6.11 that $X = {{\mathbb {P}}}(P)^{\flat }$. The same holds for Y and Q. We exhibit a few examples where ${\mathcal {O}}(f)$ is spectral. The condition in Proposition 6.18(b) says that for each $U \in \overset{\circ }{{\mathcal {K}}}(X)$ there is a smallest $V \in \overset{\circ }{{\mathcal {K}}}(Y)$ with $U \subseteq f^{-1}(V)$. Since f is continuous and U is quasi-compact it follows that $f(U) \subseteq Y$ is quasi-compact, hence ${{\,\textrm{Gen}\,}}(f(U))$ is patch closed in Y and is the intersection of all open sets containing f(U), [6, Theorem 4.1.5]. So, there is a smallest open set containing f(U) if and only if ${{\,\textrm{Gen}\,}}(f(U))$ is open, if and only if ${{\,\textrm{Gen}\,}}(f(U))$ is quasi-compact open. Here are a few situations where this is the case.

Obviously, the condition holds if f is an open map. Next assume that Y is totally ordered. Then Q is totally ordered as well and is equal to $Y^{\sharp }$, see Example 6.2(c). If $C \subseteq Y$ is a nonempty quasi-compact set then C has a largest element c, and ${{\,\textrm{Gen}\,}}(C) = {{\,\textrm{Gen}\,}}(c)$, [6, Proposition 1.6.1]. In particular, if $U \in \overset{\circ }{{\mathcal {K}}}(X)$ is nonempty then f(U) has a largest element. The set ${{\,\textrm{Gen}\,}}(c)$ is open if and only if c is the lower point of a jump or is the closed point of Y. Thus, ${\mathcal {O}}(f)$ is spectral if and only if for each $U \in \overset{\circ }{{\mathcal {K}}}(X)$ the largest point of f(U) is the closed point of Y or is the lower point of a jump.

Now let Y be any spectral space, define $X = Y_{{{\,\textrm{con}\,}}}$ to be the patch space, and $f = {{\,\textrm{con}\,}}_Y:X \rightarrow Y$ the identity map, [6, 1.3.24]. Then $C \subseteq X$ is quasi-compact open if and only if it is constructible in Y. Thus, ${\mathcal {O}}(f)$ is spectral if and only if ${{\,\textrm{Gen}\,}}(C) \subseteq Y$ is open for each constructible set $C \subseteq Y$. Equivalently, in $Y_{{{\,\textrm{inv}\,}}}$ the closure of every constructible set is constructible, i.e., $Y_{{{\,\textrm{inv}\,}}}$ is a Heyting space, [6, Definition 8.3.1].

Theorem 6.20

Notation as in  5.1. Let P be an algebraic lattice, $X \subseteq {{\mathbb {P}}}(P)$.

• (a) If P is arithmetic then ${{\mathbb {P}}}(P)$ is patch closed in P.

• (b) If P is coherent then ${{\mathbb {P}}}(P)^{\flat }$ is patch closed in P.

• (c) Suppose that X is patch closed in P. Then L is an arithmetic algebraic frame and is patch closed in P. Moreover, ${{\mathbb {P}}}(L) = {{\mathbb {P}}}(P) \cap L = X^{\sharp }$.

• (d) Suppose that X is patch closed in P and is contained in ${{\mathbb {P}}}(P)^{\flat }$. Then L is a coherent algebraic frame and is patch closed in P. Moreover, ${{\mathbb {P}}}(L)^{\flat } = {{\mathbb {P}}}(P)^{\flat } \cap L = X$.

Proof

(a). The sets $a^{\uparrow P} \setminus G^{\uparrow P}$, with $a \in {{\mathbb {K}}}(P)$ and $G \subseteq {{\mathbb {K}}}(P)$ finite, are a basis of open sets for the patch topology of P, 2.10. If $x \in P \setminus {{\mathbb {P}}}(P)$ then we must find such a and G with $x \in a^{\uparrow P} {\setminus } G^{\uparrow P}$ and $a^{\uparrow P} {\setminus } G^{\uparrow P} \cap {{\mathbb {P}}}(P) = {\emptyset }$. As x is not prime there are $y,z \in {{\mathbb {K}}}(P)$ with $y \wedge z \le x$ and $y,z \not \le x$, Lemma 6.1. Thus, $a = y \wedge z$ and $G = \{y,z\}$ meet the requirements.

(b). Item (a) shows that ${{\mathbb {P}}}(P) \subseteq P$ is patch closed. It follows from $\top _P \in {{\mathbb {K}}}(P)$ that $P^{\flat }$ is quasi-compact open,  2.10, hence ${{\mathbb {P}}}(P)^{\flat } = {{\mathbb {P}}}(P) \cap P^{\flat }$ is patch closed.

(c). Theorem 5.2(a), Theorem 4.5 and Theorem 6.13 show that $L = M$ is patch closed in P, is an algebraic lattice and isomorphic to the frame ${\mathcal {O}}(X^{\flat })$. To see that L is arithmetic we pick $a,b \in {{\mathbb {K}}}(L)$ and claim that $a \wedge _L b \in {{\mathbb {K}}}(L)$. The sets $a^{\uparrow X},b^{\uparrow X} \subseteq X$ are closed and constructible since $a^{\uparrow L}, b^{\uparrow L} \subseteq L$ are closed and constructible. Proposition 6.7 implies $(a \wedge _L b)^{\uparrow X} = a^{\uparrow X} \cup b^{\uparrow X}$ (since $X \subseteq {{\mathbb {P}}}(L)$). As $a^{\uparrow X} \cup b^{\uparrow X}$ is closed and constructible it follows that $a \wedge _L b \in {{\mathbb {K}}}(L)$. The claim about the prime elements follows from Corollary 6.9 and Corollary 6.10.

(d). By item (c) it suffices to show that $\top _L \in {{\mathbb {K}}}(L)$. We identify $L = {\mathcal {O}}(X)$, Theorem 6.13, and note that $\top _{{\mathcal {O}}(X)} = X \in \overset{\circ }{{\mathcal {K}}}({\mathcal {O}}(X))$. $\square$

Corollary 6.21

Let P be an algebraic lattice and Q a prime generated algebraic closure system. Then Q is coherent if and only if ${{\mathbb {P}}}(Q)^{\flat } \subseteq P$ is patch closed.

Proof

By Theorem 4.5 a subset $S \subseteq Q$ is patch closed in Q if and only if it is patch closed in P. Thus, Theorem 6.20(b), (d) implies that Q is coherent if and only if ${{\mathbb {P}}}(Q)^{\flat } \subseteq P$ is patch closed. $\square$

Corollary 6.22

Notation as in 5.1. Let P be an algebraic lattice and assume $X \subseteq {{\mathbb {P}}}(P)^{\flat }$. Consider the following conditions about M.

• (a) M is arithmetic.

• (b) M is coherent.

• (c) $Y^{\sharp } = {{\mathbb {P}}}(M)$.

• (d) $Y = {{\mathbb {P}}}(M)^{\flat }$.

Then (b)$\Leftrightarrow$(d)$\Rightarrow$(a)$\Leftrightarrow$(c) and the conditions are all satisfied if there is a patch closed subset $Z \subseteq P$ with $X \subseteq Z \subseteq {{\mathbb {P}}}(P)^{\flat }$.

Proof

(a)$\Rightarrow$(c). Theorem 6.20(a) shows that $Y \subseteq {{\mathbb {P}}}(M)$, and ${{\mathbb {P}}}(M) \subseteq Y^{\sharp }$ holds by Corollary 6.10. The other implications, as well as the final assertion, are clear by Theorem 6.20. $\square$

We continue with the hypotheses of Corollary 6.22. Assume that M is not arithmetic. We ask how close the elements of Y are to being prime.

Proposition 6.23

Notation as in 5.1. Let P be an algebraic lattice, $X \subseteq {{\mathbb {P}}}(L)$. For $z \in M$ the following conditions are equivalent:

• (a) $z \in Y^{\sharp }$.

• (b) If $F \subseteq {{\mathbb {K}}}(M)$ is nonempty and finite and $z \notin F^{\uparrow M}$ then $\bigwedge _M F \not \ll z$.

Proof

(a)$\Rightarrow$(b). Suppose $z \in {{\mathbb {P}}}(M)$ and $F \subseteq M$ is finite and nonempty with $z \notin F^{\uparrow M}$. Then $\bigwedge _M F \not \le z$, hence $\bigwedge _M F \not \ll z$, [8, I$-$1.2]. Now assume that $z \notin {{\mathbb {P}}}(M)$ and that there is a finite subset $F \subseteq {{\mathbb {K}}}(M)$ with $z \notin F^{\uparrow M}$, but $\bigwedge _M F \ll z$. (Note that $F \ne {\emptyset }$ since otherwise $\top = \bigwedge _M F \le z$, hence $z = \top \in {{\mathbb {P}}}(M)$.) As $z = \bigvee _M z^{\downarrow {{\mathbb {K}}}(M)}$ and the supremum is up-directed there is some $t \in z^{\downarrow {{\mathbb {K}}}(M)}$ with $\bigwedge _M F \le t$, hence $z \in t^{\uparrow M} {\setminus } F^{\uparrow M}$. The set $t^{\uparrow M} {\setminus } F^{\uparrow M}$ is patch open in M, cf. 2.10, and is disjoint from X, hence $z \notin Y = \overline{X}^{{{\,\textrm{con}\,}}}$, a contradiction.

(b)$\Rightarrow$(a). As ${{\mathbb {P}}}(M) \subseteq Y^{\sharp }$, Corollary 6.10, we assume $z \notin {{\mathbb {P}}}(M)$. Then there are $u,v \in {{\mathbb {K}}}(M)$ with $z \in (u \wedge v)^{\uparrow M} {\setminus } (u^{\uparrow M} \cup v^{\uparrow M})$, Lemma 6.1. We have to show that $C \cap X \ne {\emptyset }$ for any constructible set $C \subseteq M$ with $z \in C$. It suffices to consider basic constructible sets, say $C = c^{\uparrow M} {\setminus } F^{\uparrow M}$ with $c \in {{\mathbb {K}}}(M)$ and $F \subseteq {{\mathbb {K}}}(M)$ finite, 2.10. Then $H = F \cup \{u,v\} \subseteq {{\mathbb {K}}}(M)$ is finite and has the property that $\bigwedge _M H \le u \wedge v \le z$ and $z \notin H^{\uparrow M}$. The hypothesis says that $\bigwedge _M H \not \ll z$. Now $c \in z^{\downarrow {{\mathbb {K}}}(M)}$ implies $c \ll z$, [8, I$-$1.2], and we conclude that $\bigwedge _M H \not \le c$. Since ${{\mathbb {K}}}(M) \subseteq L$, Theorem 5.2(d), we have $c = \bigwedge _M c^{\uparrow X} \in L$ and $\bigwedge _M H \in L$. Hence there is some $p \in c^{\uparrow X}$ with $\bigwedge _M H \not \le p$, and this implies $p \in c^{\uparrow X} {\setminus } H^{\uparrow X} \subseteq C \cap X$. $\square$

The following examples illustrate the preceding results.

Example 6.24

For any set S the power set ${\mathfrak {P}}(S)$ is an algebraic frame. The compact elements are the finite subsets, hence ${\mathfrak {P}}(S)$ is arithmetic, but is coherent only if S is finite. The nontrivial prime elements are described in Example 6.2(a). We identify S with ${{\mathbb {P}}}({\mathfrak {P}}(S))^{\flat }$ via $s \mapsto S \setminus \{s\}$, hence $S^{\sharp } = {{\mathbb {P}}}({\mathfrak {P}}(S))$. The coarse lower topology of ${\mathfrak {P}}(S)$ restricts to the discrete topology on S, and $S \cup \{S\}$ is a spectral subspace of ${\mathfrak {P}}(S)$, Theorem 6.20(a). Specialization in $S^{\sharp }$ is given by $s \le S$ for all $s \in S$. The arithmetic algebraic frame ${\mathfrak {P}}(S)$ is isomorphic to ${\mathcal {O}}(S)$, S with the discrete topology.

Example 6.25

(cf. [11, p. 50, 2.4(a)]). Let P be an algebraic lattice, $a \in P$. Obviously the subset $a^{\uparrow } \subseteq P$ is an algebraic closure system. The closure operator $\eta :P \rightarrow P$ is given by $x \mapsto a \vee x$. This is always a dcpo homomorphism, but is a $\bigvee$-homomorphism only if $\bot _P = a$. (For if $\bot _P < a$ then $\eta (\bigvee _P {\emptyset }) = a > \bot _P = \bigvee _P \eta ({\emptyset })$.) The algebraic lattice P is a spectral space for the coarse lower topology and $a^{\uparrow }$ is a closed subset, hence is a spectral space as well, and the inclusion $\iota _a:a^{\uparrow } \rightarrow P$ is a spectral map.

Set $Y = {{\mathbb {P}}}(P)^{\flat }$ and $X_a = {{\mathbb {P}}}(a^{\uparrow })^{\flat }$. The inclusion $Y \cap a^{\uparrow } \subseteq X_a$ is obvious. Let $\eta$ be a $\wedge$-map. Then $Y \cap a^{\uparrow } = X_a$, Remark 6.6, and $X_a \subseteq Y$ is a closed subset. Now assume that P is distributive, equivalently a prime generated algebraic frame, [8, Theorem I$-$23.15]. Then $P \simeq {\mathcal {O}}(Y)$ and $\eta$ is a nucleus. The equality $b^{\uparrow X_a} = b^{\uparrow Y}$ holds for each $b \in a^{\uparrow }$, hence $b = \bigwedge _P b^{\uparrow Y} = \bigwedge _{a^{\uparrow }} b^{\uparrow X_a}$, and $a^{\uparrow }$ is prime generated as well.

Next consider the compact elements. It follows from Proposition 4.2 and Theorem 4.5 that ${{\mathbb {K}}}(a^{\uparrow }) = \vartheta ({{\mathbb {K}}}(P))$. It is trivially true that ${{\mathbb {K}}}(P) \cap a^{\uparrow } \subseteq {{\mathbb {K}}}(a^{\uparrow })$, and equality holds if and only if $a \in {{\mathbb {K}}}(P)$. For, assume $a \in {{\mathbb {K}}}(P)$ and pick $c \in {{\mathbb {K}}}(a^{\uparrow })$, $D \subseteq P$ up-directed with $a \le c \le \bigvee _P D$. There is some $d \in D$ with $a \le d$. Thus, $d^{\uparrow } \cap D \subseteq a^{\uparrow }$ is up-directed and $c \le \bigvee _P D = \bigvee _{a^{\uparrow }} d^{\uparrow } \cap D$. As $c \in {{\mathbb {K}}}(a^{\uparrow })$ there is some $x \in d^{\uparrow } \cap D$ with $c \le x$, hence $c \in {{\mathbb {K}}}(P)$. Conversely assume $a \notin {{\mathbb {K}}}(P)$. Then $\bot _{a^{\uparrow }} = a$ is compact in $a^{\uparrow }$, but not in P.

Consider the diagrams of Proposition 6.17 and assume P is prime generated. In general $X_a \subseteq a^{\uparrow }$ and $Y \subseteq P$ will not be spectral subspaces. Let $f_a: X_a \rightarrow Y$ be the inclusion.

As noted above, $\iota _a$ is always spectral. But $\vartheta _a$ need not be spectral. For an example, let Y be an infinite set with the cofinite topology and set $P = {\mathcal {O}}(Y)$. Then P is a prime generated frame and is algebraic since Y is a Noetherian space, i.e., ${{\mathbb {K}}}(P) = \overset{\circ }{{\mathcal {K}}}(Y) = {\mathcal {O}}(Y) = P$, [6, Proposition 8.1.5]. Pick $x,y \in Y$, set $a = Y {\setminus } \{x,y\}, b = Y {\setminus } \{y\} \in P$. Then $b \in {{\mathbb {K}}}(P) \cap a^{\uparrow }$ and $b = a \vee c = \vartheta _a(c)$ for any $c \in P$ with $x \in c$ and $y \notin c$. Clearly, there is no smallest such $c \in P$.

On the other hand, one checks that $\vartheta _a$ is spectral if Y is Boolean and $a \in {{\mathbb {K}}}(P)$.

Example 6.26

(cf. [11, p. 50, 2.4(b)]). Let P be an algebraic lattice, pick $a \in P$ and consider the principal downset $a^{\downarrow }$. The inclusion map $\varepsilon _a:a^{\downarrow } \rightarrow P$ is a $\bigvee$-morphism, a $\wedge$-morphism and an fcpo map, however not a $\bigwedge$-morphism since $a = \varepsilon _a(\bigwedge _{a^{\downarrow }} {\emptyset }) < \top _P = \bigwedge _P {\emptyset }$ if $a < \top _P$. The map $\mu : P \rightarrow a^{\downarrow }, x \mapsto a \wedge x$ is right adjoint to $\varepsilon _a$, 2.6. One checks that $a^{\downarrow {{\mathbb {K}}}(P)} = {{\mathbb {K}}}(a^{\downarrow })$, hence $a^{\downarrow }$ is an algebraic lattice as well.

The subset $a^{\downarrow } \subseteq P$ is patch closed and $\tau ^{\ell }(a^{\downarrow }) = \tau ^{\ell }(P)|_{a^{\downarrow }}$. Thus, $\varepsilon _a$ is a spectral map. It is claimed that $\mu$ is spectral as well, i.e., is coarse lower continuous and Scott continuous. We show that $\mu ^{-1}(C) \subseteq P$ is closed and constructible if $C \subseteq a^{\downarrow }$ is closed and constructible. By 2.10 it suffices to consider sets $C = c^{\uparrow a^{\downarrow }}$ with $c \in {{\mathbb {K}}}(a^{\downarrow })$. But then $c \in {{\mathbb {K}}}(P)$ and $\mu ^{-1}(C) = c^{\uparrow P}$ is closed constructible in P.

Theorem 2.11(e) yields the somewhat surprising fact that $\mu$ is a dcpo map without any assumption about distributivity, also see [6, Corollary 4.2.9]. This can also be proved using the following order-theoretic arguments. If $D \subseteq P$ is up-directed with $z = \bigvee _P D$ then $\mu (D) \subseteq a^{\downarrow }$ is up-directed as well. With $t = \bigvee _{a^{\downarrow }} \mu (D)$ we have to show $\mu (z) = t$. It is clear that $t \le \mu (z)$ and we assume $t < \mu (z)$. Then there is some $c \in {{\mathbb {K}}}(a^{\downarrow })$ with $c \not \le t$ and $c \le \mu (z)$. Hence ${{\mathbb {K}}}(a^{\downarrow }) \subseteq {{\mathbb {K}}}(P)$ and $c \le \mu (z) \le z$ imply $c \le d$ for some $d \in D$. But then $c = \mu (c) \le \mu (d) \le t$, a contradiction.

Now let P be an algebraic frame. Then $a^{\downarrow }$ is a frame as well and $\mu$ is a surjective frame homomorphism. The right adjoint $\mu _*:a^{\downarrow } \rightarrow P$ exists and is given by $b \mapsto (a \rightarrow _P b)$, cf. 2.6 and [11, p. 8]. The map $\lambda = \mu _* \circ \mu :P \rightarrow P$ is a nucleus, [11, p. 50, 2.4(b)], and $Q_a:= \lambda (P)$ is a frame and a closure system in P (in general not algebraic). The corestriction $\lambda ^*:P \rightarrow Q_a$ of $\lambda$ and the inclusion $\lambda _*:Q_a \rightarrow P$ form the adjoint pair $(\lambda ^*,\lambda _*)$. The maps $\mu \circ \lambda _*:Q_a \rightarrow a^{\downarrow }$ (restriction of $\mu$) and $\lambda ^* \circ \mu _*: a^{\downarrow } \rightarrow Q_a$ (corestriction of $\mu _*$) are mutually inverse frame isomorphisms. Thus $Q_a$ is an algebraic frame as well and the spectral spaces $a^{\downarrow }$ and $Q_a$ are homeomorphic. The topologies $\tau ^{\ell }(Q_a)$ and $\tau ^{\ell }(P)|_{Q_a}$ coincide by Remark 3.6. But $Q_a \subseteq P$ is a spectral subspace if and only if the closure system is algebraic, Theorem 4.5.

By [8, Theorem I$-$3.15] the algebraic frames P, $a^{\downarrow }$ and $Q_a$ are prime generated. As $\lambda$ is a nucleus it follows that $Q_a \in {\mathscr {N}}(P)$, Remark 6.15, and ${{\mathbb {P}}}(P) \cap Q_a = {{\mathbb {P}}}(Q_a)$, Remark 6.6. Set $Y = {{\mathbb {P}}}(P)^{\flat }$ and $U_a = {{\mathbb {P}}}(Q_a)^{\flat } = Y \cap Q_a$ and let $g_a: U_a \rightarrow Y$ be the inclusion. Then $P \simeq {\mathcal {O}}(Y)$ and $a^{\downarrow } \simeq Q_a \simeq {\mathcal {O}}(U_a)$, Theorem 6.13. In general ${{\mathbb {P}}}(P), Y, {{\mathbb {P}}}(Q_a), U_a$ are not spectral subspaces of P. However, see Theorem 6.20.

Compare the following considerations with [13, Lemma 4.6]. We claim that $\mu$ maps prime elements to prime elements. To prove this pick $p \in {{\mathbb {P}}}(P)$. If $a \le p$ then $\mu (p) = a \in {{\mathbb {P}}}(a^{\downarrow })$. Now assume $p \in P {\setminus } a^{\uparrow }$ and $b,c \in a^{\downarrow }$ with $b \wedge c \le \mu (p)$. Then $b \wedge c \le p$, which implies $b \le p$ or $c \le p$, hence $b \le \mu (p)$ or $c \le \mu (p)$. Thus, $\mu (p) \in {{\mathbb {P}}}(a^{\downarrow })^{\flat }$, proving the claim. The isomorphisms $\mu \circ \lambda _*$ and $\lambda ^* \circ \mu _*$ restrict to the mutually inverse maps $U_a \rightarrow {{\mathbb {P}}}(a^{\downarrow })^{\flat }, p \mapsto a \wedge p$ and ${{\mathbb {P}}}(a^{\downarrow })^{\flat } \rightarrow U_a, q \mapsto (a \rightarrow _P q)$. It follows that $U_a = Y {\setminus } a^{\uparrow }$ is open in Y. Thus, the open subspaces $\{U_a \mid a \in P\}$ cover Y. If $a \in {{\mathbb {K}}}(P)$ then $a \in {{\mathbb {K}}}(a^{\downarrow })$, hence ${{\mathbb {P}}}(a^{\downarrow })^{\flat }$ and $U_a$ are quasi-compact. As P is algebraic the space Y is locally quasi-compact in the sense that the quasi-compact open sets are a basis of open sets, cf. [8, Definition 0$-$5.9].

Finally consider the diagrams of Proposition 6.17.

As $\lambda _* \circ (\lambda ^* \circ \mu _*) = \mu _*$ it follows that $\lambda _*$ is a spectral map if and only if $\mu _*$ is a spectral map, if and only if $\mu$ is coherent, see Corollary 2.14. In general this is not the case. However, assume that P is arithmetic and $a \in {{\mathbb {K}}}(P)$. Then $c \in {{\mathbb {K}}}(P)$ implies $\mu (c) \in {{\mathbb {K}}}(P) \cap a^{\downarrow } = {{\mathbb {K}}}(a^{\downarrow })$ and $a^{\downarrow }$ is a coherent algebraic frame, hence $U_a$ is a spectral space by Theorem 6.20. In fact, $\{U_a \mid a \in {{\mathbb {K}}}(P)\}$ is closed under finite unions and intersections and every quasi-compact open subset is spectral, hence Y is a locally spectral space (i.e., the open spectral subspaces are a basis).

On the other hand, the equality $\lambda ^* = (\lambda ^* \circ \mu _*) \circ \mu$ shows that $\lambda ^*$ is spectral if and only if $\mu$ is spectral, which is always true as shown above, also see Proposition 6.18(b).

7 Closure systems in coherent algebraic frames

Finally we consider closure systems in frames. We start with a couple of results that are special cases of (or follow easily from) the previous sections. The main result of the section is Theorem 7.1 which says that, for a coherent algebraic frame P, the set ${\mathscr {A}}(P) \cap {\mathscr {N}}(P)$ of closure systems, cf. 3.1 and Remark 6.15, is a closure system in ${\mathfrak {P}}(P)$.

Let P be a frame and $Q \subseteq P$ a closure system. In general the closure operator is not a $\wedge$-homomorphism. For an example let P be any frame that is not totally ordered. Pick $a \in P {\setminus } {{\mathbb {P}}}(P)$, cf. Example 6.2(c), and define $Q = \langle \{a\} \rangle _P = \{a,\top _P\}$. Then Q is an algebraic closure system in P, is prime-generated since $Q = {{\mathbb {P}}}(Q)$, and is trivially coherent. But $\eta _{Q,P}$ is not a $\wedge$-homomorphism, which follows from Remark 6.6 or can be checked directly.

Now assume that the closure operator is a $\wedge$-homomorphism. Then $\eta$ is a nucleus, hence Q is a frame, $\vartheta :P \rightarrow Q$ is a frame homomorphism, and ${{\mathbb {P}}}(P) \cap Q = {{\mathbb {P}}}(Q)$, Remark 6.6. However Q need not be prime generated (Example 6.16) and ${{\mathbb {P}}}(Q)^{\flat }$ and ${{\mathbb {P}}}(P)^{\flat }$ may both be empty.

As in Remark 6.15 let ${\mathscr {N}}(P)$ be the set of prime generated closure systems Q such that $\eta _{Q,P}$ is a nucleus. The poset isomorphism

$$\begin{aligned} {\mathscr {N}}(P) \rightarrow {\mathcal {A}}({{\mathbb {P}}}(P)^{\flat },\beta ), Q \mapsto {{\mathbb {P}}}(Q)^{\flat } \end{aligned}$$

shows that $\langle {{\mathbb {P}}}(P)^{\flat } \rangle _P \simeq {\mathcal {O}}({{\mathbb {P}}}(P)^{\flat })$ is the largest element of ${\mathscr {N}}(P)$. If P is a continuous frame then $P = \langle {{\mathbb {P}}}(P)^{\flat } \rangle _P$, [8, Theorem, I$-$3.15], and P is algebraic if and only if $\overset{\circ }{{\mathcal {K}}}({{\mathbb {P}}}(P)^{\flat })$ is a basis of open sets for ${{\mathbb {P}}}(P)^{\flat }$.

If P is a coherent algebraic frame then ${{\mathbb {P}}}(P)^{\flat }$ generates P and is a patch closed subset, cf. Theorem 6.20(b). Consider ${\mathscr {A}}(P) \cap {\mathscr {N}}(P)$, the set of algebraic closure systems in ${\mathscr {N}}(P)$. We claim that each $Q \in {\mathscr {A}}(P) \cap {\mathscr {N}}(P)$ is a coherent algebraic frame. First note that $Q \subseteq P$ is patch closed by Theorem 4.5. Being prime generated, Q is a frame, Theorem 6.13, and the closure operator $\eta _{Q,P}$ is a nucleus since it is a $\wedge$-homomorphism. Corollary 6.9 implies that ${{\mathbb {P}}}(P)^{\flat } \cap Q = {{\mathbb {P}}}(Q)^{\flat }$, which is patch closed in P, and Q is coherent by Theorem 6.20(d).

But ${\mathscr {N}}(P)$ may contain coherent algebraic frames that are not algebraic closure systems in P. Examples can be constructed as follows: Let Y be a spectral space and X a spectral space that is a subspace of Y, but not a spectral subspace, see Example 4.7 or [6, Example 2.1.2], and let $e:X \rightarrow Y$ be the inclusion map. Then ${\mathcal {O}}(e):{\mathcal {O}}(Y) \rightarrow {\mathcal {O}}(X)$ is a surjective homomorphism of prime generated coherent algebraic frames and we can identify $X = {{\mathbb {P}}}({\mathcal {O}}())^{\flat }$, resp. $Y = {{\mathbb {P}}}({\mathcal {O}}(Y))^{\flat }$, via the homeomorphisms $X \rightarrow {{\mathbb {P}}}({\mathcal {O}}(X))^{\flat }, x \mapsto X {\setminus } x^{\uparrow }$, resp. $Y \rightarrow {{\mathbb {P}}}({\mathcal {O}}(Y))^{\flat }, y \mapsto Y {\setminus } y^{\uparrow }$. The right adjoint ${\mathcal {O}}(e)_*$ exists and ${\mathcal {O}}(e)_* \circ {\mathcal {O}}(e):{\mathcal {O}}(Y) \rightarrow {\mathcal {O}}(Y)$ is a closure operator by Example 3.2, even a nucleus since ${\mathcal {O}}(e)$ is a frame homomorphism and ${\mathcal {O}}(e)_*$ is a $\bigwedge$-morphism, 2.6. Let Q be the image of the nucleus. Since ${\mathcal {O}}(e)_*:{\mathcal {O}}(X) \rightarrow Q$ is an isomorphism it follows that Q is prime generated and belongs to ${\mathscr {N}}({\mathcal {O}}(Y))$. But Proposition 6.18(a) shows that ${\mathcal {O}}(e)_*$ is not a spectral map (as e is not a spectral map), hence Q is not an algebraic closure system in ${\mathcal {O}}(Y)$, Theorem 4.5.

The subset ${\mathcal {A}}({{\mathbb {P}}}(P)_{{{\,\textrm{con}\,}}}^{\flat }) \subseteq {\mathcal {A}}({{\mathbb {P}}}(P)^{\flat },\beta )$ is a closure system with closure map $X \mapsto \overline{X}^{{{\,\textrm{con}\,}}}$. The isomorphism ${\mathcal {A}}({{\mathbb {P}}}(P)^{\flat },\beta ) \rightarrow {\mathscr {N}}(P), X \mapsto \langle X \rangle _P$ maps ${\mathcal {A}}({{\mathbb {P}}}(P)_{{{\,\textrm{con}\,}}}^{\flat })$ onto the closure system ${\mathscr {A}}(P) \cap {\mathscr {N}}(P) \subseteq {\mathscr {N}}(P)$, and the corresponding closure map is given by $Q \mapsto \overline{Q}^{{{\,\textrm{con}\,}}}$, cf. Theorem 5.2.

The set ${\mathscr {C}}(P)$ of closure systems and the set ${\mathscr {A}}(P)$ of algebraic closure systems in P are closure systems in ${\mathfrak {P}}(P)$, 3.1. We do not know whether ${\mathscr {N}}(P)$ is a closure system in ${\mathfrak {P}}(P)$. However:

Theorem 7.1

If P is a coherent algebraic frame then ${\mathscr {A}}(P) \cap {\mathscr {N}}(P) \subseteq {\mathfrak {P}}(P)$ is a closure system.

Proof

We abbreviate $Z = {{\mathbb {P}}}(P)^{\flat }$. The map ${\mathcal {A}}(Z_{{{\,\textrm{con}\,}}}) \rightarrow {\mathfrak {P}}(P),Y \mapsto \langle Y \rangle _P$ is a poset embedding onto ${\mathscr {A}}(P) \cap {\mathscr {N}}(P)$, Theorem 5.2, Corollary 6.9. Consider a subset ${\mathcal {Y}}\subseteq {\mathcal {A}}(Z_{{{\,\textrm{con}\,}}})$ and define $X = \bigcap {\mathcal {Y}}$. Then $\langle X \rangle _P$ is the infimum of $\{\langle Y \rangle _P \mid Y \in {\mathcal {Y}}\}$ in ${\mathscr {A}}(P) \cap {\mathscr {N}}(P)$. We have to show that $\langle X \rangle _P$ is the infimum of $\{ \langle Y \rangle _P \mid Y \in {\mathcal {Y}}\}$ in ${\mathfrak {P}}(P)$, i.e., $\langle X \rangle _P =\bigcap _{Y \in {\mathcal {Y}}} \langle Y \rangle _P$. The inclusion $\langle X \rangle _P \subseteq \bigcap _{Y \in {\mathcal {Y}}} \langle Y \rangle _P$ holds trivially.

Let $T \subseteq Z$ be patch closed and $e:T \rightarrow Z$ the inclusion. Using Proposition 6.17 we identify the inclusion $\iota : \langle T \rangle _P \rightarrow P$ with ${\mathcal {O}}(e)_*:{\mathcal {O}}(T) \rightarrow {\mathcal {O}}(Z)$, which sends $U \in {\mathcal {O}}(T)$ to $Z \setminus \overline{T \setminus U}^{Z}$, the largest $O \in {\mathcal {O}}(Z)$ with $O \cap T = U$. Note that $T {\setminus } U$ is patch closed in Z, hence $\overline{T {\setminus } U}^{Z}$ is the set of specializations of $T {\setminus } U$. Moreover, $T {\setminus } U$ is contained in the set of specializations of $(T \setminus U)^{\min }$, its set of minimal points, [6, Proposition 4.1.2 and Theorem 4.1.3]. Thus $\overline{T {\setminus } U}^{Z} = \overline{(T {\setminus } U)^{\min }}^{Z}$ and $(\overline{T {\setminus } U}^{Z})^{\min } = (T {\setminus } U)^{\min }$.

Now pick $O \in \bigcap _{Y \in {\mathcal {Y}}} \langle Y \rangle _P$, i.e., $O = Z{\setminus } \overline{Y {\setminus } O}^Z$ for each $Y \in {\mathcal {Y}}$. The sets $\overline{Y \setminus O}^Z$ coincide for all $Y \in {\mathcal {Y}}$, hence the sets $(Y \setminus O)^{\min }$ coincide as well and are equal to $(X {\setminus } O)^{\min }$. It follows that $O = Z {\setminus } \overline{X {\setminus } O}^{Z} \in \langle X \rangle _P$. $\square$

Proposition 7.2

Let P be a coherent algebraic frame, $Z = {{\mathbb {P}}}(P)^{\flat }$. Let $Y \subseteq Z$ be patch closed, $M = \langle Y \rangle _P$. For $a \in P$ the sets $a^{\uparrow Z}$ and $a^{\uparrow Y}$ are patch closed in Z and the following conditions are equivalent:

• (a) $a \in M$,

• (b) $(a^{\uparrow Z})^{\min } \subseteq Y$.

Proof

We identify the closure operator $\eta = \eta _{M,P}:P \rightarrow P$ with the nucleus $\nu :{\mathcal {O}}(Z) \rightarrow {\mathcal {O}}(Z), U \mapsto Z {\setminus } \overline{Y {\setminus } U}^Z$, Proposition 6.17. The sets $a^{\uparrow P},Y,Z \subseteq P$ are patch closed, hence $a^{\uparrow Y}$ and $a^{\uparrow Z}$ are patch closed as well. It follows that they are the upsets in Y and Z generated by their respective sets of minimal elements, [6, Proposition 4.1.2 and Theorem 4.1.3]. Note that $a = \eta (a)$ if and only if $Z {\setminus } a^{\uparrow Z} = \nu (Z {\setminus } a^{\uparrow Z}) = Z {\setminus } \overline{a^{\uparrow Y}}^Z$.

(a)$\Rightarrow$(b). As $a \in M$, hence $a = \eta (a)$, it follows from [6, Theorem 1.5.4] that $a^{\uparrow Z} = \overline{a^{\uparrow Y}}^Z = (a^{\uparrow Y})^{\uparrow Z}$, But then $(a^{\uparrow Z})^{\min } = (a^{\uparrow Y})^{\min } \subseteq Y$.

(b)$\Rightarrow$(a). The inclusion $(a^{\uparrow Z})^{\min } \subseteq Y$ implies $(a^{\uparrow Z})^{\min } = (a^{\uparrow Y})^{\min }$, hence $\overline{a^{\uparrow Y}}^Z = a^{\uparrow Z}$, i.e., $a = \eta (a) \in M$. $\square$

Corollary 7.3

Let P be a coherent algebraic frame, $Z = {{\mathbb {P}}}(P)^{\flat }$, and $a \in P$. Then $\overline{(a^{\uparrow Z})^{\min }}^{{{\,\textrm{con}\,}}}$ is the smallest patch closed set $X \subseteq Z$ with $a \in \langle X \rangle _P$.