Topology of closure systems in algebraic lattices

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 ⟨X⟩P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle X \rangle _P$$\end{document}. The closure system ⟨Y⟩P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle Y \rangle _P$$\end{document} generated by the patch closure Y of X is the patch closure of ⟨X⟩P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle X \rangle _P$$\end{document}. If X is contained in the set of nontrivial prime elements of P then ⟨X⟩P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle X \rangle _P$$\end{document} 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.

Every topological space T carries a binary relation , which is called specialization and is defined by: x y if y ∈ {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, τ (P ), and the Scott topology, σ(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 K(P ). In particular, if T is a topological space and O(T ) is the frame of open sets then K(O(T )) = • 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 K(P ) is a decisive part of its structure. If P is an algebraic lattice then τ (P ) is a spectral topology and σ(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 ]. 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 → P belonging to a closure system Q ⊆ P is denoted by η Q,P . Conversely, the image of a closure operator η : P → P is a closure system. The set of closure systems is denoted by C (P ). The algebraic closure systems are a particularly important subset of C (P ), which is denoted by A (P ). If P is a complete lattice then C (P ) and A (P ) are both closure systems in the power set P(P ). Thus, every subset X ⊆ P generates a closure system, which is denoted by X 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 τ (Q) ⊆ τ (P )| Q , and equality holds if Q is a closure system. The relation σ(Q) ⊆ σ(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 ⊆ P is a closure system in an algebraic lattice then it is always true that K(Q) ⊆ η Q,P (K(P )), Proposition 4.2. It is algebraic if and only if it is patch closed in P , if and only if K(Q) = η Q,P (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 Y P is the patch closure of X P in P , and the compact elements of Y P are contained in X P (but need not be compact in X P ), Theorem 5.2. Section 6 is devoted to complete lattices and closure systems generated by sets of prime elements. An element p ∈ P is prime if a ∧ b ≤ p implies a ≤ p or b ≤ p, [8,. The set of prime elements is denoted by P(P ). The top element is always prime, the trivial prime element. The set of nontrivial prime elements is denoted by P(P ) and may be empty. The inclusion P(P ) ∩ Q ⊆ P(Q) holds for any closure system Q ⊆ P and may be proper. But if Q = P(Q) P then Q is isomorphic to the frame O(P(Q) ), Theorem 6.13, and the equality P(P ) ∩ Q = P(Q) holds if and only if the closure operator η Q,P is a ∧-homomorphism, Corollary 6.9. The set of closure systems satisfying these conditions is denoted by N (P ). The poset N (P ) (with inclusion) is isomorphic to the set of closed sets for the b-topology associated with τ (P ), Remark 6.15. Coherent algebraic lattices are particularly important in applications. It is shown in Theorem 6.20 that P(P ) ⊆ 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 P(P ) then X 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 N (P ) then the closure operator η Q,P is a nucleus of P . If P is a coherent algebraic frame then A (P ) ∩ N (P ), the set of algebraic closure systems in N (P ), is isomorphic (as a poset) to the set of patch closed subsets of P(P ) and is a closure system in 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].

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.

The b-topology associated with a topological space
The locally closed sets in a topological space T = (T, τ ) are a basis of open sets for the b-topology, denoted by β = β(τ ), cf. [19, 2.2] or [6, 4.5.20]. We call β(T ) = (T, β(τ )) the b-space of T . The b-topology has also been studied under the name Goldman topology or G-topology, see [3, section 4] A continuous map f : S → T is also continuous as a map of the associated b-spaces. In particular, if S is a subspace of T then β(S) ⊆ β(T ) is a subspace.

Irreducible sets, sober spaces, and sobrification
A T 0 -space T is sober if every nonempty closed and irreducible subset C is equal to {c} for a unique c ∈ 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 Sob(T ), and Sob T : T → 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 ⊆ 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 → S β is isomorphic to the sobrification.
By [2], a continuous map f : S → T of T 0 -spaces is an epimorphism in the category of T 0 -spaces if and only if f (S) ⊆ T is very dense. Thus, Sob(S) is (up to isomorphism) the largest epimorphic extension of S and is contained in any sober space containing S.

Notation and terminology for posets
Let P = (P, ≤) be a poset. Top and bottom elements in P , if they exist, are denoted by = P and ⊥ = ⊥ P . If both exist then the poset is bounded. For a subset X ⊆ P we define X = X\{ P } and X = X ∪ { P }. (Nothing happens if does not exist.) Every a ∈ P generates a principal upset, a ↑ , and a principal downset, a ↓ . For S ⊆ P we set S ↑ = a∈S a ↑ and S ↓ = a∈S a ↓ . The sets of upper, resp. lower, bounds of S are S ⇑ = a∈S a ↑ , resp. S ⇓ = a∈S a ↓ . For Q ⊆ P we define a ↑Q = a ↑ ∩ Q, S ↑Q = S ↑ ∩ Q, S ⇑Q = S ⇑ ∩ Q, and so on. The set of all upsets, resp. principal upsets, are denoted by ⇑(P ), resp. ↑(P ).
We call P a dcpo, resp. an fcpo, if P D exists for every up-directed D ⊆ P , resp. P D exists for every down-directed D ⊆ 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 ∈ P then a = a P is the set of elements that are way below a and a = a P = {x ∈ P | a x}. The set of compact elements is denoted by 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 K(P ) is always a ∨-subsemilattice of P . If the meet of two compact elements is compact (i.e., K(P ) ⊆ P is a sublattice) then P is arithmetic, cf. [8, I−4.7]. Moreover, P is coherent if 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 is up-directed for each a ∈ P and a = a ) and algebraic lattices (see the introduction).

Poset morphisms
A map ϕ : P → Q between posets is a poset morphism if it is monotonic. A poset map may have additional properties, which will always be announced explicitly.
• ϕ is a dcpo-homomorphism if Q ϕ(D) exists and is equal to ϕ( P D) for all up-directed D ⊆ P such that P D exists. The notion of fcpohomomorphisms is defined similarly.
for all a, b ∈ P such that a ∧ P b exists. Homomorphisms for , ∨ and are defined accordingly. Clearly, a -map, resp. a -map, is also a ∧-map and an fcpo-map, resp. a ∨-map and a dcpo-map.
resp. a ∧-embedding if it is a poset-embedding, a ∧-morphism and x ∧ P y exists if ϕ(x) ∧ Q ϕ(y) exists. Embeddings for , ∨, , as well as dcpoembeddings and fcpo-embeddings are defined similarly.

Adjoint pairs of poset morphisms
See [8, p. 22 ff] and [7, p. 155 ff] for the following notions and facts. Consider poset morphisms ϕ : P → Q and ψ : Q → P . We say that (ϕ, ψ) is an adjoint pair with ψ right adjoint to ϕ and ϕ left adjoint to ψ if the equivalence ϕ(a) ≤ b ⇔ a ≤ ψ(b) holds for all a ∈ P and b ∈ Q, if and only if ϕ −1 (b ↓ ) = ψ(b) ↓ for all b ∈ Q, if and only if ψ −1 (a ↑ ) = ϕ(a) ↑ or all a ∈ P . Each member of the pair (ϕ, ψ) determines the other one uniquely. The map ϕ has a right adjoint ϕ * if and only if for all b ∈ Q the set ϕ −1 (b ↓ ) has a largest element (which then is equal to ϕ * (b)). Similarly, ψ has a left adjoint ψ * if and only if for all a ∈ P the set ψ −1 (a ↑ ) has a smallest element (which then is equal to ψ * (a)), if and only if the map ↑(ψ) is defined, cf. 2.5. Let (ϕ, ψ) be an adjoint pair. [7, p. 159 The same holds for ϕ.
If P is complete then ϕ : P → Q has a right adjoint, resp. left adjoint, if and only if it is a -morphism, resp. a -morphism, [8, Corollary 0−3.5].
(a) ψ is a -morphism and ϕ is a -morphism.
Proof. It suffices to prove the assertions about ψ; the proofs for ϕ are similar. (a). Pick S ⊆ Q such that Q S exists. Then ψ( Q S) ≤ ψ(s) for all s ∈ S. Pick a ∈ P with a ≤ ψ(s) for all s ∈ S. Then ϕ(a) ≤ s for all s ∈ S, hence ϕ(a) ≤ Q S. It follows that a ≤ ψ( Q S), i.e., ψ( Q S) = P ψ(S).
(b). Note that ψ is a poset-embedding by 2.6(c) and is a -morphism by (a). Assume S ⊆ Q and a = P ψ(S) exists. We show that ϕ(a) = Q S.
It follows that Q S exists and is equal to ϕ(a). Clearly, ψ is a ∧-embedding and an fcpo-embedding.
(c). Again, ψ is a poset-embedding by 2.6(c) and is a -map by hypothesis. Now suppose S ⊆ Q and a = P ψ(S) exists. By 2.6(a) we have ψ(s) ≤ a ≤ ψ • ϕ(a) for all s ∈ S. Since ψ is a poset-embedding it follows that If ψ is a ∨-map, resp. a dcpo-map, then use subsets S ⊆ Q with |S| = 2, resp. up-directed sets S ⊆ Q.

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 {x} = x ↑ . Continuous maps between T 0 -spaces are monotonic for specialization.
Consider a T 0 -space T and a nonempty irreducible subset C. If C has the generic point c, cf. 2.3, then C = c ↑ and T C exists and is equal to c, [  (g) Let T be sober and S ⊆ T a subspace. Then S is sober if and only if T C ∈ S for all nonempty irreducible C ⊆ S.

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 ≤, resp. the inverse partial order ≤ inv , cf. [6, p. 589]. Every open set of a lower topology is a downset for ≤, every open set of an upper topology is an upset. If (T, τ ) is a T 0 -space then τ is a lower topology for . The coarse lower topology τ (P ) and the coarse upper topology τ u (P ) (for the definition see [6, p. 589] ) are intrinsic topologies. The set ↑(P ) (see 2.5 for the notation) is called the canonical subbasis of closed sets for the coarse lower topology. If τ is any lower topology then ↑(P ) is the set of nonempty closed and irreducible sets with generic point. We need two more intrinsic topologies: The coarse upper topology (resp. the lower Scott topology) is the coarse lower topology (resp. the Scott topology) for the inverse partial order ≤ inv . Thus properties of τ u (P ) and σ (P ) follow from properties of τ (P ) and σ(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 τ be an upper topology on P . Then item (c) says that τ ⊆ σ(P ) if and only if D τ 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 τ (P ) and σ(P ) in the lattice of topologies on P is the Lawson topology, [8, Definition III−1.5], which is denoted by λ(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 \F ↑ with U ∈ O(σ(P )) and Let ϕ : P → 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 ϕ is continuous then we say that ϕ is coarse lower continuous, resp. Scott continuous, and so on.

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 cit.. Thus the sets a ↑ with a ∈ K(P ) are a subbasis of closed sets for τ (P ). Usually this is a proper subset of the canonical subbasis ↑(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 λ(P ). For any subset X ⊆ P the patch closure of X is denoted by X con . The basic constructible , are a basis of open sets for the patch topology. Since F ↑ \G ↑ = a∈F a ↑ \G ↑ the sets a ↑ \ G ↑ (with a ∈ K(P ) and G ⊆ K(P ) finite) are a basis as well, and the sets a ↑ \b ↑ (where a, b ∈ K(P )) are a subbasis. Note that K(P ) is dense in P for the patch topology since a basic open set a ↑ \ G ↑ is nonempty if and only if a ∈ a ↑ \G ↑ .
Theorem 2.11. Let ϕ : P → Q be a poset map. Then: (a) Let ϕ : (P, τ P ) → (Q, τ Q ) be continuous, where τ P and τ Q are lower topologies on P and Q. If τ P ⊆ σ (P ) then ϕ is an fcpo map.  To prove lower Scott continuity we show that ϕ is an fcpo-map, cf. item (b). So, assume D ⊆ P is down-directed and P D exists. It is claimed that

b). Assume that ϕ is an fcpo-map and let
Remark 2.12. Let X and Y be spectral spaces and f : X → 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. Proof. (a). Use Corollary 2.13.
Assume ϕ * is spectral and a ∈ K(P ). Then ϕ(a) ↑ ⊆ Q is closed and constructible, hence ϕ(a) ∈ K(Q). Now let ϕ be coherent. If a ∈ K(P ) then ϕ(a) ∈ K(Q), hence a ↑ and ϕ(a) ↑ are closed and constructible. Thus ϕ −1 * (C) is closed and constructible if C ⊆ P is closed and constructible.

Products of posets
Let P and Q be bounded posets. By [8, Lemma III−1.3] the coarse lower topology on P × Q is the product of the coarse lower topologies on P and Q. The projection π P : P × Q → P (similarly the projection π Q ) has both right and left adjoints and (π P ) * (a) = (a, Q ) and π * P (a) = (a, ⊥ Q ). For Thus K(P × Q) = K(P ) × K(Q) and π P , π * P , π Q , π * Q are all coherent. It follows that P × Q is an algebraic lattice if so are P and Q.
If Q = P then the diagonal map Δ : P → P × P, x → (x, x) is both a -homomorphism and a -homomorphism and is coherent. Moreover, Δ * is coherent. The Scott topology of products is more complicated. Usually σ(P × Q) is finer than the product of σ(P ) and σ(Q), [8, p. 197 Define D 1 and D 2 to be the sets of first and second components of elements of D and let C = {x 1 ∧ y 2 | x, y ∈ D} ⊆ P , which is up-directed as well. Then

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 η : P → P is a closure operator if it is idempotent and inflationary (i.e., a ≤ η(a)), resp. a kernel operator if it is idempotent and deflationary (i.e., η(a) ≤ a), [8, Definition 0−3.8]. It suffices to discuss closure operators since a poset map η : P → P is a closure operator if and only if η viewed as a map P inv → P inv (the inversely ordered poset) is a kernel operator.
Let Q ⊆ P and ι = ι P,Q : Q → P the inclusion map. Then Q is a closure system if ι has a left adjoint, equivalently if, for all a ∈ P , the set a ↑Q has a smallest element, cf. 2.6. The left adjoint ι * : P → Q is denoted by ϑ = ϑ Q,P and is called the closure map. The composition η Q,P = ι P,Q • ϑ Q,P is a closure operator. Conversely, if η : P → P is a closure operator then Q = η(P ) is a closure system with η = η Q,P , and ϑ Q,P is the corestriction of η. The correspondence between closure systems and closure operators of P is bijective. Clearly, η(a) ↑Q = a ↑Q and η(a) = P a ↑Q = Q a ↑Q where a ∈ P . The inclusion ι is a -embedding, Proposition 2.7(b), and is coarse lower continuous, Theorem 2.11(c). The closure map is a -map, hence a dcpo map, Proposition 2.7(a), hence is Scott continuous, Theorem 2.11(e). A closure system Q ⊆ P , and the corresponding closure operator, are algebraic, or inductive, if ι 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 C (P ), resp. A (P ). Consider a subset X ⊆ P such that P a ↑X exists for all a ∈ P . Then η : P → P, a → P a ↑X is a closure operator, X P := η(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 ↑X = ∅, hence P b ↑X does not exist since there is no top element in P . For the second example, let N * be the inversely ordered set of natural numbers; its elements are denoted by 0 We say that X is dense in P if X P exists and is equal to P . If P is a complete lattice and Q ⊆ P then one shows easily that Q is a closure system if and only if P S ∈ Q for all S ⊆ Q.
In particular, Q is a complete lattice as well. It follows that C (P ) and A (P ) are closure systems (not necessarily algebraic) in the complete lattice P(P ). If X ⊆ P then η C (P ),P(P ) (X) = X P , and its elements are the infima of subsets S ⊆ X. Example 3.2. Let ψ * : P → Q be the left adjoint of a poset map ψ : Q → P . It follows from 2.6 that ψ • ψ * : P → P is a closure operator with corresponding closure system ψ(Q), and ψ * •ψ is a kernel operator. Let C ⊆ P and D ⊆ Q be closure systems with inclusion maps ι C , ι D and closure maps ϑ C , ϑ D . Assume that ψ(D) ⊆ C and let ϕ : D → C be the restriction of ψ. Then ϕ has a left adjoint and ϕ * • ϑ C = ϑ D • ψ * . Moreover, ψ(D) is a closure system in P , and Left adjoint maps are -homomorphisms, Proposition 2.7(a). Now assume that ψ is a dcpo map and D is an algebraic closure system, i.e., ι D is a dcpo map.
is also algebraic in P . To show this one may assume C = ψ(D), i.e., ϕ is surjective, and then ι C = ι C • ϕ • ϕ * = ψ • ι D • ϕ * is a dcpo map, 2.6(c). The following lemmas record elementary facts about closure systems, generating sets and the maps γ X,P (without proof).

Lemma 3.4.
Let P be a poset, S, X ⊆ P and assume that Q := X P exists. Then: (a) P a ↑X = P a ↑Q for all a ∈ P .   Proposition 3.7. Consider a poset P with a closure system Q ⊆ P . Then: Proof. (a). If a ∈ P \Q then a < η(a), hence a belongs to a ↑P \ η(a) ↑P , which is β-open and is disjoint from Q.
(c). The closure system is algebraic if and only if ι is a dcpo map, if and only if ι is Scott continuous, if and only if σ(P )| Q = σ(Q) (by (b)). Scott continuity of ι is equivalent to Lawson continuity (note that ι is coarse lower continuous and use Theorem 2.11(g)).
(d). Let D ⊆ Q be up-directed such that Q D exists. We have to show that P D exists and is equal to Q D. If this is false then there is some a ∈ P with D ⊆ a ↓P , but Q D ≤ a. The set a ↓P is closed for σ(P ), hence for λ(P ). Thus a ↓Q is closed in Q for λ(P )| Q and is quasi-compact (as Q is quasi-compact). The sets P \d ↑P with d ∈ D are open for λ(P ) and cover a ↓Q .
(f) follows from (d) and (e).  Proof. (a). The sets F ↑ (where F ⊆ P is finite) are a basis of closed sets for τ (P ). If C is irreducible and C ⊆ F ↑ = x∈F x ↑ then C ⊆ x ↑ for some x ∈ F . Conversely, assume that C ⊆ x ↑ for some x ∈ F whenever F ⊆ P is finite and x ∈ H, and C is irreducible.
(b). The equality C τ (P ) = y∈C ⇓ y ↑ = C ↑(P ) follows from (a). If P C exists then the set is equal to ( P C) ↑ , Lemma 3.4(d).

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
Proof. Assume Q is a closure system. Then it is closed in P under all infima. Proposition 3.7(a) shows that Q is β(τ (P ))-closed in P . As τ (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 τ (P )| Q and is closed in P under finite infima. It is claimed that P S ∈ Q for any subset S ⊆ Q. If S = ∅ then P = P S ∈ Q. For S = ∅ let T ⊆ 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 ⇓ = T ⇓ . As Q is sober it follows from 2.8(g) that

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 ⊆ 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 ∈ Q there is an up-directed set D ⊆ b Q with Q D = b. By hypothesis, the set b P is up-directed and b = P b P . Thus, ϑ(b P ) is up-directed and, since ϑ is a dcpo-

Proposition 4.2.
Let P be an algebraic lattice and Q ⊆ P a closure system. Then K(Q) ⊆ ϑ(K(P )).
Proof. Pick a ∈ K(Q) and write a = P a ↓K(P ) . Then since ϑ is a dcpo map. As the supremum is up-directed and a ∈ K(Q) there is some c ∈ a ↓K(P ) with a ≤ ϑ(c) ≤ ϑ(a) = a. Proof. The inclusion Q ⊆ A is obvious. First consider any a ∈ P \A. There is some c ∈ a ↓K(P ) with η(c) ≤ a. The set c ↑ \η(c) ↑ is patch open (by 2.10), contains a and is disjoint from Q (by 3.1). It follows that Q con ⊆ A. For the reverse inclusion pick a ∈ A and let C be a patch open set containing a. We may assume that C = c ↑ \F ↑ with c ∈ K(P ) and F ⊆ P finite, cf. 2.10. It suffices to show that η(c) ∈ C. Note that c ≤ η(c) ≤ a (as a is absorbing) and η(c) / ∈ F ↑ (since otherwise x ≤ η(c) ≤ a for some x ∈ F , hence a / ∈ C, a contradiction). (d)⇒(c). For z ∈ P \ Q we must find a patch open set C ⊆ P with z ∈ C and C ∩ Q = ∅. Since z / ∈ Q it follows that z < η(z) = P η(z) ↓K(P ) and u ≤ z for some u ∈ η(z) ↓K(P ) . As z = P z ↓K(P ) , ϑ preserves suprema and u ≤ η(z) = ϑ(z) it follows that ϑ(u) ≤ ϑ(z) = Q ϑ(z ↓K(P ) ). By hypothesis ϑ(u) ∈ K(Q), hence there is some v ∈ z ↓K(P ) with ϑ(u) ≤ ϑ(v). The set v ↑P \η(v) ↑P is patch-open in P , contains z (since v ∈ z ↓K(P ) , u ≤ z, and u ≤ η(u) ≤ η(v)) and is disjoint from Q (as v ↑Q = η(v) ↑Q ). (e)⇒(a). Since σ(P ) is an upper topology its specialization poset is P inv . Pick D ⊆ Q up-directed, hence down-directed in P inv . Then P D is the generic point of D σ(P ) , [6, Proposition 4.2.1(ii)]. Soberness of (Q, σ(P )| Q ) implies P D ∈ Q (cf. 2.8(g)), i.e., P D = Q D.
Example 4.6. Let P be an algebraic lattice, Q ⊆ P an algebraic closure system. Theorem 4.5 shows that the inclusion ι : Q → P is a spectral map, hence η is spectral if and only if ϑ is spectral. (Clearly, ϑ spectral implies η = ι • ϑ spectral. If η is spectral and Q ⊆ P is a spectral subspace then the corestriction ϑ : P → 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 N ∪ {ω}, and define Q = {ω, 0} = {ω} P . Note that P is an algebraic lattice and Q is an algebraic closure system. The closure operator η is given by ω → ω and a → 0 otherwise. It is a homomorphism of bounded lattices and preserves all suprema, but is not an fcpo-map since N = ω and η( N) = ω < 0 = η(N). It follows from Corollary 2.14(a) that the closure operator is not a spectral map. In fact, {0} ⊆ P is closed for the coarse lower topology, but η −1 ({0}) = N is not closed. Example 4.7. Let P be an algebraic lattice and Q ⊆ P a closure system. Then K(Q) ⊆ ϑ(K(P )), Proposition 4.2. We exhibit a closure system that shows how equality can fail.
Let P be the totally ordered set 2 · ω + 1, which is an algebraic lattice since every non-limit ordinal is compact. The subset Q = N∪{2·ω} is a closure system in P , but is not algebraic since P N = ω < Q N = 2 · ω. Note that Q is isomorphic to ω + 1, hence is an algebraic lattice. The only element that is not compact is 2 · ω. The closure map is given by a → a if a ∈ N and a → 2 · ω otherwise. The element ω + 1 ∈ P is not a limit ordinal, hence is compact. But ϑ(ω + 1) = 2 · ω is not compact.

Closure systems and generating subsets
An algebraic lattice P is complete, hence every subset X ⊆ 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 P(P ) → P(P ), X → X con and P(P ) → P(P ), X → X 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 = X P , and if P is an algebraic lattice then we also define Y = X con and M = Y P .  Proof. (a). To show that P \ M is patch open in P , pick z ∈ P \M . Then z < η M,P (z) = P η M,P (z) ↓K(P ) , hence there is some a ∈ η M,P (z) ↓K(P ) with a ≤ z. Note that z ↑Y = η M,P (z) ↑Y ⊆ a ↑Y = η M,P (a) ↑Y , cf. 3.1. The set a ↑P is closed and constructible in P and Y ⊆ P is patch closed, hence a ↑Y is closed and constructible in Y . Since z = P z ↓K(P ) and the supremum is up-directed it follows that z ↑Y = b∈z ↓K(P ) b ↑Y . The sets b ↑Y are closed and constructible in Y , the intersection is down-directed for inclusion and is disjoint from the . First we claim that L con ⊆ P is closed under finite meets. As P ∈ L it suffices to prove a∧b ∈ L con if a, b ∈ L con . The map ∧ : P ×P → P is spectral, Corollary 2.16, hence sends patch closed sets to patch closed sets. As L ⊆ P is closed under infima it follows that ∧ restricts to a map L×L → L. The equality L × L con = L con × L con , cf. [6, Theorem 2.2.1], implies ∧(L con × L con ) = L con .
Next we claim that L con is a closure system in P . For, consider any nonempty subset S ⊆ L con and let T ⊆ P be the set of infima F with F ⊆ S finite. Then  (a) Q con is an algebraic closure system and an algebraic lattice.
(c) If K(Q) = K(Q con ) then Q = Q con .
(d) Q is dense in Q con for β(σ(P )). Thus, Q con with the Scott topology is the sobrification of (Q, σ(P )| Q ). (e) The elements of Q 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 ∈ K(Q) and write a = Q con a ↓K(Q con ) . Then K(Q con ) ⊆ Q, cf. Theorem 5.2(d), implies a = Q a ↓K(Q con ) . Since a ↓K(Q con ) is up-directed there is some b ∈ a ↓K(Q con ) with a ≤ b, hence a = b ∈ K(Q con ).
(c). Pick b ∈ Q con and write b = . We may assume that Q con = P . The sets (e). If D ⊆ Q is up-directed then P D ∈ Q con by (a). Conversely, every element a ∈ Q con is equal to Q con a ↓K(Q con ) = P a ↓K(Q con ) , where a ↓K(Q con ) is up-directed and is contained in Q, Theorem 5.2(d).

Remark 5.4.
Let P be an algebraic lattice. As noted in 3.1 the set C (P ) of closure systems and the set A (P ) of algebraic closure systems in P are closure systems in P(P ). The subset A(P con ) ⊆ P(P ) of patch closed subsets of P is a closure system as well. Theorem 4.5 shows that A (P ) = A(P con ) ∩ C (P ), and it follows that η A (P ),P(P ) • η C (P ),P(P ) = η A (P ),P(P ) = η A (P ),P(P ) • η A(Pcon),P(P ) , i.e., X P con = X con P for X ∈ P(P ).

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 ∧-semilattice, which will always be the case in our considerations, then p ∈ P is prime if a ∧ b ≤ p implies a ≤ p or b ≤ p for all a, b ∈ P , cf. [8, Proposition I−3.12]. The set of prime elements is denoted by P(P ). A top element, if it exists, is always prime and is called the trivial prime element and P(P ) (which may be empty) is the set of nontrivial prime elements. We show that P(P ) and P(P ) are both closed for the b-topology β(τ (P )), Proposition 6.3. A closure system Q ⊆ P is called prime generated if Q = X P , where X ⊆ P(Q). Let P be a complete lattice and Q ⊆ P a closure system. It is always true that P(P ) ∩ Q ⊆ P(Q), and the inclusion may be proper. If Q is prime 17 Page 20 of 33 N. Schwartz Algebra Univers. generated then P(P ) ∩ Q = P(Q) if and only if the closure operator is a ∧homomorphism, Corollary 6.9. Moreover, if Q is prime generated by X then it is isomorphic to the frame O(X ), 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 ∈ P is prime if and only if a ∧ b ≤ p implies a ≤ p or b ≤ p for all a, b ∈ 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 ∈ P with x ∧ y ≤ p, but x ≤ p and y ≤ p. Since x = x ↓K(P ) and y = y ↓K(P ) there are a ∈ x ↓K(P ) and b ∈ y ↓K(P ) with (c). Let P be a ∧-semilattice and X ⊆ P a dense subset. Then X is totally ordered if and only if P is totally ordered, if and only if P = P(P ).
(d). Consider ∧-semilattices P and Q with top elements. One checks that P(P × Q) = P(P ) × { Q } ∪ { P } × P(Q). Thus the projection maps π P and π Q , as well as their right adjoint maps, send prime elements to prime elements, 2.15. Moreover, ∧ : P × P → P, (a, b) → a ∧ b maps prime elements to prime elements. Proposition 6.3. Let P be a ∧-semilattice with the coarse lower topology. Then P(P ) and P(P ) are closed in P for β = β(τ (P )). Thus, P(P ) and P(P ) are sober if P is complete. There are a, b ∈ P with a, b ≤ x and a ∧ is a β-neighborhood of x and is disjoint from P(P ). -The subset P ⊆ P is τ (P )-open, hence is β-closed. Thus P(P ) = P(P ) ∩ P is β-closed. -If P is complete then τ (P ) is sober, Proposition 3.8(b). Hence P(P ) and P(P ) are sober by 2.3. Example 6.4. Let P be a complete lattice. Then the infimum of a downdirected 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 = N ∪ {ω} ∪ {a, b, c} with the following partial order: The set N ∪ {ω} 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, N ⊆ P(P ) and N = ω / ∈ P(P ).

Vol. 84 (2023)
Topology of closure systems Page 21 of 33 17 Lemma 6.5. Let P and Q be ∧-semilattices, ϕ : P → Q a poset map with left adjoint ϕ * . If ϕ * is a ∧-homomorphism then ϕ maps prime elements to prime elements.
Proof. If p ∈ P(P ) and Remark 6.6. Let P be a ∧-semilattice with top element and Q ⊆ P a ∧subsemilattice, e.g., a closure system. The inclusion P(P )∩Q ⊆ P(Q) is obvious and may be proper. For example, consider a power set P(S) where |S| ≥ 3, cf. Example 6.2(a). There are distinct elements A, B ∈ P(S) \ P(P(S)). We define Q = {A, B} P(S) = {A ∩ B, A, B, S}. Then A, B ∈ P(Q)\P(P(S)). Now assume that Q is a closure system. If the closure map ϑ = ι * (equivalently, the closure operator, cf. Remark 3.6) is a ∧-homomorphism then P(P ) ∩ Q = P(Q) by Lemma 6.5. But note that the equality P(P ) ∩ Q = P(Q) may be true without η being a ∧-homomorphism. For an example let K be a field and V a vector space, U (V ) ⊆ P(V ) the algebraic closure system of subspaces, Example 6. Proof. The inequalities η Q,P (a) ∧ η Q,P (b) ≤ η Q,P (a), η Q,P (b) imply The second inclusion follows from a ∧ b ≤ η Q,P (a) ∧ η Q,P (b).
and the claim follows from 3.1.
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 P(P ) is not totally ordered and let Q = P(P ) P . Both inclusions are equalities, and η Q,P is a ∧-homomorphism. Now consider the trivial generating set Q of Q. Then 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 η Q,P does not depend on the particular generating set. Proof. If X β = Y β then X P = X β P = Y β P = Y P , Proposition 3.7(a). Conversely, suppose X P = Y P . We may assume that X and Y are βclosed, cf. loc.cit., and claim that X = Y . If y ∈ Y \X then y ∈ X P implies y = P y ↑X , where y ↑X = ∅. Since X is β-closed there is a finite nonempty set F ⊆ P such that y ↑P \ F ↑P is a β-open neighborhood of y and is disjoint from X. But then y ↑X ⊆ F ↑X = ( P F ) ↑X (where we use X ⊆ P(P )). It follows that P F ≤ y, and y ∈ P(P ) yields y ∈ F ↑P , a contradiction. Proof. (a). Trivially, A ⊆ a ↑X . Now pick p ∈ X\A. Since A is closed for τ (P )| X there is a finite set F ⊆ P with A ⊆ F ↑P ⊆ ( P F ) ↑P and p / ∈ F ↑P . It follows that P F ≤ a, hence P F ≤ p (since p ∈ P(P )), and we see that a ≤ p.
(b). The map γ X,P : P → O(X, ↑(X, P )), a → X\a ↑X is an isomorphism of posets, Lemma 3.5. Thus it suffices to show O(X, ↑(X, P ))) = O(X). The inclusion O(X, ↑(X, P )) ⊆ O(X) holds trivially, Example 3.3, and equality follows from item (a). 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 ⊆ P(P ) is generating then P(P ) is the sobrification of X, Corollary 6.12, and the canonical map O(P(P ) ) → O(X) is an isomorphism. Remark 6.15. Let P be a complete lattice and define N (P ) to be the set of prime generated closure systems Q ⊆ P such that η Q,P is a ∧-homomorphism. If Q ∈ N (P ) then Q = P(Q) P and P(Q) ⊆ P(P ) , Corollary 6.9. Thus Proposition 6.11 shows that N (P ) → A(P(P ) , β), Q → P(Q) and A(P(P ) , β) → N (P ), X → X P are mutually inverse isomorphisms of posets. Each element of N (P ) is a spatial frame by Theorem 6.13. Example 6.16. Consider a spatial (= prime generated) frame P . For each Q ∈ N (P ) the closure operator η Q,P : P → P is a nucleus and Q is a spatial frame. Conversely, if ν : P → P is a nucleus such that ν(P ) is a spatial frame then ν(P ) ∈ N (P ), Corollary 6.9. Identifying Q ∈ N (P ) with the nucleus η Q,P we consider 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 ν : P → P such that ν(P ) is not spatial, hence is not in 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 and let f : X → Y be the restriction of ϕ. Then the following diagrams are commutative: Proof. Note that the left adjoint ϕ * 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 ϕ is continuous for the coarse lower topology, hence f is continuous as well.
We continue with the notation and hypotheses of Proposition 6.17. It follows from Theorem 6.13 that P and Q are frames and The composition ϕ • ϕ * is a closure operator, 2.6(a). In fact, it is a nucleus of the frame Q since ϕ * is a frame homomorphism and ϕ preserves all infima.
If P and Q are algebraic lattices then they are spectral spaces and ϕ and ϕ * are maps between spectral spaces. We ask whether they are spectral maps. We know from 2.6 and Theorem 2.11(c), (e) that ϕ is coarse lower continuous and ϕ * is Scott continuous. Proof. (a). If ϕ is a spectral map then its restriction f is trivially spectral.
Conversely assume that f is spectral.
holds trivially, and we must prove equality. Assume the inclusion is proper. Then there is some It follows that (b). Pick a ∈ K(P ), i.e., a ↑P is closed and constructible, cf. 2.10, and let b ∈ Q be the smallest element with a ≤ ϕ * (b). It suffices to show that b ∈ K(Q) since then (ϕ * ) −1 (a ↑P ) = b ↑Q is closed and constructible. As b = Q b ↓K(Q) and ϕ * preserves all suprema it follows that a ≤ ϕ * (b) = P ϕ * (b ↓K(Q) ). The supremum is up-directed and a ∈ K(P ), hence there is some Now assume ϕ * is spectral. For a ∈ K(P ) the set (ϕ * ) −1 (a ↑P ) is closed and constructible in Q, hence is equal to G ↑Q with G ⊆ K(Q) finite, cf. 2.10. It follows that a ≤ ϕ * (y) for all y ∈ G, thus a ≤ P ϕ * (G) = ϕ * ( Q G) (note that ϕ * is a frame homomorphism, Proposition 6.17). We define b = Q G and claim that this is the smallest c ∈ Q such that a ≤ ϕ * (c). So, pick any z ∈ Q with a ≤ ϕ * (z). Then z ∈ (ϕ * ) −1 (a ↑P ) = G ↑Q ⊆ b ↑Q , i.e., b ≤ z. Example 6.19. We continue with Proposition 6.18(b) and assume X ⊆ P and Y ⊆ Q are patch closed. Since P = X P = P(P ) P and X = X τ (P ) it follows from Proposition 6.11 that X = P(P ) . The same holds for Y and Q.
We 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 , see Example 6.2(c). If C ⊆ Y is a nonempty quasi-compact set then C has a largest element c, and Gen(C) = Gen(c), [ The following examples illustrate the preceding results. Example 6.24. For any set S the power set P(S) is an algebraic frame. The compact elements are the finite subsets, hence 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 P(P(S)) via s → S \ {s}, hence S = P(P(S)). The coarse lower topology of P(S) restricts to the discrete topology on S, and S ∪ {S} is a spectral subspace of P(S), Theorem 6.20(a). Specialization in S is given by s ≤ S for all s ∈ S. The arithmetic algebraic frame P(S) is isomorphic to O(S), S with the discrete topology. Obviously the subset a ↑ ⊆ P is an algebraic closure system. The closure operator η : P → P is given by x → a ∨ x. This is always a dcpo homomorphism, but is a -homomorphism only if ⊥ P = a. (For if ⊥ P < a then η( P ∅) = a > ⊥ P = P η(∅).) The algebraic lattice P is a spectral space for the coarse lower topology and a ↑ is a closed subset, hence is a spectral space as well, and the inclusion ι a : a ↑ → P is a spectral map.
Set Y = P(P ) and X a = P(a ↑ ) . The inclusion Y ∩ a ↑ ⊆ X a is obvious. Let η be a ∧-map. Then Y ∩ a ↑ = X a , Remark 6.6, and X a ⊆ Y is a closed subset. Now assume that P is distributive, equivalently a prime generated and a ↑ is prime generated as well.
Next consider the compact elements. It follows from Proposition 4.2 and Theorem 4.5 that K(a ↑ ) = ϑ(K(P )). It is trivially true that K(P )∩a ↑ ⊆ K(a ↑ ), and equality holds if and only if a ∈ K(P ). For, assume a ∈ K(P ) and pick Conversely assume a / ∈ K(P ). Then ⊥ a ↑ = a is compact in a ↑ , but not in P . Consider the diagrams of Proposition 6.17 and assume P is prime generated. In general X a ⊆ a ↑ and Y ⊆ P will not be spectral subspaces. Let f a : X a → Y be the inclusion.
As noted above, ι a is always spectral. But ϑ a need not be spectral. For an example, let Y be an infinite set with the cofinite topology and set P = O(Y ). Then P is a prime generated frame and is algebraic since Y is a Noetherian space, i.e., K(P ) = . Pick x, y ∈ Y , set a = Y \{x, y}, b = Y \{y} ∈ P . Then b ∈ K(P ) ∩ a ↑ and b = a ∨ c = ϑ a (c) for any c ∈ P with x ∈ c and y / ∈ c. Clearly, there is no smallest such c ∈ P . On the other hand, one checks that ϑ a is spectral if Y is Boolean and a ∈ K(P ). Example 6.26 (cf. [11, p. 50, 2.4(b)]). Let P be an algebraic lattice, pick a ∈ P and consider the principal downset a ↓ . The inclusion map ε a : a ↓ → P is amorphism, a ∧-morphism and an fcpo map, however not a -morphism since a = ε a ( a ↓ ∅) < P = P ∅ if a < P . The map μ : P → a ↓ , x → a ∧ x is right adjoint to ε a , 2.6. One checks that a ↓K(P ) = K(a ↓ ), hence a ↓ is an algebraic lattice as well.
The subset a ↓ ⊆ P is patch closed and τ (a ↓ ) = τ (P )| a ↓ . Thus, ε a is a spectral map. It is claimed that μ is spectral as well, i.e., is coarse lower continuous and Scott continuous. We show that μ −1 (C) ⊆ P is closed and constructible if C ⊆ a ↓ is closed and constructible. By 2.10 it suffices to consider sets C = c ↑a ↓ with c ∈ K(a ↓ ). But then c ∈ K(P ) and μ −1 (C) = c ↑P is closed constructible in P .
Theorem 2.11(e) yields the somewhat surprising fact that μ 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 ⊆ P is up-directed with z = P D then μ(D) ⊆ a ↓ is up-directed as well. With t = a ↓ μ(D) we have to show μ(z) = t. It is clear that t ≤ μ(z) and we assume t < μ(z). Then there is some c ∈ K(a ↓ ) with c ≤ t and c ≤ μ(z). As λ * • (λ * • μ * ) = μ * it follows that λ * is a spectral map if and only if μ * is a spectral map, if and only if μ is coherent, see Corollary 2.14. In general this is not the case. However, assume that P is arithmetic and a ∈ K(P ). Then c ∈ K(P ) implies μ(c) ∈ K(P ) ∩ a ↓ = K(a ↓ ) and a ↓ is a coherent algebraic frame, hence U a is a spectral space by Theorem 6.20. In fact, {U a | a ∈ 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 λ * = (λ * • μ * ) • μ shows that λ * is spectral if and only if μ is spectral, which is always true as shown above, also see Proposition 6.18(b).

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 A (P ) ∩ N (P ) of closure systems, cf. 3.1 and Remark 6.15, is a closure system in P(P ).
Let P be a frame and Q ⊆ P a closure system. In general the closure operator is not a ∧-homomorphism. For an example let P be any frame that is not totally ordered. Pick a ∈ P \P(P ), cf. Example 6.2(c), and define Q = {a} P = {a, P }. Then Q is an algebraic closure system in P , is primegenerated since Q = P(Q), and is trivially coherent. But η Q,P is not a ∧homomorphism, which follows from Remark 6.6 or can be checked directly. Now assume that the closure operator is a ∧-homomorphism. Then η is a nucleus, hence Q is a frame, ϑ : P → Q is a frame homomorphism, and P(P ) ∩ Q = P(Q), Remark 6.6. However Q need not be prime generated (Example 6.16) and P(Q) and P(P ) may both be empty.
As in Remark 6.15 let N (P ) be the set of prime generated closure systems Q such that η Q,P is a nucleus. The poset isomorphism N (P ) → A(P(P ) , β), Q → P(Q) shows that P(P ) P O(P(P ) ) is the largest element of N (P ). If P is a continuous frame then P = P(P ) P , [8, Theorem, I−3.15], and P is algebraic if and only if • K(P(P ) ) is a basis of open sets for P(P ) . If P is a coherent algebraic frame then P(P ) generates P and is a patch closed subset, cf. Theorem 6.20(b). Consider A (P )∩N (P ), the set of algebraic closure systems in N (P ). We claim that each Q ∈ A (P )∩N (P ) is a coherent algebraic frame. First note that Q ⊆ P is patch closed by Theorem 4.5. Being prime generated, Q is a frame, Theorem 6.13, and the closure operator η Q,P is a nucleus since it is a ∧-homomorphism. Corollary 6.9 implies that P(P ) ∩Q = P(Q) , which is patch closed in P , and Q is coherent by Theorem 6.20(d).
But 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 the image of the nucleus. Since O(e) * : O(X) → Q is an isomorphism it follows that Q is prime generated and belongs to N (O(Y )). But Proposition 6.18(a) shows that O(e) * is not a spectral map (as e is not a spectral map), hence Q is not an algebraic closure system in O(Y ), Theorem 4.5.
The subset A(P(P ) con ) ⊆ A(P(P ) , β) is a closure system with closure map X → X con . The isomorphism A(P(P ) , β) → N (P ), X → X P maps A(P(P ) con ) onto the closure system A (P ) ∩ N (P ) ⊆ N (P ), and the corresponding closure map is given by Q → Q con , cf. Theorem 5.2.
The set C (P ) of closure systems and the set A (P ) of algebraic closure systems in P are closure systems in P(P ), 3.1. We do not know whether N (P ) is a closure system in P(P ). However: