Nuclear ranges in implicative semilattices

Abstract

A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice $\mathrm{N }A$ that is isomorphic to the system ${\mathcal {N}}A$ of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed.

1 Introduction

Closure is a vital theme in quite diverse mathematical disciplines (see [16] for a survey and historical background). A convenient framework for closure theory is that of ordered sets (posets): a closure or hull operation on a poset A is an isotone (order-preserving), inflationary (extensive) and idempotent self-map on A, or in succinct terms, a unary operation $g$ on A with

$$\begin{aligned} x\le gy \Leftrightarrow gx \le gy. \end{aligned}$$

(C)

Extensions to the even more comprehensive categorical theory of reflections, monads [1, Ch. V-20] and closure operators [10, 12] are possible but will not concern us here. Closure may be described by several equivalent structures, the most obvious ones being the so-called closure ranges [15, 16], elsewhere also termed partial ordinals [2], closure subsets [9], or closure systems [11]. While by a closure range of a poset A we mean a subset C such that above each element of A there is a least element in C, we reserve the name “closure system” for subsets of power sets ${{\mathcal {P}}}X$ that are closed under arbitrary intersections (with $\bigcap \emptyset = X$), hence closure ranges in ${{\mathcal {P}}}X$. Sending each closure operation to its range (the fixpoint set), one obtains an isomorphism between the pointwise ordered set $\mathrm{C}A$ of all closure operations and the set ${{{\mathcal {C}}}}A$ of all closure ranges, ordered by dual inclusion.

Taking into account any binary operation on A that is isotone in both arguments, one calls $g$ multiplicative if $g$ preserves that operation. An early investigation of this rather general situation with many examples is due to Varlet [44, 45]. Specifically, motivated by the familiar notion of Kuratowski closure in topology, one considers closure operations on boolean algebras that preserve finite joins. This approach, intertwining topology with algebraic logic and lattice theory, was pursued by McKinsey and Tarski in their groundbreaking papers The algebra of topology [34] and On closed elements in closure algebras [35], which had a great number of followers. In the same vein as the lattices of open resp. closed sets in topological spaces are the fixpoint sets of topological interior resp. closure operators on power sets, the range of any interior operation on a boolean algebra is a Heyting algebra, and all Heyting algebras arise this way; see Esakia [20] for a categorical equivalence.

Another obvious choice is the binary meet operation of $\wedge$-semilattices with top, in this paper merely referred to as semilattices. At first glance, this choice does not look very promising, because on a boolean algebra the only $\wedge$-preserving closure operations are the unary join-operations $\gamma _a : x \mapsto a\vee x$. But $\wedge$-preserving closure operations, nowadays often called nuclei, turned out to be of fundamental importance in pointfree topology, logic, topos theory [27, 28], and other branches of mathematics. Perhaps the first account of that theme is Bergmann’s 1952 paper [5]. Restricting the isomorphism between $\mathrm{C}A$ and ${{{\mathcal {C}}}}A$ to the poset $\mathrm{N }A$ of all nuclei on a semilattice A leads to an isomorphism between $\mathrm{N }A$ and the subsemilattice ${{{\mathcal {N}}}}A$ of ${{{\mathcal {C}}}}A$ consisting of all nuclear ranges (also termed nuclear systems [11] or strong ideals [40]).

In the paper at hand and its successor [17] we investigate nuclei on implicative semilattices (cf. [6, 7, 36, 44, 45]), that is, semilattices with a binary operation $\rightarrow$, called residuation, formal implication or relative pseudocomplementation (according to the respective interpretation), such that

$$\begin{aligned} x\wedge y \le z \Leftrightarrow y \le x \!\rightarrow \!z. \end{aligned}$$

Other common notations for $x\!\rightarrow \!z$ are $x*z$ [30, 36, 45] or z : x [9], but often it is more convenient to use the symbols $x^z$ for $x\!\rightarrow \!z$ and $x^{yz}$ for $(x^y)^z$ in order to avoid parentheses in iterated applications [8, 45]. According to that convention, it is consistent to write $x^{\bot }$ for the pseudocomplement ${x^* = x\!\rightarrow \!\bot }$ (provided a bottom element $\bot$ exists). In fact, $^{\!\bot }$ is a kind of orthocomplementation, in view of De Morgan’s law $(x\vee y)^{\bot } = x^{\bot \!} \wedge y^{\bot }$ in Heyting algebras. Notice also the “powerful” equation

$$\begin{aligned}x^{zz}\wedge x^z = z \end{aligned}$$

which assures that implicative semilattices are distributive: $x\wedge y \le z$ implies $z = x'\wedge y'$ for some $x'\ge x$ and $y'\ge y$ (take $x' = x^{zz}$, $y' = x^z$). The exchanged notation $z^x$ instead of $x^z$ would be more suggestive, in view of the rules

$$\begin{aligned} x\!\rightarrow \!\,(y\wedge z)&= (x\!\rightarrow \!y)\wedge (x\!\rightarrow \!z), \\ (x\vee y)\!\rightarrow \!z&= (x\!\rightarrow \!z)\wedge (y\!\rightarrow \!z), \\ (x\wedge y) \!\rightarrow \!z&= x\!\rightarrow \!\,(y\!\rightarrow \!z), \end{aligned}$$

which in the $z^x$ notation, oppressing the symbol $\wedge$ and writing $+$ for $\vee$, would turn into the familiar exponential rules $(yz)^x = y^x z^x$, $z^{x+y} = z^x z^y$, $z^{xy} = (z^y)^x$. However, this exchanged notation seems not to have prevailed in the lattice-theoretical literature, while it is common in the wider context of cartesian closed categories (see, e.g., [1, Ch. VII]).

Implicative semilattices are also called relatively pseudocomplemented or Brouwerian (see, e.g., Köhler [30, 31]); some authors use the latter term for the dual structures (McKinsey and Tarski [35]). Picado, Pultr and Tozzi [40] speak of Heyting semilattices, while we reserve that term for bounded implicative semilattices (possessing a least element $\bot$). A Heyting lattice is then both a Heyting semilattice and a lattice, and the associated algebra with the operations $\vee , \wedge , \bot , \top$ and $\rightarrow$ is a Heyting algebra; for the origins of this concept, see the pioneering work of Glivenko [24] and Heyting [25] related to Brouwer’s intuitionistic logic. Immense research has been devoted to that theme and its role in logic, algebra and topology (see, e.g., Esakia [20]).

Some of the results we shall prove for nuclei on semilattices are known for the case of lattices or at least of complete lattices, but the lack of certain joins or meets often requires new methods. By general properties of adjoint maps (see Sect. 2), binary meets in implicative semilattices distribute over all existing joins, and the complete Heyting lattices are the frames or locales [26, 27, 38, 39, 43], satisfying for all subsets Y the distributive law

$$\begin{aligned}x\wedge \mathop {\textstyle \bigvee }Y = \mathop {\textstyle \bigvee }\{ x\wedge y\,{|}\,y\in Y\}. \end{aligned}$$

Further examples of Heyting lattices are all products of bounded chains. The chain $\omega \,$ of natural numbers is not an implicative semilattice, missing a top element. On the contrary, the dual chain $\omega ^{op}$ is a $\mathop {\textstyle \bigvee }$-complete implicative lattice (but not a Heyting lattice), and ${{{\mathcal {C}}}}\omega ^{op} = {{{\mathcal {N}}}}\omega ^{op}$ is the closure system of all subsets containing the top element. Thus, it is a boolean frame. The following six structures all describe the same objects if their carriers are finite:

$$\begin{aligned}&\displaystyle \text {nonempty (bounded) distributive lattices, frames, locales,}&\\&\displaystyle \text {Heyting semilattices, Heyting lattices, Heyting algebras.}&\end{aligned}$$

Nuclei on Heyting algebras are also referred to as modal operators (Beazer and Macnab [4], Macnab [32, 33]). Our perspective is slightly different from the classical one: we consider an implicative semilattice A with top element $\top$ as a general algebra with the binary meet operation $\wedge$, the nullary operation $\top$, and the family $\alpha = (\alpha _a : a\in A)$ of unary operations

$$\begin{aligned}\,\alpha _a : A \mathop {\longrightarrow }A \ \text{ with } \ \alpha _a y = a\!\rightarrow \!y, \end{aligned}$$

which are related to the unary meet operations

$$\begin{aligned}\lambda _a : A \mathop {\longrightarrow }A \ \text{ with } \ \lambda _a x = a\wedge x \end{aligned}$$

via the adjoining equivalence

$$\begin{aligned}\lambda _a x \le y \Leftrightarrow x \le \alpha _a y. \end{aligned}$$

The subalgebras of $(A, \wedge ,\top ,\alpha )$ are what Köhler calls total subalgebras [31]. In accordance with a general ideal concept in universal algebra, we call them left $\rightarrow$-ideals, or l-ideals for short (Picado, Pultr and Tozzi merely speak of ideals [40]). Those l-ideals which are closure ranges are said to be nuclear, because they are exactly the ranges of nuclei. Under the changed perspective the appropriate alternative to the usual homomorphisms are residuated (that is, coadjoint) and top-preserving $\wedge$-homomorphisms between semilattices. We call them r-morphisms and their adjoints l-morphisms. A basic observation will be that the image of an implicative semilattice under an r- or l-morphism is implicative, too. For implicative semilattices, the unary meet operations $\lambda _a$ induce r-morphisms from A onto $\downarrow \!a = \{ x \in A \,{|}\, x \le a\}$, and their adjoints $\alpha _a$ are nuclei inducing l-morphisms from ${\downarrow }\,a$ into A.

In the complete case, the r-morphisms are nothing but the frame homomorphisms, whereas the l-morphisms are the locale morphisms or localic maps [38], the natural morphisms in pointfree topology, corresponding to continuous maps between topological spaces [26, 27, 38]. The tools of r- and l-morphisms provide extensions of results from the realm of frames/locales to arbitrary Heyting algebras or even to implicative semilattices. Sometimes the existence of certain joins or meets is indispensable. But one also finds substantial results on nuclei in the non-complete setting, for example in the work of Macnab [32, 33] on Heyting algebras; see also Varlet [44, 45].

Section 2 provides the necessary fundaments concerning adjunctions and closure operations. In Section 3 we introduce some useful weak variants of nuclei and see that nuclei and their ranges are transported forth and back by suitable adjoint maps. Section 4 contains the relevant connections between nuclei, prenuclei and weak nuclei on implicative semilattices and their ranges. Certain completeness properties turn out to be necessary and sufficient in order that for each prenucleus there is a least nucleus above it, and for each weak nucleus there is greatest nucleus below it. Section 5 is devoted to three important kinds of l-ideals in implicative semilattices: the basic open l-ideals ${\mathfrak {a}}a = \{ a\!\rightarrow \!x \,{|}\, x\in A\}$, the boolean l-ideals ${\mathfrak {b}}a = \{ x\!\rightarrow \!a \,{|}\, x\in A\}$, and the basic closed l-ideals ${\mathfrak {c}}a = \{ x \in A \,{|}\, x\ge a\}$. The first two kinds are always nuclear, whereas the ${\mathfrak {c}}a$’s are closure ranges only in lattices. The nuclear l-ideals ${\mathfrak {b}}a$ form a meet-dense subset of ${{{\mathcal {N}}}}A$ and are exactly those l-ideals which are boolean lattices. The basic open and the basic closed l-ideals are complementary in the frame ${{{\mathcal {T}}}\! }A$ of all l-ideals, and together they generate ${{{\mathcal {N}}}}A$ via joins of finite meets if A is a Heyting algebra [33].

In the second part [17] we use the results derived in this first part for a thorough study of the algebraic structure of ${{{\mathcal {T}}}\! }A$ (cf. Köhler [31]) and of ${{{\mathcal {N}}}}A$ (cf. Picado, Pultr and Tozzi [40]). Other applications occur in [19].

2 Closure operations, closure ranges, and adjoint maps

The letter A denotes a (partially) ordered set (poset), $\le$ its order relation, $\ge$ the dual order, and $A^{op}$ the opposite or dually ordered set. A least element (bottom) is denoted by $\bot$ or $\bot A$, and a greatest element (top) by $\top$ or $\top \!A$. A poset possessing a top is said to be topped. For $Y \subseteq A$,

$$\begin{aligned} {\uparrow }\,Y = \{ x \in A \,{|}\, \exists \ y\in Y \, (x \ge y)\} \ \text{ and } \ {\downarrow }\,Y = \{ x \in A \,{|}\, \exists \ y\in Y \, (x\le y)\} \end{aligned}$$

are the upset and the downset generated by Y, respectively. ${\uparrow }\,y = {\uparrow }\{y\}$ resp. ${\downarrow }\,y = {\downarrow }\{y\}$ is the principal upset resp. principal downset generated by $y\in A$. An equation $x = \mathop {\textstyle \bigvee }Y$ means that x is the least upper bound (join, supremum) of Y; dually, meets (infima) are defined as greatest lower bounds and denoted by $\mathop {\textstyle \bigwedge }Y$. The poset A is complete if all subsets have joins (or equivalently, all subsets have meets), and $\mathop {\textstyle \bigvee }$-complete if all nonempty subsets have joins (or equivalently, all lower bounded subsets have meets). In a $\vee$ -semilattice, the binary joins $x\vee y = \mathop {\textstyle \bigvee }\{ x,y \}$ exist for all elements x, y; dually, in $\wedge$-semilattices all binary meets $x \wedge y$ exist.

If maps are applied to elements, we omit parentheses and write $fa$ or $f_a$ for the image of an element a under a map $f$, and also $fX$ for the image of a subset X of the domain, while the preimage of a subset Y of the codomain is denoted by $f^{\leftarrow }Y$. We write $A^A$ for the pointwise ordered set of all self-maps or unary operations $f$ on A, and we put

$$\begin{aligned} A_{f} = \{ a\in A \,|\, fa = a \}, \ A_{f} x = A_{\,f} \cap {\uparrow \!x} \text{ for } x\in A. \end{aligned}$$

The fixpoint set $A_{f}$ coincides with the range $fA$ whenever $f$ is idempotent ($ff= f$). The map $f$ is isotone if $x\le y$ implies $fx \le fy$, and inflationary or an inflation if $x\le fx$. By a preclosure operation we mean an isotone inflation, and by a weak closure operation [41] an idempotent inflation. A retraction (in [23]: projection) is an isotone idempotent map, and a closure or hull operation is an idempotent preclosure operation. Dual closure operations are called coclosure or kernel operations. An easy verification shows:

Lemma 2.1

For all preclosure operations $j$ and all weak closure operations $g$, passing to the ranges inverts the order: $j\le g$ is equivalent to $A_{g} \subseteq A_{j}$.

We call a subset C of A a closure range if for each $a\in A$ there is a least $c\in C$ with $a\le c$, but apply the terms closure operator and closure system only to the case of power set lattices $A = {{\mathcal {P}}}X$. A coclosure range or kernel  range in A is a closure range in the dual poset $A^{op}$. The term “closure range” is justified by the fact that associating with each closure operation its range yields an isomorphism between $\mathrm{C}A$, the pointwise ordered set of closure operations on A, and ${{{\mathcal {C}}}}A$, the set of all closure ranges, ordered by dual inclusion [2, 16, 44]. Closure ranges are subsets that are closed under all existing meets; the converse holds in complete lattices. A basic construction of closure operations leans on the existence of certain meets [16]:

Lemma 2.2

Let S be a any subset of A. If for each $x\in A$ the set $S \cap {\uparrow \!x}$ has a meet $mx$ then the so defined map $m$ is a closure operation on A, and its range is the closure range generated by S.

Closure operations are connected with adjoint maps. Given two posets A, B and two maps $h: A \mathop {\longrightarrow }B$ and $f: B \mathop {\longrightarrow }A$ related by the equivalence

$$\begin{aligned}hx \le y \Leftrightarrow x\le fy \end{aligned}$$

one calls $f$ the (right or upper) adjoint of $h$, and $h$ the coadjoint (left or lower adjoint) of $f$. In that situation, the resulting equations $fhf= f$ and $hfh=h$ ensure that $fh$ is a closure operation, and $hf$ a kernel operation. In accordance with [38] we chose the letter $h$ because often $h$ will be a homomorphism between semilattices or lattices. Observe that in [9] and elsewhere $f$ stands for the left adjoint, while right adjoints are often denoted by $g$.

Every map $h$ from A to B factors through its surjective corestriction $h_0 \!$ from A onto $hA$ and the inclusion map $h^0$ from $hA$ into B. By the characteristic equivalence (C), $g$ is a closure operation iff $g^0$ is adjoint to $g_0$. A map is left adjoint iff it is residuated, that is, preimages of principal downsets are principal downsets, and (right) adjoint iff it is residual, that is, preimages of principal upsets are principal upsets. Residuated maps preserve all existing joins, and residual maps all existing meets; the converse holds for maps between complete lattices. For more results on adjoint maps and closure operations refer to [9, 15, 16, 18, 23]. Note the following straightforward equivalences:

Lemma 2.3

A map $h: A \mathop {\longrightarrow }B$ with adjoint $f: B \mathop {\longrightarrow }A$ is surjective iff $f$ is injective iff $hf = id_B$ iff $hx = \mathop {\textstyle \bigvee }Y$ for all subsets $Y\!$ of B with $x = \mathop {\textstyle \bigvee }fY$. Thus, for a residuated surjection $h$, if A is $(\mathop {\textstyle \bigvee }$-)complete then so is $B = hA$.

Lemma 2.4

A map $h: A \mathop {\longrightarrow }B$ with adjoint $f$ is injective iff $f$ is surjective, and then $C \subseteq A$ is a principal upset iff its preimage $f^{\leftarrow }C$ is a principal upset.

Proof

Suppose $f^{\leftarrow }C = {\uparrow \!y}$. Then $y \in f^{\leftarrow }C$, $fhfy = fy \in C$, $hfy \in f^{\leftarrow }C ={\uparrow \!y}$, and so $y = hfy$ (as $hfy \le y$). Now, injectivity of $h$ gives $fhx = x$, and then

$$\begin{aligned} fy\le & {} x \Rightarrow \! y \le hx \Rightarrow \! hx \in f^{\leftarrow }C\\&\Rightarrow &\! x = fhx \in C \Rightarrow \! y \le hx \Rightarrow \! fy \le fhx = x \end{aligned}$$

confirms the equation $C = {\uparrow }{f{y}}$. $\square$

Closure operations and their ranges are transported by adjoint maps:

Proposition 2.5

Let $h\! : A \! \mathop {\longrightarrow }B$ be residuated and $f\! : B \mathop {\longrightarrow }A$ its adjoint.

• (1) For a closure operation $j$ on B with range C, $fjh$ is a closure operation on A with range $fC$. Hence, adjoint maps send closure ranges to closure ranges.

• (2) For a closure operation $g$ on A, the restriction $g'$ of $hgf$ to $hA$ is a closure operation on $hA$ if one of the following conditions is fulfilled:

  • (a) $fB$ is an upset in A, (b) $fh$ commutes with $g$, (c) $fhgf= gf$.

  In these cases the range of $g' \!$ is the preimage of $A_{g}\!$ under $f\!\upharpoonright \! hA$. In particular, preimages of closure ranges under injective adjoint maps whose range is an upset are again closure ranges.

Proof

(1) Since $f, j$ and $h$ are isotone, so is $fjh$, and $id_B \! \le j$ gives the inequality $id_A \le fh\le fjh$; and $hf\le id_B$ yields $fjhfjh\le fjjh= fjh$, so that the map $fjh$ is idempotent. Further, $jhx = \bot (C\cap \,{\uparrow }\,hx)$ implies $fjhx = \bot (fC\cap \,{\uparrow }\,x)$, whence $fC$ is the range of $fjh$.

(2) (a) implies (c): if $fB$ is an upset then for $y \in B$ there is some $z\in B$ with $gfy = fz$, whence $fhgfy = fhfz = fz = gfy$. That (b) implies (c) is also clear by the equation $fhf= f$. And if (c) holds then the restriction $g'$ of $hgf$ to $hA$ is a closure operation on $hA$ by the following equivalences:

$$\begin{aligned} hx \le hgfhy \Leftrightarrow fhx \le fhgfhy = gfhy \Leftrightarrow gfhx \le gfhy \Leftrightarrow hgfhx \le hgfhy. \end{aligned}$$

Further,

$$\begin{aligned} g' hA = \{ y \in hA\,\, \! {|} hgfy \le y \} \!=\! \{ y \in hA\,\, \! {|} gfy \le fy \} \!= hA \cap f^{\leftarrow \!} A _{g}.\quad \qquad \square \end{aligned}$$

Partial completeness properties ensure the existence of ${\overline{g}}$, the least closure operation above $g$, or of $g^{\circ }\!$, the greatest closure operation below $g$:

Proposition 2.6

Let $g$ be a unary operation on A such that for each $x\in A$ the set $A_{g} x$ has a meet $mx$.

• (1) The so defined map $m\in A^A$ is a closure operation.

• (2) If $g$ is a preclosure operation then $m= {\overline{g}}$.

• (3) If $g$ is a weak closure operation then $m= g^{\circ }$.

Proof

(1) is clear by Lemma 2.2.

(2) $g\le m$ follows from $gx \le a = ga$ for $x\le a \in A_{g}$. And if some $j\in \mathrm{C}A$ fulfils $g\le j$ then $mx \le jx$, as $x\le a = jx$ and $ga \le ja = jjx = jx = a \in A_{g}x$.

(3) $m\le g$ holds, as $a = gx$ entails $x \le a = ga \in A_{g}$, hence $mx \le a = gx$. And if some $j\in \mathrm{C}A$ fulfils $j\le g$ then $jx \le a$ for all $a \in A_{g} x$, since $x \le a = ga$ entails $jx \le jga \le gga = ga = a$. Thus, $jx \le mx$ for all $x\in A$. $\square$

We call a map $g\in A^A$ lower complete, briefly l-complete, if each of the sets $A_{g}x$ has a meet. Under weak assumptions on A, l-completeness is not only sufficient but also necessary in order that ${\overline{g}}$ resp. $g^{\circ }$ exists.

Theorem 2.7

A preclosure operation $g$ on a topped poset A is l-complete iff there is a least closure operation ${\overline{g}}$ above $g$, and then ${\overline{g}} x = \bot A_{g}x$.

Proof

Suppose $g$ is a preclosure operation on A such that ${\overline{g}}$ exists. Then $a\in A_{{\overline{g}}}$ implies $a \le ga \le {\overline{g}} a = a$, that is, $a\in A_{g}$. Conversely, assume $a \in A_{g}$. The map $j$ on A defined by $jx = a$ if $x\le a$ and $jx = \!\top A$ if $x \not \le a$ is easily seen to be a closure operation with $g\le j$. Thus, ${\overline{g}} \le j$; in particular, ${\overline{g}} a \le ja = a \in A_{{\overline{g}}}$. This proves the equations $A_{g} = A_{{\overline{g}}}$ and ${\overline{g}} x = \bot A_{g} x$. $\square$

Similar constructions of closure operations $j\ge g$ with $ja = a$ show that 2.7 remains true for $\vee$- or $\wedge$-semilattices instead of topped posets.

Theorem 2.8

A weak closure operation $g$ on a $\vee$-semilattice is l-complete iff there is a greatest closure operation $g^{\circ }\!$ below $g$, and then $g^{\circ } x = \mathop {\textstyle \bigwedge }A_{g}x$.

Proof

Suppose $g$ is a weak closure operation on a $\vee$-semilattice A such that $g^{\circ }$ exists. For any $x \in A$, $g^{\circ } x$ is a lower bound of $A_{g} x$, since $x \le a = ga$ implies $g^{\circ } x \le g^{\circ } a \le ga = a$. Given any lower bound b of $A_{g} x$, put $jy = b \vee y$ if $x \le y$ and $jy = y$ if $x \not \le y$. Then the so defined map $j$ on A is inflationary and satisfies $j\le g$ (indeed, $jy = b \vee y \le gy$ if $x \le y$ and so $x \le gy \in A_{g}x$, while $jy = y \le gy$ if $x \not \le y$). If $x \le y \le jz$ then $x \le z$ (otherwise, $x \not \le z = jz$, in contrast to $x \le y \le j z$) and so $jy = b\vee y \le jz = b \vee z$, while $y \le jz$ together with $x\not \le y$ entails $jy = y \le jz$. Hence, $j$ is a closure operation on A with $j\le g^{\circ }$, and therefore $b \le b \vee x = jx \le g^{\circ } x$. Thus, $g^{\circ } x = \mathop {\textstyle \bigwedge }A_{g} x$. $\square$

3 Nuclei and their generalizations

A nucleus on a $\wedge$-semilattice is a closure operation that preserves binary meets (cf. [6, 7]). At first glance, the term “nucleus” looks a bit strange, because $\mathfrak {nucleus}$ is the latin word for kernel; but it is justified by the one-to-one correspondence between nuclei and congruence kernels of residuated $\wedge$-homomorphisms (see the end of this section for more details). We reserve the term interior operation for kernel operations that preserve finite meets (having in mind the prototypes of topological interior operators).

Example 3.1

A peach P has a kernel K and a hull H, its skin.

Starting from a fixed inner point p of K, the peach P is partially ordered by $x\le y$ if x lies closer to p than y on a radial ray, or $x = y$. In fact, P is a $\wedge$-semilattice with $x\wedge y = p$ if x and y are incomparable; but, clearly, P has no greatest element. Adding a universal upper bound $\top$ yields a complete lattice $P^{\top } = P\,\cup \{ \top \}$, which however is not implicative. K deserves its name, being in fact a kernel range in  $P\,$; its kernel operation $k$ maps x to the nearest point of K on the ray from p to x. Whereas H is not a closure range, $H\cup \{ p\}$ is a closure range indeed. The closure or hull operation $h$ associated with $H \cup \{ p\}$ maps each x distinct from p to the nearest point of H on the ray from p through x and leaves p fixed. Both $k$ and $h$ preserve binary meets, and so does the extension $h^{\top }$ of $h$ to $P^{\top }$ with $h^{\top } \top = \top$. Thus, $h$ and $h^{\top }$ are nuclei, and their range is $H \cup \{ p\}$ resp. $H^{\top } \cup \{p\}$, whereas the extension $K^{\top } = K \cup \{ \top \}$ of the kernel K is the range of the interior operation $k^{\top }$.

The pointwise formed meet of two nuclei is again a nucleus, and the same holds for arbitrary meets, provided they exist (as in the complete case). Thus, the nuclei always form a $\wedge$-semilattice $\mathrm{N }A$ (with $id_A$ as bottom, and the constant map $x \mathop {\longmapsto }\top A$ as top if $\top A$ exists). It is a challenging task to find out under what circumstances $\mathrm{N }A$ becomes an implicative (or Heyting) semilattice or lattice when A is one. A thorough analysis of this and related problems reveals that some major results in the theory of nuclei on semilattices depend on suitable completeness hypotheses (See [17]). It is our main purpose to detect where completeness assumptions are indispensable, and where they may be circumvented by alternate arguments.

We call a closure range C in a $\wedge$-semilattice A a nuclear range if for all $x,y\in A$ and $z\in C$ with $x\wedge y\le z$ there exists a $c\in C$ with $x\le c$ and $c \wedge y \le z$. Recall that by a semilattice we always mean a $\wedge$-semilattice with top. A nonempty subset C of a semilattice is a nuclear range iff for all $x,y\in A$ and $z\in C$ with $x\wedge y\le z$ there exists a least $c\in C$ with $x\le c$ and $c \wedge y \le z$. The following description of nuclear ranges, justifying our terminology, is due to Varlet [44], who speaks of multiplicative closure :

Proposition 3.2

Associating with each nucleus on a $\wedge$-semilattice A its range, one obtains an isomorphism between $\,\mathrm{N }A$, the $\wedge$-semilattice of nuclei, and ${{{\mathcal {N}}}}A$, the $\wedge$-semilattice of nuclear ranges, ordered by dual inclusion.

For a categorical treatment of nuclei on semilattices, a suitable morphism class is formed by so-called r-morphisms, that is, residuated semilattice homomorphisms preserving top elements. Their adjoints are called l-morphisms, having a left adjoint that preserves finite meets, and sometimes referred to as localizations (Bezhanishvili and Ghilardi [6]). In the category of locales, they are the localic maps (Johnstone [27], Picado and Pultr [38]).

An injective r- resp. l-morphism will be called an r- resp. l-embedding, and a surjective r- resp. l-morphism an r- resp. l-surjection. By an r- resp. l-domain of a semilattice we mean the domain of an inclusion map that is an r- resp. l-morphism. From basic connections between adjoint maps and closure operations (see [16, Ch. 3]) and the fact that composites of maps preserve finite meets if the factors do, one derives the following facts:

Proposition 3.3

Let $h: A \mathop {\longrightarrow }B$ be an r-morphism having the range D, and $f: B \mathop {\longrightarrow }A$ its adjoint l-morphism with range C. Then $g= fh$ is a nucleus with range C, and $k= hf$ is an interior operation whose range D is isomorphic to C under the restriction ${i }= h_0 \!\upharpoonright \! C$. Thereby, one obtains a factorization $h= k^0 {i }g_0$ into an r-embedding $k^0$, an isomorphism i and an r-surjection $g_0$. An analogous mono-iso-epi-factorization $f= g^0 {i }^{-\!1} k_0$ into l-morphisms holds in the opposite direction; see Figure 1.

Furthermore, $h$ is a nucleus iff $f$ is an interior operation iff $h= fh$ iff $f= hf$.

Fig. 1

Factorization of r- and l-morphisms

Corollary 3.4

The r-surjections are up to isomorphisms the surjective corestrictions of nuclei on semilattices, and the l-embeddings are up to isomorphisms the inclusion maps of nuclear ranges (l-domains) in semilattices.

Corollary 3.5

The poset of all r-morphisms between semilattices A and B is dual to the poset of all l-morphisms from B to A, but also isomorphic to the poset of all isomorphisms between l-domains of A and r-domains of B.

The r-domains of frames are the subframes, while the l-domains of locales are the sublocales in the sense of [38]. In view of Corollary 3.4, afficionados of category theory alternately refer to the extremal r-epimorphisms, that is, r-surjections, as sublocales; for the sake of distinction, l-domains of locales are sometimes called sublocale sets [37].

It is useful to take into account several generalizations of nuclei, some of which also occur (under other names) in the ample work of Simmons on frames (see, e.g., [43]). Let A be a $\wedge$-semilattice. By a subnucleus on A we mean an inflation $g$ on A satisfying

$$\begin{aligned}x \wedge gy \le g(x\wedge y). \end{aligned}$$

Following Banaschewski [3], we call an isotone subnucleus a prenucleus (some authors reserve that term for $\wedge$-preserving inflations; Simmons [43] calls prenuclei stable inflators). By a weak nucleus we mean an idempotent subnucleus. The nuclei are not only the idempotent prenuclei but also the isotone weak nuclei. Being $\wedge$-preserving and idempotent, any nucleus g satisfies

$$\begin{aligned}g(gx \wedge gy) = gx \wedge gy \le g(x\wedge y). \end{aligned}$$

We call an inflation fulfilling that condition a pseudonucleus. Pseudonuclei play a “central” role in the determination of the center (the boolean part) of the frame of all l-ideals, as demonstrated in the second part [17].

In Figure 2 we display the hierarchy among the operations introduced before. The bold framed classes are closed under composition and pointwise meets. The second property also holds for the class of nuclei, which however is not closed under composition, and $jg\! \in \! \mathrm{N }A$ is not equivalent to $gj\! \in \! \mathrm{N }A$.

Fig. 2

Generalizations of nuclei

Lemma 3.6

For $g, j\in \mathrm{N }A$, the following equivalences and implications hold:

$$\begin{aligned} jg\! \in \! \mathrm{N }A \Leftrightarrow jgj\!=\! jg\Leftrightarrow gjg\!=\! jg\Leftarrow jg\!=\! gj\! \Rightarrow gj\!=\! jgj\Leftrightarrow gj\!=\! gjg\Leftrightarrow gj\! \in \! \mathrm{N }A. \end{aligned}$$

On the other hand, if $g\in \mathrm{N }A$ and $j\in \mathrm{N }A_{g}$ then $g^0 j\,g_0 \in \mathrm{N }A$.

The last claim and the next proposition follow from Proposition 2.5.

Proposition 3.7

Let $h: A \mathop {\longrightarrow }B$ be an r-morphism with adjoint $f: B \mathop {\longrightarrow }A$.

• (1) For a nucleus $j$ on B with range C, $fjh$ is a nucleus on A with range $fC$. Thus, l-morphisms send nuclear ranges to nuclear ranges.

• (2) If $fB$ is an upset then for $g\in \mathrm{N }A$ the restriction $g'$ of $hgf$ to $hA$ is a nucleus on $hA$, and the range of $g'$ is the preimage of $A_{g}$ under $f\!\upharpoonright \! hA$. Thus, the preimage maps of l-embeddings whose range is an upset send nuclear ranges to nuclear ranges.

Proposition 3.8

Let $h: A \mathop {\longrightarrow }B$ be an r-morphism with adjoint $f: B \mathop {\longrightarrow }A$.

• (1) If A is an implicative semilattice then so is $hA \simeq fB$, and $f\!\upharpoonright \! hA$ is an l-embedding adjoint to $h_0 : A \mathop {\longrightarrow }hA$ and preserves the operation $\rightarrow$.

• (2) If B is an implicative semilattice then so is $fB \simeq hA$, and $h\!\upharpoonright \! fB$ is an r-embedding coadjoint to $f_0 : B \mathop {\longrightarrow }fB$ but need not preserve $\rightarrow$.

Proof

Being surjective, $h_0$ has the injective adjoint $f\!\upharpoonright \! hA$, and dually for $f_0$.

(1) $hx \wedge hy \le hz \Leftrightarrow h(x\wedge y) \le hz \Leftrightarrow x \wedge y \le fhz \Leftrightarrow x\le y \!\rightarrow \!fhz$

$\Rightarrow hx \le h(y \!\rightarrow \!fhz) \Rightarrow hx \wedge hy \le h((y\!\rightarrow \!fhz) \wedge y) \le hfhz = hz$.

Thus, $h(y\!\rightarrow \!fhz) = hy \!\rightarrow \!hz$ in $hA$. As $fh$ is a nucleus, $f\!\upharpoonright \! hA$ preserves $\rightarrow \,$:

$$\begin{aligned} x \le f(hy \!\rightarrow \!hz) \Leftrightarrow hx \le hy \!\rightarrow \!hz \Leftrightarrow x \le y \!\rightarrow \!fhz = fhy \!\rightarrow \!fhz. \end{aligned}$$

(2) $fx \wedge fy \le fz \Leftrightarrow hfx \wedge hfy \le hfz \Leftrightarrow hfx \le hfy \!\rightarrow \!hfz$

$\Rightarrow fx\le f( hfy \!\rightarrow \!hfz)$ $\Rightarrow fx \wedge fy \le f( (hfy \!\rightarrow \!hfz) \wedge hfy) \le fhfz = fz$.

Thus, $f( hfy \!\rightarrow \!hfz )$ is $fy \!\rightarrow \!fz$, the relative pseudocomplement in $fB$.

An embedding of a three-element chain in a four-element boolean lattice is an r-embedding that does not preserve $\rightarrow \,$. $\square$

Let us add a few words about congruences. An equivalence relation R on A is isotone if $x\le y$ implies $R {\downarrow x} \subseteq {\downarrow R}y$, a weak closure equivalence if each equivalence class has a top, and a closure equivalence if both conditions hold. An equivalence relation R on a $\wedge$-semilattice is a weak congruence if $x \!\mathrel {R}y$ implies $x\wedge z \in {\downarrow \!R}(y\wedge z)$, and a congruence if $x \!\mathrel {R}y$ implies $x\wedge z \in R(y\wedge z)$.

Proposition 3.9

Sending each map $g$ to the equivalence relation R defined by $x \!\mathrel {R}y \Leftrightarrow gx = gy$, one obtains bijective correspondences between

• (1) weak closure operations and weak closure equivalences,

• (2) closure operations and closure equivalences,

• (3) weak nuclei and weak congruences that are weak closure equivalences,

• (4) nuclei and congruences that are closure equivalences,

• (5) nuclei on frames and frame congruences.

Proof

Except (3), these equivalences are known (see [9, 22, 38, 41]), so we only prove (3). If $g$ is a weak nucleus and $x \!\mathrel {R}y$, then ${t = g(y\wedge z) = \!\top \! \!\mathrel {R}(y\wedge z)}$ satisfies $x\wedge z \le gx \wedge z = gy \wedge z \le gt = t$ and $t \!\mathrel {R}y\wedge z$, so R is a weak congruence, and a weak closure equivalence by (1).

Conversely, if R is assumed to be a weak congruence and a weak closure equivalence then for $y,z \in A$ and $x = gy = ggy$ we have $gx = gy$, that is, $x \!\mathrel {R}y$, so there is a $t\in A$ satisfying $x \wedge z \le t \!\mathrel {R}y\wedge z$, that is, $gy \wedge z \le t \le gt = g(y\wedge z)$. Thus, $g$ is a subnucleus, and a weak nucleus by (1). $\square$

4 Nuclei, prenuclei and weak nuclei on implicative semilattices

From now on, we assume that

$$\begin{aligned} A \text { is an implicative semilattice with top element } \top \text { and residuation } \rightarrow . \end{aligned}$$

Nuclei on frames play an important role in pointfree topology. A comprehensive investigation of that concept, also in the non-complete case, is due to Macnab [4, 32, 33], who called nuclei on Heyting algebras modal operators and gave a nice description of them by a single equation, which extends to implicative semilattices: a map $g$ on A is a nucleus iff

$$\begin{aligned}x\!\rightarrow \!gy = gx \!\rightarrow \!gy. \end{aligned}$$

Any nucleus g on A fulfils the inequality (which may fail to be an equality)

$$\begin{aligned}g(x\!\rightarrow \!y) \le gx \!\rightarrow \!gy, \end{aligned}$$

A subset of A is said to be left residually closed or l-closed if it is invariant under all the unary residuation operations $\alpha _a$ with $\alpha _a x = a\!\rightarrow \!x$. If A is regarded as a general algebra $(A, \wedge , \top , \alpha )$ with $\alpha = (\alpha _a \,|\, a\in A)$ then the subalgebras are the l-closed subsemilattices. Köhler [31] calls them total subalgebras, while Picado, Pultr and Tozzi [40] call them merely ideals. In fact, in a general algebra with a distinguished binary operation (“a multiplication”) m, a left (m-)ideal is a subalgebra containing m(x, y) whenever it contains y. In order to avoid ambiguities, we call total subalgebras of implicative semilattices left $\rightarrow$-ideals or briefly l-ideals, referring to the binary residuation $\rightarrow \,$. The analogy to classical algebra is obvious; for example, the intuitionistic rule of importation and exportation, $(x\wedge y) \!\rightarrow \!z = x\!\rightarrow \!(y\!\rightarrow \!z)$, mimics the associative law $(xy)\mathop {*}z = x\mathop {*}(y\mathop {*}z)$; however, l-ideals are rarely two-sided. Order-theoretical filters (i.e. subsemilattices that are upsets) are l-ideals, but not conversely. Filters of implicative semilattices may be characterized as nonempty subsets F with $y\in F$ whenever $x\in F$ and $x\!\rightarrow \!y \in F$.

An l-ideal is said to be nuclear if it is a closure range. This terminology is justified by the next two propositions. In proofs we often write $x^y$ for $x\!\rightarrow \!y$.

Proposition 4.1

Let $g$ be any unary operation on A.

• (1) If $g$ is a subnucleus then $g(x\!\rightarrow \!y) \le x\!\rightarrow \!gy$, and $A_{g}\!$ is l-closed.

• (2) $g$ is a prenucleus iff $g$ is a preclosure operation with $g(x\!\rightarrow \!y)\le x\!\rightarrow \!gy$.

• (3) If $g$ is a prenucleus or a pseudonucleus then $A_{g}$ is an l-ideal.

• (4) $g$ is a nucleus iff $A_{g}$ is a nuclear l-ideal and $gx =\mathop {\textstyle \bigwedge }A_{g}x$.

Proof

(1) $x\wedge g(x^y) \le x^{yy} \wedge g(x^y) \le g(x^{yy}\wedge x^y) = gy\,$ gives $g(x^y)\le x^{gy}$. And for $y\in A_{g}$, $g(x^y) \le x^{gy}$ entails $g(x^y) \le x^y \le g(x^y)$, hence $x^y\in A_{g}$.

(2) If $g$ is a preclosure operation with $\,g(x^y)\le x^{gy}$ for all $x,y \in A$ then $y\le x^{\,x\wedge y}$ yields $gy \le g(x^{\,x\wedge y}) \le x^{\,g(x\wedge y)}$, hence $x\wedge gy \le g(x\wedge y)$.

(3) By (1), $A_{g}$ is l-closed. For $y,z\in A_{g}$ we get $y\wedge z = y \wedge gz \le g(y\wedge z) \le gy \wedge gz = y\wedge z$ if $g$ is a prenucleus, and $y\wedge z = gy \wedge gz = g( gy \wedge gz ) = g(y\wedge z)$ if $g$ is a pseudonucleus. In any case, $y\wedge z \in A_{g}$, whence $A_{g}$ is an l-ideal.

(4) We use the bijection between closure operations and closure ranges. By (3), if $g\in \mathrm{N }A$ then $A_{g}$ is a nuclear l-ideal with $gx = \mathop {\textstyle \bigwedge }A_{g}x$. On the other hand, if that holds then $g$ is a closure operation (Lemma 2.2). For $a = g(x \wedge y)$, $x\wedge y \le a \in A_{g}$ entails $x^a \in A_{g}$, $gy \le g(x^a) = x^a$ and so $x\wedge gy \le a$. Thus, $g$ is a subnucleus and, being a closure operation, a nucleus. $\square$

We supplement Lemma 2.2 by the following construction of nuclei:

Proposition 4.2

For any l-closed subset C of A such that each of the sets $C\cap {\uparrow \!x}$ has a meet $mx$, the operation ${n}_C = m$ on A is a nucleus. Every nucleus $g$ comes in that manner from a unique nuclear l-ideal, namely $C = A_{g}$.

Proof

By Lemma 2.2, $m$ is a closure operation. For $z\in C$, $x\wedge y \le z$ implies

$y\le x^z \! \in C$, $my \le x^z$ and $x \mathop {\wedge } my \le z$.

Thus, $x \mathop {\wedge } my \le z$ for $z \in C \mathop {\cap } {\uparrow \!(x \mathop {\wedge } y)}$, and so $x\wedge my \le m(x\wedge y)$. The rest is clear by Proposition 4.1 (4). $\square$

Let us denote by ${{\mathcal {S}}}l A$ the (inclusion-ordered) collection of all

$$\begin{aligned} \text { l-domains }= \text { l-closed closure ranges }= \text { nuclear l-ideals } = \text { nuclear ranges.} \end{aligned}$$

Corollary 4.3

${{{\mathcal {N}}}}A$ is a semilattice dual to ${{\mathcal {S}}}l A$, and isomorphic to $\mathrm{N }A$ by virtue of the mutually inverse bijections $C \mathop {\longmapsto }{n}_C$ and $g \mathop {\longmapsto }A_g$.

This correspondence was observed by several authors [6, 32, 33, 40, 45], at least in the setting of Heyting algebras or locales. Macnab calls nuclear ranges in Heyting algebras modal subalgebras, while Picado, Pultr and Tozzi [40] speak of strong ideals. Note that in implicative semilattices satisfying the descending chain condition all l-ideals are nuclear.

In a bounded chain A, which is a Heyting lattice anyway, every closure operation is a nucleus, every weak closure operation is a pseudonucleus, every subset containing $\top$ is an l-ideal, every closure range is nuclear, and the union of two nuclear ranges is their join in ${{\mathcal {S}}}l A$ (meet in ${{{\mathcal {N}}}}A$). But ${{\mathcal {S}}}l A$ resp. ${{{\mathcal {N}}}}A$ need not be a lattice and neither pseudocomplemented nor dually pseudocomplemented, nor distributive.

Example 4.4

Consider the chain ${\mathbb {N}} = \omega ~{\diagdown }{\ }\{ 0 \}$ of positive integers and the bounded rational chain

$$\begin{aligned}A = \{ \pm \tfrac{1}{n} \,|\, n \in {\mathbb {N}}\} \simeq \omega \oplus \omega ^{op}, \end{aligned}$$

a simple example of a non-complete Heyting lattice. In the $\vee$-semilattice ${{\mathcal {S}}}l A$, which is dual to the semilattice ${{{\mathcal {N}}}}A$ of nuclear ranges, the set $\{ B,C\}$ with

$$\begin{aligned} B&= \{ \tfrac{1}{n} \,|\, n\in {{\mathbb {N}}}\} \cup \{ -\tfrac{1}{2n-1} \,|\, n\in {{\mathbb {N}}}\}\\ C&= \{ \tfrac{1}{n} \,|\, n\in {{\mathbb {N}}}\} \cup \{ -\tfrac{1}{2n} \,|\, n\in {{\mathbb {N}}}\} \end{aligned}$$

has no greatest lower bound. Indeed, each of the finite nuclear l-ideals

$$\begin{aligned}F_k = \{ \tfrac{1}{n} \,|\, n\le k\} \ (k\in {\mathbb {N}}) \end{aligned}$$

is a lower bound of $\{ B,C\}$, and their union is the filter

$$\begin{aligned}F = \{ \tfrac{1}{n} \,|\, n\in {{\mathbb {N}}}\} = B\mathop {\cap } C, \end{aligned}$$

which fails to be a closure range. The complementary l-ideal

$$\begin{aligned}D = \{ -\tfrac{1}{n} \,|\, n\in {{\mathbb {N}}}\} \cup \{ 1 \} \end{aligned}$$

is nuclear but has no pseudocomplement in ${{\mathcal {S}}}l A$, as F contains no greatest l-domain; and D has no pseudocomplement in ${{{\mathcal {N}}}}A$ either: the l-domains

$$\begin{aligned}C_k = F \cup \{ -\tfrac{1}{n} \,|\, n\ge k\} \ (k\in {\mathbb {N}}) \end{aligned}$$

fulfil $C_k \mathop {\cup } D = \! A$, but there is no least l-domain E satisfying $E \mathop {\cup } D = A$. The semilattice ${{{\mathcal {N}}}}A = ({{\mathcal {S}}}l A) ^{op}$ is not distributive: though B is contained in $A = C \cup D$, there are no l-domains $C'\subseteq C$ and $D'\subseteq D$ with $B = C' \cup D'$. This example also witnesses that pseudonuclei need not be nuclei, and that preimages of nuclear l-ideals under injective l-morphisms need not be nuclear. Indeed, the map $g$ with $gx = 1$ for $x\in D$ and $gx = x$ otherwise is easily seen to be a pseudonucleus but not a nucleus; and the map $f$ with $fx = x$ for $x > 0$ and $fx = x/2$ for $x < 0$ is an injective l-morphism. The range of $g$ is the filter F, which is an l-ideal but not a closure range. The range of $f$ is the l-domain C, and the preimage of the l-domain B under $f$ is the filter F.

Now, we are in a position to determine for many unary operations the least nucleus above or the greatest nucleus below them.

Theorem 4.5

For any $g\in A^A$ such that each of the sets $A_{g}x$ has a meet $mx$, the map $m\in A^A$ is a closure operation on A. Furthermore,

• (1) if $g$ is a subnucleus then $m$ is a nucleus,

• (2) if $g$ is a prenucleus then $m= {\overline{g}}$ is the least nucleus above $g$,

• (3) if $g$ is a weak nucleus then $m= g^{\circ }$ is the greatest nucleus below $g$.

Proof

The general closure claim holds by Proposition 2.6 (1).

(1) follows from Proposition 4.1 (1) and Proposition 4.2.

(2) By Proposition 2.6 (2), $m$ is the least closure operation above $g$.

(3) By Proposition 2.6 (3), m is the greatest closure operation below $g$. $\square$

Corollary 4.6

For each prenucleus on a frame there is a least nucleus above it, and for each weak nucleus on a frame there is a greatest nucleus below it.

The first of these two results was observed by Banaschewski [3], the second by Beazer and Macnab [4]. Combination of Theorem 4.5 with Theorems 2.7 and 2.8 leads to the following conclusions.

Theorem 4.7

• (1) A prenucleus $g$ on an implicative semilattice A is l-complete iff ${\overline{g}}$ exists and is the least nucleus above $g$; in that case, ${\overline{g}} x = \bot A_{g}x$.

• (2) A weak nucleus $g$ on an implicative lattice A is l-complete iff $g^{\circ }$ exists and is the greatest nucleus below $g$; in that case, $g^{\circ } x = \mathop {\textstyle \bigwedge }A_{g}x$.

5 The ABC of nuclei and nuclear ranges

As before, A always denotes an implicative semilattice. For a better understanding of the relationships between nuclei and their ranges it is helpful to focus on three specific kinds of l-ideals, called basic open, boolean and basic closed (cf. [4, 19, 40], and for the complete case [26, 27, 32, 38, 43]).

For a topology ${{\mathcal {O}}}$, regarded as a frame (complete Heyting lattice), each open set $U\in {{\mathcal {O}}}$ gives rise to an induced topology ${{\mathcal {O}}}_U = \{ U\cap V \,|\, V\in {{\mathcal {O}}}\}$, which is isomorphic to a sublocale (l-domain, nuclear l-ideal) of ${{\mathcal {O}}}$, viz.

$$\begin{aligned}{\mathfrak {a}}U = \{ U\!\rightarrow \!V = (U^\mathrm{c} \cup V)^{\circ } \,|\, V\in {{\mathcal {O}}}\} \end{aligned}$$

where $^\mathrm{c}$ means set-theoretical complementation and $^{\circ }$ topological interior. Motivated by these prototypical examples, consider for any element a of A the unary residuation ${\alpha }_a \!: A \mathop {\longrightarrow }A$, which is defined by

$$\begin{aligned}\alpha _a x = a^x = a\!\rightarrow \!x, \end{aligned}$$

adjoint to the unary meet operation $\lambda _a = a\wedge -$, and a nucleus by the rules $x\le a^x = a^{a^x}$ and $a^{x\wedge y} = a^x\wedge a^y$. Consequently, its range

$$\begin{aligned} {\mathfrak {a}}a = \{ a^x \,|\, x\in A\} = \{ x\in A \,|\, a^x = x \} = \{ x\in A \,|\, a^{xx} = \!\top \} \end{aligned}$$

is a nuclear l-ideal. Thus, the surjective corestriction of $\alpha _a$ is not only adjoint to the restriction of $\lambda _a$ to ${\mathfrak {a}}a$ but also coadjoint to the inclusion map of ${\mathfrak {a}}a$ in A. Moreover, $\alpha _a$ preserves $\rightarrow \,$. The following facts are found in [40].

Theorem 5.1

The map ${\mathfrak {a}}_A : A \mathop {\longrightarrow }{{\mathcal {S}}}l A, \, a \!\longmapsto \! {\mathfrak {a}}a$ satisfies ${\mathfrak {a}}(a \mathop {\wedge } b) = {\mathfrak {a}}a \mathop {\cap } {\mathfrak {a}}b$ and is an embedding preserving all existing joins and finite meets (though ${{\mathcal {S}}}l A$ need not be a $\wedge$-semilattice). Hence, $\alpha _A = \alpha$ is a dual embedding of A in $\mathrm{N }A$. The sets ${\mathfrak {a}}a$ are exactly those nuclear ranges whose nuclei preserve the operation $\rightarrow$ and have residual surjective corestrictions.

The second kind of nuclei to be considered are the maps $\beta _a \in A^A$ with

$$\begin{aligned}\!\beta _a x = x^{aa} = (x\!\rightarrow \!a)\!\rightarrow \!a \end{aligned}$$

(denoted by $w_a$ in Macnab’s work [4, 33, 43]). For an early proof that each $\beta _a$ is a nucleus see Varlet [45]. The range of $\beta _a$ is

$$\begin{aligned}{\mathfrak {b}}a = \{ x^a \,|\, x\in A\} = \{ y\in A \,|\, y = y^{aa}\}, \end{aligned}$$

which is therefore a nuclear l-ideal. However, the map ${\mathfrak {b}}_{A}$ from A to ${{{\mathcal {N}}}}A$ with ${\mathfrak {b}}_{A} a = {\mathfrak {b}}a$ is not isotone unless A is a boolean lattice (see Theorem 5.9). Nevertheless, ${\mathfrak {b}}_{A}$ has a universal property for so-called l-continuous maps, defined by the requirement that preimages of principal upsets are l-ideals (see [14, 16] for general background). If A has a bottom $\bot$ then ${\mathfrak {b}}\bot$ is the booleanization of A; that it is a boolean lattice is the content of the famous Glivenko-Frink Theorem [21, 24]. Parts of the next more general theorem are known (cf. [9, 44, 45]), at least in the complete case [26, 27, 38].

Theorem 5.2

For any $a\in A$, the range ${\mathfrak {b}}a$ of the nucleus $\beta _a$ is the least l-ideal containing a. The sets ${\mathfrak {b}}a$ are exactly those (nuclear) l-ideals which are boolean lattices. Thus, all l-ideals are unions of boolean ones.

Proof

$\top = a^a$ is the greatest and a the least element of ${\mathfrak {b}}a$, since $a =a^{aa}\in {\mathfrak {b}}a$ and $a\le x^a$ for all $x\in A$. The join of x and y in ${\mathfrak {b}}a$ is $x\vee _a y = (x^a \wedge y^a)^a$; indeed, $x,y \le (x^a \wedge y^a)^a$, and if $x,y\le z\in {\mathfrak {b}}a$ then $z^a \le x^a \wedge y^a$ and so $(x^a\wedge y^a)^a\le z^{aa} = z$. The residuation in ${\mathfrak {b}}a$ is induced from that in A. And as $x\vee _a x^a = (x^a \wedge x^{aa})^a = a^a = \!\top$, $x^a$ is the complement of x in ${\mathfrak {b}}a$.

Now let C be an arbitrary l-ideal of A. Then $a\in C$ implies $x^{aa}\in C$ for all $x\in A$, which shows that ${\mathfrak {b}}a$ is the least (nuclear) l-ideal containing a. And if C is a boolean lattice with $a = \!\bot C$ then, for each $y\in C$, the element $y^a\in {\mathfrak {b}}a \subseteq C$ coincides with the complement z of y in C, because $y\wedge z =a$ implies $z\le y^a$, and $y\vee z = \!\top$ (join in C) entails

$$\begin{aligned} y^a = y^a \wedge \!\top = (y^a \wedge y) \vee (y^a \wedge z) = a \vee (y^a \wedge z) \le z, \end{aligned}$$

hence $y^a = z$. As complements in C are unique, we get $y = y^{aa} \in {\mathfrak {b}}a$. Thus, $C = {\mathfrak {b}}a$. $\square$

${{{\mathcal {T}}}\! }A$, the closure system of all l-ideals, is always a frame [31, 40]; hence, complements in ${{{\mathcal {T}}}\! }A$ are pseudocomplements. From Theorem 5.2, we deduce:

Corollary 5.3

A subset ${{\mathcal {Y}}}$ of ${{\mathcal {S}}}l A$ has a meet in ${{\mathcal {S}}}l A$ iff $\,\mathop {\textstyle \bigcap }{{\mathcal {Y}}}\in {{\mathcal {S}}}l A$, and then $\mathop {\textstyle \bigcap }{{\mathcal {Y}}}$ is that meet. Thus, not only finite joins but also all existing meets and complements in ${{\mathcal {S}}}l A$ coincide with those in ${{{\mathcal {T}}}\! }A$.

The third kind of nuclei to be considered exist only under the proviso that there are enough binary joins. For each $a\in A$, the principal upset

$$\begin{aligned}{\mathfrak {c}}a = {\uparrow \!a} \end{aligned}$$

is an l-ideal but need not be nuclear, that is, a closure range. However, in $\vee$-semilattices, and only in these, each ${\mathfrak {c}}a$ is the range of the nucleus $\gamma _a$ with $\gamma _a x = a\vee x$. Consider the topological case of an open set U, that is, a member of a topology ${{\mathcal {O}}}$, regarded s a frame. Here, the nuclear range ${\mathfrak {a}}U$ is isomorphic to the induced topology ${{\mathcal {O}}}_U$, while the nuclear range ${\mathfrak {c}}U$ is isomorphic to the induced topology ${{\mathcal {O}}}_C$ on a closed subset C, namely the set-theoretical complement of U. Therefore, the sets ${\mathfrak {a}}a$ are generally called basic open and the sets ${\mathfrak {c}}a$ basic closed ; in the case of frames/locales, the ${\mathfrak {c}}a$’s form a closure system, and the prefix “basic” is omitted. The letters $\alpha$ and ${\mathfrak {a}}$ remind of the Greek $\alpha \nu o\iota \kappa \tau o\varsigma$ and the Latin $\mathfrak {apertus}$ for open, while $\gamma \varepsilon \nu \varepsilon \sigma \iota \varsigma$ is Greek for generation, and $\mathfrak {clusus}$ is Latin for closed, whence we chose the letters $\gamma$ and ${\mathfrak {c}}$. (In [4, 33, 43], v stands for $\alpha$ and u for $\gamma$, while in [27] u has the meaning of $\alpha$, and c is our $\gamma$; in [38] ${\mathfrak {o}}$ stands for ${\mathfrak {a}}$).

The complementarity between open and closed sets in spaces is reflected by the next proposition (for weaker statements see [4, 27, 33, 38, 40]).

Proposition 5.4

${\mathfrak {c}}a$ is the complement and so the pseudocomplement of ${\mathfrak {a}}a$ in ${{{\mathcal {T}}}\! }A$, hence also in ${{\mathcal {S}}}l A$ and in ${{{\mathcal {N}}}}A$ if A is a lattice.

Proof

For the least l-ideal $\{ \top \}$, one obtains ${\mathfrak {a}}a \cap {\mathfrak {c}}a = \{ \top \}$, because x lies in ${\mathfrak {a}}a \cap {\mathfrak {c}}a$ iff $a\le x = a^x$, which implies $x =a^x =\!\top$. If ${\mathfrak {a}}a \cup {\mathfrak {c}}a \subseteq C$ for some l-ideal C then each $x\in A$ satisfies $a^x\in {\mathfrak {a}}a \subseteq C$, $a^{xx} \in {\mathfrak {c}}a \subseteq C$, and therefore $x = a^x \wedge a^{xx} \in C$. Thus, A is the only upper bound of $\{ {\mathfrak {a}}a, {\mathfrak {c}}a\}$ in ${{{\mathcal {T}}}\! }A$. $\square$

Proposition 5.5

For any $a\in A$, the following conditions are equivalent:

• (a) $a\vee x$ exists for all $x\in A$.

• (b) ${\mathfrak {c}}a$ is a nuclear range.

• (c) ${\mathfrak {c}}a$ is the complement of ${\mathfrak {a}}a$ in ${{\mathcal {S}}}l A$ resp. in ${{{\mathcal {N}}}}A$.

• (d) ${\mathfrak {a}}a$ has a complement in ${{\mathcal {S}}}l A$ resp. in ${{{\mathcal {N}}}}A$.

Thus, A is a lattice iff for all $a\in A$, ${\mathfrak {a}}a$ and ${\mathfrak {c}}a$ are complementary elements of  ${{\mathcal {S}}}l A$, or equivalently, $\alpha _a$ and $\gamma _a$ are complementary elements of $\mathrm{N }A$.

Proof

(a)$\Leftrightarrow$(b): y is the join of a and x iff y is the least element of ${\mathfrak {c}}a$ above x.

(b)$\Leftrightarrow$(c)$\Leftrightarrow$(d): Use Corollary 5.3 and Proposition 5.4. $\square$

Macnab [33] showed that for any nucleus $g$ on a Heyting algebra the composite map $\beta _a g$ is a boolean nucleus determined by the value $ga$, and established a series of interesting equations for the boolean nuclei $\beta _a$ via $\gamma$. In the case of semilattices, the nuclei $\gamma _a$ need not exist. Instead, one has the identities displayed in the next two lemmas.

Lemma 5.6

For $a,c\in A$, the nucleus $\beta _{{\overline{c}}}$ with ${\overline{c}} = \beta _a c$ fulfils $\beta _{{\overline{c}}} x = (x^a\! \wedge c^a)^a$.

Proof

Put $b = c^a$. Then $b^a = {\overline{c}}$, $\ \beta _{{\overline{c}}}x = (x^{b^a})^{b^a} \! = ((x\wedge b)^a \wedge b)^a = (x^a \wedge b)^a$, since $(x \wedge b)^a \wedge x \wedge b \le a$ implies $(x \wedge b)^a \wedge b \le x^a \le (x \wedge b)^a$. $\square$

Lemma 5.7

Let a be an element of A and $g$ a prenucleus on A. Then:

• (1) $(gx)^a = x^a \wedge (ga)^a$ for $x\ge a$.

• (2) $\beta _a g\beta _a = \beta _b$ for $b = \beta _a ga$. Hence, $\beta _b = \beta _a \vee g$ in $\mathrm{N }A$ if $g\in \mathrm{N }A$.

Proof

(1) $a \le x\le gx$ yields $ga \le gx$, $(gx)^a \le x^a \wedge (ga)^a$. On the other hand, $gx \mathop {\wedge } x^a \le g(x \wedge x^a) \le ga$ implies $gx \mathop {\wedge } x^a \wedge (ga )^a \le a$, i.e. $x^a \wedge (ga)^a \le (gx)^a$.

(2) By (1) for $x^{aa}$ instead of x and Lemma 5.6 for $c = ga$, $x^{aa} \ge a$ yields $\beta _a g\beta _a x = (g(x^{aa}))^{aa} = (x^{aaa} \! \wedge (ga)^a)^a = (x^a \! \wedge (ga)^a)^a = \beta _{{\overline{c}}} x = \beta _b x$. $\square$

Lemma 5.7 together with Proposition 3.2 leads to a result that was obtained by Macnab [33] for the case of modal operators on Heyting algebras:

Theorem 5.8

Let $a \in A$, $g\in \mathrm{N }A$ and $b = \beta _a ga$. Then $g\ge \beta _a \Leftrightarrow g= \beta _b$. The boolean nuclear ranges of A form an upset in ${{{\mathcal {N}}}}A$, i.e. a downset in ${{\mathcal {S}}}l A$.

We finish this part with diverse characterizations of boolean lattices in terms of basic open, boolean, and basic closed nuclear ranges (cf. [27, 39]).

Theorem 5.9

For a Heyting semilattice A, the following are equivalent:

(b$_0$):

A is a boolean lattice.

(a$_1$):

The nuclear ranges are the basic open l-ideals.

(b$_1$):

The nuclear ranges are the boolean l-ideals.

(c$_1$):

The nuclear ranges are the basic closed l-ideals.

(a$_2$):

${\mathfrak {a}}_A : A \mathop {\longrightarrow }{{{\mathcal {N}}}}A$ is a dual isomorphism.

(b$_2$):

${\mathfrak {b}}_A : A \mathop {\longrightarrow }{{{\mathcal {N}}}}A$ is an isomorphism.

(c$_2$):

${\mathfrak {c}} _A : A \mathop {\longrightarrow }{{{\mathcal {N}}}}A$ is an isomorphism.

(a$_3$):

${\mathfrak {a}}_A$ is complementary to ${\mathfrak {b}}_A$ in $({{{\mathcal {N}}}}A)^A$.

(b$_3$):

${\mathfrak {b}}_A$ is isotone.

(c$_3$):

${\mathfrak {c}}_A$ coincides with ${\mathfrak {b}}_A$.

Proof

(b$_0$)$\Rightarrow$(c$_2$)$\Rightarrow$(c$_1$): If A is a boolean lattice then so is each ${\mathfrak {c}}a$; and conversely, each $C \in {{{\mathcal {N}}}}A$ satisfies $C = {\mathfrak {c}}b$ for $b = \bot C$ (indeed, for $b\le a$, one obtains $a = a \vee b = {\lnot \,}{\lnot \,}a \vee b = {\lnot \,}a \!\rightarrow \!b \in C$).

(c$_1$)$\Rightarrow$(a$_1$): By Proposition 5.5, (c$_1$) entails that A is a lattice, each C in ${{{\mathcal {N}}}}A$ is of the form ${\mathfrak {c}}b$, and its complement ${\mathfrak {a}}b$ equals some ${\mathfrak {c}}c$; by uniqueness of complements, it follows that $C = {\mathfrak {a}}c$. Thus, each $C\in {{{\mathcal {N}}}}A$ is basic open.

(a$_1$)$\Rightarrow$(a$_2$): By Theorem 5.1, ${\mathfrak {a}}_A$ is an embedding of A in ${{\mathcal {S}}}l A = ({{{\mathcal {N}}}}A)^{op}$.

(a$_2$)$\Rightarrow$(c$_1$) is shown by a similar argument as for (c$_1$)$\Rightarrow$(a$_1$), using Proposition 5.5 and the fact that (a$_2$) forces A to be a lattice, as A and ${{{\mathcal {N}}}}A$ are dually isomorphic semilattices.

(c$_1$)$\Rightarrow$(c$_3$): ${\mathfrak {b}}a = {\mathfrak {c}}b$ implies $a = \bot {\mathfrak {b}}a = \bot {\mathfrak {c}}b = b$, hence ${\mathfrak {b}}a = {\mathfrak {c}}a$.

(c$_3$)$\Rightarrow$(a$_3$) is clear by Proposition 5.4, and (a$_3$)$\Rightarrow$(b$_3$) is obvious.

(b$_3$)$\Rightarrow$(b$_2$): ${\mathfrak {b}}a \subseteq {\mathfrak {c}}a$ holds anyway, and if ${\mathfrak {b}}_A : A \mathop {\longrightarrow }{{{\mathcal {N}}}}A$ is isotone then $a \le b$ implies $b\in {\mathfrak {b}}b \subseteq {\mathfrak {b}}a$ ($\,{{{\mathcal {N}}}}A$ carries the reverse inclusion order!), whence ${\mathfrak {b}}_A$ agrees with the embedding ${\mathfrak {c}}_A$. Now, $\bot \le a$ implies $a\in {\mathfrak {b}}a \subseteq {\mathfrak {b}}\bot$. Thus, $A = {\mathfrak {b}}\bot$, and by Theorem 5.8 each nuclear range is boolean, so ${\mathfrak {b}}_A$ is an isomorphism.

The trivial implications (b$_2$)$\Rightarrow$(b$_1$)$\Rightarrow$(b$_0$) close the circuit

(b$_0$)$\Rightarrow$(c$_2$) $\Rightarrow$(c$_1$)$\Rightarrow$(a$_1$)$\Rightarrow$(a$_2$)$\Rightarrow$(c$_3$)$\Rightarrow$(a$_3$)$\Rightarrow$(b$_3$)$\Rightarrow$(b$_2$)$\Rightarrow$(b$_1$)$\Rightarrow$(b$_0$).

$\square$

It is quite surprising that isotonicity of the map ${\mathfrak {b}}_A$ resp. $\beta _A$ is already sufficient (and necessary) for A to be a boolean lattice. Many of the above implications and equivalences remain valid for arbitrary implicative semilattices (possibly missing a least element).