Difference hierarchies and duality with an application to formal languages

TL;DR

This paper explores the structure of difference hierarchies in lattices and their applications to formal languages, providing canonical decompositions and new insights into regular language representations.

Contribution

It introduces a canonical minimum length decomposition of Boolean elements in lattices relative to filter completion and extends this to applications in formal language theory.

Findings

Canonical difference chain decompositions exist for Boolean elements over lattices.

Boolean elements over (co-)Heyting algebras have canonical difference chains.

Regular languages can be characterized by Boolean combinations of universal sentences with specific predicates.

Abstract

The notion of a difference hierarchy, first introduced by Hausdorff, plays an important role in many areas of mathematics, logic and theoretical computer science such as descriptive set theory, complexity theory, and the theory of regular languages and automata. From a lattice theoretic point of view, the difference hierarchy over a bounded distributive lattice stratifies the Boolean algebra generated by it according to the minimum length of difference chains required to describe the Boolean elements. While each Boolean element is given by a finite difference chain, there is no canonical such writing in general. We show that, relative to the filter completion, or equivalently, the lattice of closed upsets of the dual Priestley space, each Boolean element over the lattice has a canonical minimum length decomposition into a Hausdorff difference. As a corollary each Boolean element over a (co-)Heyting algebra has a canonical difference chain. With a further generalization of this result involving a directed family of adjunctions with meet-semilattices, we give an elementary proof of the fact that a regular language is given by a Boolean combination of purely universal sentences using arbitrary numerical predicates if and only if it is given by a Boolean combination of purely universal sentences using only regular numerical predicates.

Equations

$$b=a_1-(a_2-(\dots (a_{n-1}-a_n)...))$$

$$\mathcal{B}{\Pi }_1[\mathcal{N}]\cap \mathsf{Reg}=\mathcal{B}{\Pi }_1[\mathsf{Reg}].$$

$$D^{-}\cong \mathsf{Clopen}(X).$$

$$x^{-}=\{b\in D^{-}\mid \exists a,a^{\prime }\in D\text{ with }a\in x\text{ and }{a}^{\prime }\notin x\text{ and }a-a^{\prime }\le b\},$$

$$\forall p\in P\forall q\in Q\left(f(p)\le q⟺p\le g(q)\right).$$

$$f(p)=\bigwedge \{q\in Q\mid p\le g(q)\}\text{and}g(q)=\bigvee \{p\in P\mid f(p)\le q\}.$$

$$\forall a,b,c\in D(a/b\le c⟺a\le b\vee c).$$

$$a/b=\bigwedge \{c\mid a\le b\vee c\}.$$

$$D^{-}⟶D,b\mapsto \lceil b\rceil =\bigwedge \{c\in D\mid b\le c\}.$$

$$a/b\le c⟺a\le b\vee c⟺a-b\le c.$$

$$d\le c⟺$$

$$⟺$$

$$\uparrow K=\bigcap \{W\subseteq X\mid K\subseteq W\text{ and W clopen upset}\}.$$

$$a_1-(a_2-(\dots -(a_{n-1}-a_n)...)),$$

$$a_1-(a_2-(\cdots -(a_{2m-1}-a_{2m})...))=(a_1-a_2)\vee (a_3-(a_4-(\cdots -(a_{2m-1}-a_{2m})\dots ))$$

$$a_1-(a_2-(\cdots -(a_{2m-1}-a_{2m})...))=\underset{n=1}{\overset{m}{\bigvee }}(a_{2n-1}-a_{2n})$$

$$\pi =\mathcal{P}(\mathbb{N}\cup \{x\})\cup \{C\cup \{y\}\mid C\subseteq \mathbb{N}\cup \{x\}\text{ is cofinite}\}.$$

$$V=W-W^{\prime }$$

$$V=\bigcup _{i=1}^m(U_i-W_i),$$

$$K_1=\uparrow V,K_2=\uparrow (\uparrow V-V).$$

$$K_i=\{x\in X\mid {\mathrm{deg}}_V(x)\ge i\}=\uparrow \{x\in X\mid {\mathrm{deg}}_V(x)=i\}.$$

$$V=G_1-(G_2-(\cdots -(G_{2p-1}-G_{2p})\dots )).$$

$$K_1\subseteq G_1,K_2\subseteq G_2\text{ and }{G}_1-G_2\subseteq K_1-K_2.$$

$$K_1=\uparrow V,K_{2i}=\uparrow (K_{2i-1}-V),\text{and}{K}_{2i+1}=\uparrow (K_{2i}\cap V),$$

$$K_n=\{x\in X\mid {\mathrm{deg}}_V(x)\ge n\}=\uparrow \{x\in X\mid {\mathrm{deg}}_V(x)=n\}.$$

$$V=\bigcup _{i=1}^m(K_{2i-1}-K_{2i})=K_1-(K_2-(\dots (K_{2m-1}-K_{2m})...)),$$

$$V=\bigcup _{i=1}^p(G_{2i-1}-G_{2i})=G_1-(G_2-(\dots (G_{2p-1}-G_{2p})...)).$$

$$K_{2n-1}\subseteq G_{2n-1},K_{2n}\subseteq G_{2n},\text{and}\bigcup _{i=1}^n(G_{2i-1}-G_{2i})\subseteq \bigcup _{i=1}^n(K_{2i-1}-K_{2i}).$$

$$\bigcup _{i=1}^n(G_{2i-1}^{\prime }-G_{2i}^{\prime })=\left(\bigcup _{i=1}^n(G_{2i-1}-G_{2i})\right)\cap K_{2n}\subseteq \left(\bigcup _{i=1}^n(K_{2i-1}-K_{2i})\right)\cap K_{2n}=\varnothing$$

$$K_{2n+1}=\uparrow (K_{2n}\cap V)=\uparrow V^{\prime }\subseteq G_{2n+1}^{\prime }\subseteq G_{2n+1}$$

Full text

1 Introduction

Hausdorff introduced the notion of a difference hierarchy in his work on set theory [12]. Subsequently, the notion has played an important role in descriptive set theory as well as in complexity theory. More recently, it has seen a number of applications in the theory of regular languages and automata [10, 2]. From a lattice theoretic point of view, the difference hierarchy over a bounded distributive lattice $D$ stratifies the Booleanization, $B$, of the lattice in question. The Booleanization of $D$ is the (unique up to isomorphism) Boolean algebra containing $D$ as a bounded sublattice and generated (as a Boolean algebra) by $D$. The stratification is made according to the minimum length of difference chains required to describe an element $b\in B$:

[TABLE]

where $a_{1}\geq a_{2}\geq\ldots\geq a_{n-1}\geq a_{n}$ are elements of $D$. One difficulty in the study of difference hierarchies is that in general elements $b\in B$ do not have canonical associated difference chains.

Stone duality [20] represents any bounded distributive lattice as the simultaneously compact and open subsets of an associated topological space known as the Stone dual space of the lattice. Priestley duality [19] is a rephrasing of this duality which uses the Stone space of the Booleanization equipped with a partial order to represent the lattice as the closed and open upsets of the associated Priestley space. Priestley duality provides an elucidating tool for the study of difference hierarchies. For one, the minimum length of difference chains for an element $b\in B$ has a nice description relative to the Priestley dual space $X$ of $D$ as the length of the longest chain of points $x_{1}\leq x_{2}\leq\dots\leq x_{n}$ in $X$ so that $x_{i}$ belongs to the clopen corresponding to $b$ if and only if $i$ is odd. Further, if we allow difference chains of closed upsets of the Priestley space, rather than clopen upsets, then every element $b\in B$ has a canonical difference chain which is of minimum length. In particular, if the lattice $D$ is a co-Heyting algebra, then the canonical difference chain of closed upsets consists of closed and open upsets and thus every $b\in B$ has a canonical difference chain in $D$. We present this material, which is closely related to work by Leo Èsakia on skeletal subalgebras of closure algebras [6], in Section 3.

In Section 5, we consider a situation where $B$ is equipped with a directed family of adjunctions. Using Stone duality in the form of canonical extensions, we generalize the results of Section 3. In turn, the results of Section 5 are used in Section 7 for an application in the setting of logic on words. More precisely, we give an elementary proof of the fact that a regular language is given by a Boolean combination of purely universal sentences using arbitrary numerical predicates if and only if it is given by a Boolean combination of purely universal sentences using only regular predicates. That is, expressed in a formula, we give an elementary proof of the equality

[TABLE]

This result was first proved by Macial, Péladeau and Thérien in [14]. For more details, see [22].

Before each of the main sections 3, 5, and 7, we include the background needed. Namely, in Section 2 we introduce the basics on lattices and duality, Section 4 is an introduction to canonical extensions, and Section 6 contains the preliminaries on recognition and logic on words.

2 Preliminaries on lattices and duality

Semilattices, lattices, and Boolean algebras

A semilattice is an idempotent commutative monoid. A bounded distributive lattice is a structure $(D,\wedge,\vee,0,1)$ so that $(D,\wedge,1)$ and $(D,\vee,0)$ both are semilattices and the operations $\vee$ and $\wedge$ distribute over each other. The relation $a\leq b$ if and only if $a\wedge b=a$ if and only if $a\vee b=b$ is a partial order on $D$ and one can recover the operations $\vee$ and $\wedge$ as the binary supremum and infimum, and the constants $1$ and [math] as the maximum and minimum elements of the poset, respectively. A Boolean algebra is a bounded distributive lattice equipped with a unary operation $\neg$ satisfying the identities $0=a\wedge\neg a$ and $1=a\vee\neg a$. The fundamental example of a Boolean algebra is a powerset with the set-theoretic operations, obtained from the inclusion order and the negation $\neg$ given by complementation. Thus any subset of a powerset closed under the bounded lattice operations is a bounded distributive lattice. We view the classes of bounded distributive lattices and of Boolean algebras as categories in which the morphisms are the algebraic homomorphisms, that is, the maps that preserve all the basic operations.

Stone duality

Stone duality [20] shows that all bounded distributive lattices are, up to isomorphism, of the above form. In fact, it does more than that as it provides a category of topological spaces dually equivalent to the category of bounded distributive lattices thus yielding an embedding of each bounded distributive lattice into a certain sublattice of the lattice of open sets of the corresponding space. We work here with the equivalent formulation due to Priestley [19], which uses ordered compact Hausdorff spaces rather than the non-Hausdorff spaces of the classical Stone duality.

Recall that a prime filter of a lattice $D$ is a non-empty upset $x$ of $D$ that is closed under binary meets and such that whenever $a\vee b\in x$, we have $a\in x$ or $b\in x$. The Priestley dual space of $D$ consists of the set $S(D)$ of prime filters of $D$ equipped with the topology generated by the sets $\widehat{a}=\{x\in S(D)\mid a\in x\}$ and their complements, where $a\in D$. Further, this space is ordered by inclusion of prime filters. One can show that the resulting ordered topological space $(X,\pi,\leq)$ is compact and totally order disconnected (TOD). That is, if $x\nleq y$ in $X$ then there is a clopen upset $V$ of $X$ with $x\in V$ and $y\not\in V$. Totally order disconnected compact spaces are called Priestley spaces and the appropriate structure preserving maps are the continuous and order preserving maps. Indeed, given a bounded lattice homomorphism, one can show that the preimage map restricted to prime filters is a continuous and order preserving map between the corresponding Priestley spaces. In the other direction, given a Priestley space $(X,\pi,\leq)$ the collection ${\sf UpClopen}(X,\pi,\leq)$ of subsets of $X$ that are simultaneously upsets and closed and open (called clopen) forms a lattice of sets. Further, given a continuous and order preserving map between Priestley spaces, the inverse image map restricted to the clopen upsets is a bounded lattice homomorphism. These functors account for the dual equivalence of the category of bounded distributive lattices and the category of Priestley spaces. On objects, this means that $D\cong{\sf UpClopen}(S(D))$ (via the map $a\mapsto\widehat{a}$) for any bounded distributive lattice $D$ and $X\cong S({\sf UpClopen}(X))$ (via the map $x\mapsto\{V\in{\sf UpClopen}(X)\mid x\in V\}$) for any Priestley space $X$. In addition, the double dual of a morphism, on either side of the duality, is naturally isomorphic to the original. For more details see [4, Chapter 11].

Booleanization

The Booleanization, or free Boolean extension, $D^{-}$, of a bounded distributive lattice $D$ is a Boolean algebra with a bounded lattice embedding $D\to D^{-}$, satisfying a universal property. Namely, given any bounded lattice homomorphism $h\colon D\to B$ into a Boolean algebra, there is a unique extension $h^{-}\colon D^{-}\to B$ so that the following diagram commutes:

D$$D^{-}$$B$$h$$h^{-}

The fact that Boolean algebras are bounded distributive lattices with an additional operation satisfying some set of equations implies, for general algebraic/categorical reasons, that a free Boolean extension where $D\to D^{-}$ is only guaranteed to be a homomorphism exists. The Booleanization has a few extra properties: first of all the homomorphism $D\to D^{-}$ is in fact an embedding. Secondly, given any embedding of $D$ into any Boolean algebra, the Boolean algebra generated by the image is isomorphic to $D^{-}$. The former property follows as soon as one shows the existence of at least one embedding of $D$ into some Boolean algebra. The latter property follows as one can show that a homomorphism of Boolean algebras which is injective on a distributive lattice that generates the domain must be injective.

It is well known that every distributive lattice can be embedded in a Boolean algebra. For one, it is a consequence of Stone duality since the Stone map $a\mapsto\widehat{a}$ is a lattice embedding into the powerset of the dual space. However, showing that the Stone map is an embedding requires a non-constructive principle, so one may ask whether such an embedding is available in a constructive manner. A first attempt was made by MacNeille [15], but there was a gap in his proof. A corrected version of MacNeille’s argument was subsequently provided by Peremans [17]. Later yet, Grätzer and Schmidt [11] and then Chen [3] also provided such constructive embeddings. In particular, Grätzer and Schmidt provide a simple proof by showing that any bounded distributive lattice $D$ embeds in the Boolean algebra of finitely generated congruences of $D$.

In the setting of Priestley duality, we have seen that $D$ is isomorphic to the lattice of clopen upsets of its dual space $X$. It thus follows that $D^{-}$ is isomorphic to the Boolean subalgebra of $\mathcal{P}(X)$ generated by ${\sf UpClopen}(X)$. One can show that this is the Boolean algebra of all clopen subsets of $X$. That is,

[TABLE]

In fact, if $(X,\pi,\leq)$ is the Priestley space of $D$, then $(X,\pi,=)$ is the Priestley space of $D^{-}$ (and $(X,\pi)$ is the Stone space of $D^{-}$). This is a consequence of the fact, due to Nerode [16], that each prime filter $x$ of $D$ extends to a unique prime filter $x^{-}$ of $D^{-}$ given by

[TABLE]

where $a-a^{\prime}$ is shorthand for $a\wedge\neg a^{\prime}$. Accordingly, given an element $b\in D^{-}$ we write $\widehat{b}$ for the corresponding clopen of $X$. As follows from Nerode’s result, in terms of the prime filters of $D$, we have $\widehat{b}=\{x\in S(D)\mid\exists a,a^{\prime}\in D\text{ with }a\in x,a^{\prime}\not\in x,\text{ and }a-a^{\prime}\leq b\}$.

Adjunctions

Let $P$ and $Q$ be posets. Given maps $f:P\to Q$ and $g:Q\to P$, we say that $(f,g)$ is an adjoint pair222Sometimes called a covariant Galois connection. provided

[TABLE]

Note that in this case, $f$ and $g$ uniquely determine each other since

[TABLE]

Accordingly, we also call $f$ the lower adjoint of $g$ and $g$ the upper adjoint of $f$. Thus a function has a lower, respectively upper, adjoint provided the appropriate minima, respectively maxima, exist. Also, one can show that lower adjoints preserve all existing suprema, while upper adjoints preserve all existing infima. In the case that the posets $P$ and $Q$ are complete lattices this gives a simple criterion for the existence of adjoints.

Proposition 2.1**.**

A map between complete lattices has a lower adjoint if and only if it preserves arbitrary meets, and it has an upper adjoint if and only if it preserves arbitrary joins.

Finally, we remark that if $(f,g)$ is an adjoint pair, then one can show that $fgf=f$ and $gfg=g$ and that the operation $c=gf$ is always a closure operator on $P$ in the sense that $p\leq c(p)=cc(p)$ holds for all $p\in P$. For the reader who would like to see the proofs of these facts we refer to [4, Chapter 7].

Heyting and co-Heyting algebras

Heyting algebras are the algebras for intuitionistic propositional logic in the same sense that Boolean algebras are the algebras for classical propositional logic. A Heyting algebra is a bounded distributive lattice equipped with an additional binary operation, $\to$, which models the intuitionistic implication. The order dual notion (obtained by interchanging the meet and join as well as the [math] and $1$ of the lattice) is called a co-Heyting algebra. We will focus on co-Heyting algebras here as this is more convenient for the sequel, but any result about the one notion has a corresponding order dual result about the other notion.

Definition 2.2**.**

A co-Heyting algebra is an algebra $(D,\wedge,\vee,0,1,/)$ such that $(D,\wedge,\vee,0,1)$ is a bounded distributive lattice and the operation $/$ is the lower adjoint of the operation $\vee$, in the sense that $((\ )/b,b\vee(\ ))$ is an adjoint pair for each $b\in D$. That is, the operation $/$ is uniquely determined by the following property:

[TABLE]

As stated above, these are precisely the order duals of Heyting algebras. That is, $(D,\wedge,\vee,0,1,/)$ is a co-Heyting algebra if and only if $(D,\vee,\wedge,1,0,\to)$ is a Heyting algebra, where $a\to b=b/a$. Notice that the adjunction property connecting $\vee$ and $/$ implies that the co-Heyting operation on a distributive lattice, if it exists, is unique and is given by

[TABLE]

Thus one may think of a co-Heyting algebra as a special kind of distributive lattice, namely one for which $\bigwedge\{c\mid a\leq b\vee c\}$ exists for all $a,b\in D$ and for which the thus defined operation satisfies the adjunction property in the definition of co-Heyting algebras. One can show that the class of co-Heyting algebras forms a variety, but here we are rather interested in a description of these algebras involving the Booleanization of the underlying bounded distributive lattice. It is not difficult to see that the following proposition states that a bounded distributive lattice $D$ is a co-Heyting algebra if and only if the inclusion of $D$ in its Booleanization has a lower adjoint. See [7, Proposition 3] for the corresponding fact for Heyting algebras. We include a proof for self-containment.

Proposition 2.3**.**

Let $D$ be a bounded distributive lattice. Then $D$ is a co-Heyting algebra if and only if there is a ceiling function

[TABLE]

Proof.

Let $D$ be a bounded distributive lattice and let $a,b,c\in D$. Consider these as elements, first of the co-Heyting algebra $D$ and then as elements of the Boolean algebra $D^{-}$. We have

[TABLE]

Accordingly, we see that $D$ is a co-Heyting algebra if and only if, the ceiling function above is well defined for all elements of $D^{-}$ of the form $a-b$ with $a,b\in D$. However, since the elements of $D^{-}$ are given by finite joins of elements of this form and since the ceiling function preserves joins (being a lower adjoint of the inclusion of $D$ in $D^{-}$) we can easily verify that this is sufficient to show that the ceiling function is globally defined by extending by join. To this end, let $d=\bigvee_{i=1}^{n}(a_{i}-b_{i})$ be an arbitrary element of $D^{-}$ and let $c\in D$, then we have

[TABLE]

so that $\lceil d\rceil:=\bigvee_{i=1}^{n}a_{i}/b_{i}$ yields the desired ceiling function. ∎

Since Heyting and co-Heyting algebras may be seen as certain bounded distributive lattices, it is useful to understand which Priestley spaces they correspond to. This was worked out by Èsakia [5] independently of Priestley’s work: a bounded distributive lattice $D$ is a Heyting algebra if and only if, for any clopen $V\subseteq X$ of the Priestley dual of $D$, the generated downset ${\downarrow}V$ is again clopen. This of course is equivalent to the order dual characterization of co-Heyting algebras but we include a proof to illustrate the correspondence between the algebraic and topological formulations.

Theorem 2.4** ([5]).**

Let $D$ be a bounded distributive lattice and $X$ its Priestley dual space. Then $D$ admits a co-Heyting structure if and only if for each $V\subseteq X$ clopen, ${\uparrow}V$ is again clopen. When this is the case, the map $\lceil\ \rceil\colon D^{-}\to D$ is naturally isomorphic to the map on clopen subsets of $X$ given by $V\mapsto{\uparrow}V$.

Proof.

This is a simple consequence of Proposition 2.3. Note that by total order disconnectedness of $X$, for any closed (and thus compact) $K\subseteq X$, we have

[TABLE]

Now we see easily that for $V\subseteq X$ clopen, there is a least clopen upset (i.e., element of $D$) above $V$ if and only if the collection \{W\subseteq X\mid V\subseteq W\text{ and W clopen upset}\} has a minimum and this, in turn, happens if and only if ${\uparrow}V$ is clopen. ∎

Closed subsets and closed upsets in Priestley spaces

We highlight a few useful facts about closed subsets and upsets in Priestley spaces. First we note that closed subspaces of Priestley spaces, that is, closed subsets equipped with the inherited topology and order, are again Priestley spaces. This is because the TOD property is inherited by any subspace of a TOD space, and compactness is inherited by any closed subspace of a compact space. In fact, if a Priestley space $X$ is dual to a bounded distributive lattice $D$, it is not difficult to see that the closed subspaces of $X$ correspond to the lattice quotients of $D$, cf. [4, Section 11.32].

The following fact about upsets of closed sets will be used extensively in the sequel. For a subset $S\subseteq P$ of a poset $P$, we use $\min(S)$ to denote the set of minimal elements of $S$.

Proposition 2.5**.**

Let $X$ be a Priestley space and $K\subseteq X$ a closed subset. Then, ${\uparrow}K={\uparrow}\min(K)$ and this is a closed subset of $X$.

Proof.

As seen in(2) in the proof of Theorem 2.4, ${\uparrow}K$ is a closed subset of $X$ whenever $K\subseteq X$ is closed. Now consider $X$ as the dual space of a bounded distributive lattice $D$. Then the points of $X$ are the prime filters of $D$. Let $x$ be any element of $K$ and let $C$ be a maximal chain of prime filters contained in $K$ with $x\in C$. Since $C$ is a chain, it is easy to show that $x_{0}=\bigcap_{x\in C}x$ is again a prime filter. Also, if $W=\widehat{a}$ is any clopen upset of $X$ with $K\subseteq W$, then $a\in y$ for all $y\in K$ and thus $a\in x_{0}$. It follows that $x_{0}\in W$ for all clopen upsets $W$ of $X$ with $K\subseteq W$ and thus $x_{0}\in{\uparrow}K$. Now by maximality of $C$ it follows that $x_{0}\in\min(K)$ and $x_{0}\leq x$. Thus ${\uparrow}K={\uparrow}\min(K)$. ∎

3 The difference hierarchy and closed

upsets

Let $D$ be a bounded distributive lattice and $D^{-}$ its Booleanization. Since $D^{-}$ is generated by $D$ as a Boolean algebra and because of the disjunctive normal form of Boolean expressions, every element of $D^{-}$ may be written as a finite join of elements of the form $a-b$ with $a,b\in D$. A fact, that is well known but somewhat harder to see is that every element of $D^{-}$ is of the form

[TABLE]

for some $a_{1},\dots,a_{n}\in D$. The usual proof is by algebraic computation and is not particularly enlightening. It is also a consequence of our results here. We begin with a technical observation which is straightforward to verify.

Proposition 3.1**.**

Let $B$ be a Boolean algebra and let $a_{1},\ldots,a_{2m}$ be a decreasing sequence of elements of $B$. Then, the following equality holds:

[TABLE]

where the join is disjoint, and by induction we obtain

**

[TABLE]

where the joinands are pairwise disjoint.

One problem with difference chain decompositions of Boolean elements over a bounded distributive lattice, which makes them difficult to understand and work with, is that, in general, there is no ‘most efficient’ such decomposition. We give an example of a Boolean element over a bounded distributive lattice that illustrates the problem of non-canonicity of difference chains.

Example 3.2**.**

We specify the lattice via its Priestley space. Consider $X={\mathbb{N}}\cup\{x,y\}$ equipped with the topology of the one point compactification of the discrete topology on ${\mathbb{N}}\cup\{x\}$. That is, the frame of opens of $X$ is:

[TABLE]

The space $(X,\pi)$ is compact since any open cover must contain a neighborhood of $y$ and it must be cofinite. Covering the remaining points only requires a finite number of opens.

The order relation on $X$ is as depicted. That is, the only non-trivial order relation in $X$ is $x\leq y$.

1$$2$$\dots$$y$$x

We argue that $(X,\pi,\leq)$ is TOD. Given $x_{1},x_{2}\in X$ with $x_{1}\nleq x_{2}$, we have two cases. Either $x_{2}\in{\mathbb{N}}\cup\{x\}$ and then $\{x_{2}\}^{c}$ is a clopen upset containing $x_{1}$ but not $x_{2}$, or $x_{2}=y$ and then $x_{1}\in{\mathbb{N}}$ and $\{x_{1}\}$ is a clopen upset containing $x_{1}$ but not $x_{2}$. It follows that $X$ is a Priestley space.

The clopen upsets of $X$ are the finite subsets of ${\mathbb{N}}$ and the cofinite subsets of $X$ containing $y$ and they form the lattice $D$ dual to $X$. Note that $V=\{x\}$ is clopen in $X$ and thus $V\in D^{-}$. On the other hand, any clopen upset $W$ of $X$ containing $V$ must be cofinite. We can write

[TABLE]

where $W^{\prime}=W-\{x\}$ is also a clopen upset of $X$. It follows that there is no canonical choice for $W$ as ${\uparrow}V=\{x,y\}$ is not open and thus not in $D$.

However, if we look for difference chains for $V$ relative to *the lattice of closed subsets of *$X$, then we have a canonical choice of difference chain, namely $V=K_{1}-K_{2}$ where $K_{1}={\uparrow}V$ and $K_{2}=K_{1}-V$. This is a completely general phenomenon as we shall see next. Finally, we observe that since $V$ is a clopen upset but ${\uparrow}V$ is not, $D$ is not a co-Heyting algebra.

In this section we show that we may write each element of the Booleanization of a bounded distributive lattice canonically as a difference of closed upsets (cf. Theorem 3.7), and that, in the case of a co-Heyting algebra, this provides a canonical difference chain of the form (3) for each element of its Booleanization (cf. Corollary 3.9).

Recall that a subset $S\subseteq P$ of a poset is said to be convex provided $x\leq y\leq z$ with $x,z\in S$ implies $y\in S$.

Definition 3.3**.**

If $P$ is a poset, $S\subseteq P$, and $p\in P$, then we say that $p_{1}<p_{2}<\dots<p_{n}$ in $P$ is an alternating sequence of length $n$ for $p$ (with respect to $S$) provided

1. (a)

$p_{i}\in S$ for each $i\in\{1,\dots,n\}$ which is odd; 2. (b)

$p_{i}\not\in S$ for each $i\in\{1,\dots,n\}$ which is even; 3. (c)

$p_{n}=p$.

Further, we say that $p\in P$ has degree $n$ (with respect to $S$), written $\deg_{S}(p)=n$, provided $n$ is the largest natural number $k$ for which there is an alternating sequence of length $k$ for $p$. In particular, if there is no alternating sequence for $p$ with respect to $S$ (i.e. if $p\in P-{\uparrow}S$) then $\deg_{S}(p)=0$. Notice that an element of finite degree is of odd degree if and only if it belongs to $S$. Also, if $S$ is convex, then every element of $S$ has degree $1$, while every element of ${\uparrow}S-S$ has degree $2$.

In general, there will be non-empty subsets of posets with respect to which no element has finite degree. However, that is not the case for clopen subsets of Priestley spaces.

Proposition 3.4**.**

Let $X$ be a Priestley space and $V$ a clopen subset of $X$. Then every element of $X$ has finite degree with respect to $V$. There are elements of degree [math] if and only if $\,{\uparrow}V\neq X$, and the positive degrees achieved with respect to $V$ form an initial segment of the positive integers.

Proof.

Any element of the Booleanization of a bounded distributive lattice $D$ may be written as a finite disjunction of differences of elements from $D$ (using disjunctive normal form). Accordingly, if $V$ is a clopen subset of a Priestley space $X$, then there is an $m$ so that we may write

[TABLE]

where $U_{i},W_{i}\subseteq X$ are clopen upsets of $X$. In particular, since each $U_{i}-W_{i}$ is convex, by the Pigeonhole Principle, there is no alternating sequence with respect to $V$ of length strictly greater than $2m$, and thus, every element of $X$ has degree at most $2m$ with respect to $V$.

Also, picking an alternating sequence for an element $x\in X$ whose length is the maximum possible, it is clear that the $k$-th element of such a sequence has degree $k$. Thus the set of positive degrees that are achieved form an initial segment of the positive integers. Finally, $x\in X$ has degree [math] if and only if $x\not\in{\uparrow}V$. ∎

Lemma 3.5**.**

Let $X$ be a Priestley space and $V$ a clopen subset of $X$. Define subsets of $X$ as follows:

[TABLE]

Then, for each $i\in\{1,2\}$, $K_{i}$ is closed and

[TABLE]

Proof.

By Proposition 3.4, every element of $X$ has a finite degree. Also if $x\leq y$, then it is clear that $\deg_{V}(x)\leq\deg_{V}(y)$. Furthermore, by Proposition 2.5, we have that ${\uparrow}K={\uparrow}\min(K)$ for any closed set $K$. Now since both $V$ and ${\uparrow}V-V$ are closed, it suffices to show that the elements of $\min(V)$ have degree $1$, and the elements of $\min({\uparrow}V-V)$ have degree $2$. Note that it is clear that $\deg_{V}(x)=1$ for any $x\in\min(V)$. Now suppose $x\in\min({\uparrow}V-V)$. Since $x\in{\uparrow}V$, there is $x^{\prime}\in V$ with $x^{\prime}\leq x$. Since $x\not\in V$, this is an alternating sequence of length $2$ for $x$. On the other hand, if $x_{1}<x_{2}<\dots<x_{n}=x$ is an alternating sequence for $x$, then $x_{2}\in{\uparrow}V-V$ and thus $x\not\in\min({\uparrow}V-V)$ unless $n=2$ and $x_{2}=x$. Thus $\deg_{V}(x)=2$ for any $x\in\min({\uparrow}V-V)$. ∎

Corollary 3.6**.**

Let $X$ be a Priestley space, $V$ a clopen subset of $X$, and $G_{1}\supseteq G_{2}\supseteq\dots\supseteq G_{2p}$ a sequence of closed upsets in $X$ satisfying

[TABLE]

If $K_{1}$ and $K_{2}$ are as defined in Lemma 3.5, then

[TABLE]

Proof.

By (4), we have $V\subseteq G_{1}$. Also, since $G_{1}$ is an upset we have $K_{1}={\uparrow}V\subseteq G_{1}$. Now, since $G_{1}-G_{2}\subseteq V$ we have $G_{1}-V\subseteq G_{2}$ and as $G_{2}$ is an upset, it follows that ${\uparrow}(G_{1}-V)\subseteq G_{2}$. Also, $K_{1}\subseteq G_{1}$ implies $K_{2}={\uparrow}(K_{1}-V)\subseteq{\uparrow}(G_{1}-V)$ and thus, $K_{2}\subseteq G_{2}$. In particular, we have $G_{1}-G_{2}\subseteq V-K_{2}\subseteq K_{1}-K_{2}$. ∎

An iteration of Lemma 3.5 leads to the main result of this section.

Theorem 3.7**.**

Let $X$ be a Priestley space and $V$ a clopen subset of $X$. Define a sequence of subsets of ${\uparrow}V$ as follows:

[TABLE]

for $i\geq 1$ (see Figure 1). Then, $K_{1}\supseteq K_{2}\supseteq\dots$ is a decreasing sequence of closed upsets of $X$ and, for every $n\geq 1$, we have

[TABLE]

In particular,

[TABLE]

where $2m-1=\max\{\deg_{V}(x)\mid x\in V\}$.

Proof.

First we note that if (5) holds, then (6) holds since $K_{2i-1}-K_{2i}$ will consist precisely of the elements of $V$ of degree $2i-1$ and since each element of $V$ has an odd degree less than or equal to the maximum degree achieved in $V$. For the first statement and for (5), the proof proceeds by induction on the parameter $i$ used in (6). For $i=1$, we have $K_{1}=K_{2i-1}={\uparrow}V$ and $K_{2}=K_{2i}={\uparrow}(K_{2i-1}-V)={\uparrow}({\uparrow}V-V)$ and thus $K_{1}\supseteq K_{2}$ are closed sets satisfying (5) by Lemma 3.5.

For the inductive step, suppose the statements hold for $n\leq 2i$ and notice that $K_{2i+1}={\uparrow}(K_{2i}\cap V)$ and $K_{2i+2}={\uparrow}(K_{2i+1}-V)$ are in fact the sets $K_{1}$ and $K_{2}$ of Lemma 3.5 when we apply it to the Priestley space $X^{\prime}=K_{2i}$ and its clopen subset $V^{\prime}=K_{2i}\cap V$. Thus, to complete the proof, it suffices to notice that, for every $x\in X^{\prime}$, we have $\deg_{V}(x)=\deg_{V^{\prime}}(x)+2n$. ∎

The corresponding generalization of Corollary 3.6 goes as follows, and it proves the canonicity of writing (6).

Proposition 3.8**.**

Let $X$ be a Priestley space and $V\subseteq X$ be a clopen subset of $X$. Let $G_{1}\supseteq G_{2}\supseteq\dots\supseteq G_{2p}$ be a decreasing sequence of closed upsets of $X$ satisfying

[TABLE]

Then, taking $(K_{i})_{i\geq 1}$ as defined in Theorem 3.7, we have $p\geq m$ and, for every $n\in\{1,\dots,p\}$, the following inclusions hold:

[TABLE]

Proof.

We proceed by induction on $n$. The case $n=1$ is the content of Corollary 3.6. Now suppose that (8) holds for a certain $n\in\{1,\dots,p\}$. As in the proof of Theorem 3.7, we consider the new Priestley space $X^{\prime}=K_{2n}$ and its clopen subset $V^{\prime}=X^{\prime}\cap V$. Then, setting $G_{i}^{\prime}=X^{\prime}\cap G_{i}$ for each $i\in\{1,\dots,p\}$, we obtain a decreasing sequence of clopen subsets of $X^{\prime}$ that form a difference chain for $V^{\prime}$. However, notice that by the induction hypothesis we have

[TABLE]

so that the first $2n$ sets do not contribute to the writing of $V^{\prime}$ as a difference. It follows that the sequence $G^{\prime}_{2n+1}\supseteq G^{\prime}_{2n+2}\supseteq\dots\supseteq G^{\prime}_{2p}$ is also a difference chain of closed upsets of $X^{\prime}$ for $V^{\prime}$. Now applying Corollary 3.6 to this sequence, we see that

[TABLE]

and

[TABLE]

We also obtain that $(G_{2n+1}-G_{2n+2})\cap K_{2n}=G^{\prime}_{2n+1}-G^{\prime}_{2n+2}\subseteq K_{2n+1}-K_{2n+2}$. On the other hand, by Theorem 3.7, we have

[TABLE]

We thus conclude that

[TABLE]

as required for the inductive step. ∎

We are now ready to derive the normal form for elements of $D^{-}$ whenever $D$ is a co-Heyting algebra. This is a straightforward consequence of Theorem 2.4, Theorem 3.7 and Proposition 3.8.

Corollary 3.9**.**

Let $D$ be a co-Heyting algebra and $b\in D^{-}$. Define a sequence of elements in $D$ (recall Proposition 2.3) as follows:

[TABLE]

for $i\geq 1$. Then, the sequence $\{a_{i}\}_{i\geq 0}$ is decreasing, and there exists $m\geq 1$ such that $a_{2m+1}=0$ and

[TABLE]

Moreover, for every other writing

[TABLE]

as a difference chain with $c_{1}\geq\dots\geq c_{2p}$ in $D$, we have $p\geq m$, $c_{i}\geq a_{i}$ for $i\in\{1,\dots,2p\}$, and for each $n\leq m$ we have $\bigvee_{i=1}^{n}(c_{2i-1}-c_{2i})\leq\bigvee_{i=1}^{n}(a_{2i-1}-a_{2i})$.

Notice that every finite distributive lattice is a co-Heyting algebra, so that the above corollary applies to any finite distributive lattice. Combined with the fact that every bounded distributive lattice is the direct limit of its finite sublattices and that the Booleanization is the direct limit of the Booleanizations of these finite bounded sublattices, we also have the following.

Corollary 3.10**.**

Every Boolean element over any bounded distributive lattice may be written as a difference chain of elements of the lattice.

As shown by Corollary 3.9, the case of a co-Heyting algebra is particularly simple. In order to be able to apply the ideas of this section in a broader setting, we need a generalization which is easier to work with in the pointfree setting afforded by canonical extensions.

4 Preliminaries on canonical extensions

Here we provide the required information on canonical extensions. For further details, please see [7] or [8].

Canonical extensions

Let $D$ be a bounded distributive lattice and $X$ its dual Priestley space. Then, Priestley duality implies that the Stone map

[TABLE]

is an embedding of $D$ into the complete lattice $\mathcal{U}(X,\leq)$ of upsets of the poset underlying $X$. An embedding into a complete lattice is called a completion, and canonical extension, first introduced by Jónsson and Tarski [13], comes about from the fact that the above completion can be uniquely characterized in abstract terms among all the completions of $D$. Indeed, it is the unique completion $e:D\hookrightarrow C$ (up to isomorphism) satisfying the following two properties:

1. (dense) Each element of $C$ is a join of meets and a meet of joins of elements in the image of $D$;
2. (compact) For $S,T\!\subseteq\!D$ with $\bigwedge\!e(S)\!\leq\!\bigvee\!e(T)$ in $C$, there are finite $S^{\prime}\!\subseteq\!S$ and $T^{\prime}\!\subseteq\!T$ with $\bigwedge\!e(S^{\prime})\!\leq\!\bigvee\!e(T^{\prime})$.

Thus, instead of working with the the dual space of a bounded distributive lattice $D$, we will work with its canonical extension, denoted $D^{\delta}$. It comes with an embedding $D\hookrightarrow D^{\delta}$, which is compact and dense in the above sense. As stated above this means that $D^{\delta}$ is isomorphic to $\mathcal{U}(X,\leq)$, where $X$ is the Priestley space of $D$, and that the embedding $e$ is naturally isomorphic to the Stone map $a\mapsto\widehat{a}$. In what follows, to lighten the notation, we will assume (WLOG) that the embedding $e$ is an inclusion so that $D$ sits as a sublattice in $D^{\delta}$.

Filter and ideal elements

Since $D$ sits in $D^{\delta}$ as the clopen upsets sit in $\mathcal{U}(X,\leq)$, where $X$ is the Priestley space of $D$, we see that the join-closure of $D$ in $D^{\delta}$ will correspond to the lattice of open upsets of $X$. One can show that these are in one-to-one correspondence with the ideals of $D$. Thus we denote the join-closure of $D$ in $D^{\delta}$ by $I(D^{\delta})$ and call the elements of $I(D^{\delta})$ ideal elements of $D^{\delta}$. Similarly the meet-closure of $D$ in $D^{\delta}$ will correspond to the lattice of closed upsets of $X$ and one can show that these are in one-to-one correspondence with the filters of $D$. Accordingly we denote the meet-closure of $D$ in $D^{\delta}$ by $F(D^{\delta})$ and call its elements filter elements of $D^{\delta}$. Note that, relative to the concepts of filter and ideal elements, the density property of $D^{\delta}$ states that every element of $D^{\delta}$ is a join of filter elements and a meet of ideal elements.

We are particularly interested in the closed upsets of $X$, and thus in $F(D^{\delta})$, since they provide canonical difference chains, as we have just seen in the previous section. We point out another abstract characterization of $F(D^{\delta})$ which will be useful to us. $F(D^{\delta})$ is the free down-directed meet completion of $D$ and thus it is uniquely determined by the following two properties [9, Proposition 2.1]:

1. (filter dense)    Each element of $F(D^{\delta})$ is a down-directed meet of elements from $D$;
2. (filter compact) For $S\!\subseteq\!F(D^{\delta})$ down-directed and $a\!\in\!D$, if $\bigwedge\!S\!\leq\!a$, then there is $s\!\in\!S$ with $s\!\leq\!a$.

Notice, that in the particular case of a Boolean algebra $B$, the order on the dual space is trivial and thus $B^{\delta}$ is isomorphic to the full powerset of the dual space $X$ of $B$. Also, the ideal elements of $B^{\delta}$ correspond to all the opens of $X$ while the filter elements of $B^{\delta}$ correspond to all the closed subsets of $X$.

A ceiling function at the level of canonical extensions

Consider a situation where we have a Boolean algebra $B$ and a bounded sublattice $D$ of $B$. We saw in Proposition 2.3, that if $D$ is a co-Heyting algebra and $B$ is its Booleanization, then there is a lower adjoint $\lceil\ \rceil:B\to D$. Here we will show, that on the level of canonical extensions any embedding has a lower adjoint with nice properties for filter elements.

It is a part of the theory of canonical extensions that the embedding of $D$ in $B$ extends to a complete embedding of the canonical extensions $D^{\delta}\hookrightarrow B^{\delta}$ [8, Theorem 3.2] which restricts to embeddings for the filter elements as well as for the ideal elements [8, Theorem 2.19]. Since the embedding is a complete embedding it has both an upper and a lower adjoint, see Proposition 2.1. We will be interested in the lower adjoint, which we will denote by $\overline{(\ )}\colon B^{\delta}\to D^{\delta}$. Thus we have, for $v\in B^{\delta}$,

[TABLE]

Since $D^{\delta}\hookrightarrow B^{\delta}$, we may think of $D^{\delta}$ as a (complete) sublattice of $B^{\delta}$ and of the embedding as the identity. If we do this, then we may see $\overline{(\ )}$ as a closure operator on $B^{\delta}$ taking values inside $D^{\delta}$, cf. the comment following Proposition 2.1. We are particularly interested in the restriction of this closure operator to the filter elements.

Proposition 4.1**.**

Let $B$ be a Boolean algebra and $D$ a bounded sublattice of $B$, and let $\overline{(\ )}\colon B^{\delta}\to D^{\delta}$ be the lower adjoint of the embedding $D^{\delta}\hookrightarrow B^{\delta}$. Then the following properties hold:

1. (a)

for each $u\in B^{\delta}$, $\overline{u}$ is the least element of $D^{\delta}$ which lies above $u$; 2. (b)

the map $\overline{(\ )}\colon B^{\delta}\to D^{\delta}$ sends filter elements to filter elements; 3. (c)

the map $\overline{(\ )}\colon F(B^{\delta})\to F(D^{\delta})$ preserves down-directed meets.

Proof.

We think of the embedding $D^{\delta}\hookrightarrow B^{\delta}$ as an inclusion. Thus (a) follows by the definition of adjoints: for $v\in D^{\delta}$ we have $\overline{u}\leq v$ if and only if $u\leq v$. For (b), let $v\in F(B^{\delta})$ and let $u\in D^{\delta}$ with $v\leq u$. Order dually to $F(D^{\delta})$, the free directed join completion of $D$ is given by the subframe of ideal elements, $I(D^{\delta})$. By the density property of canonical extension, every element of $D^{\delta}$ is a join of filter elements and a meet of ideal elements. For each $y\in I(D^{\delta})$ with $u\leq y$, we have $v\leq y=\bigvee\{a\in D\mid a\leq y\}$ and thus, by compactness, there is $a_{y}\in D$ with $v\leq a_{y}\leq y$. Now we get

[TABLE]

Since $\bigwedge\{a_{y}\mid y\in I(D^{\delta}),u\leq y\}\in F(D^{\delta})\subseteq D^{\delta}$, we have

[TABLE]

For (c), let $S$ be a down-directed subset of $F(B^{\delta})$ with $v=\bigwedge S$. By (b), $\overline{v}\in F(D^{\delta})$, and thus $\overline{v}=\bigwedge\{a\in D\mid v\leq a\}$. Let $a\in D$ with $v\leq a$, then $\bigwedge S\leq a$ and by the filter compactness property of $F(B^{\delta})$, there is $w_{a}\in S$ with $w_{a}\leq a$. Therefore we have

[TABLE]

On the other hand, since $v\leq w$ for each $w\in S$, by monotonicity of the closure operator, we also have $\overline{v}\leq\bigwedge\{\overline{w}\mid w\in S\}$ and thus the closure operator, restricted to $F(B^{\delta})$, preserves down-directed meets. ∎

Remark 4.2**.**

Notice that if $B$ is the Booleanization of $D$, and $X$ is the Priestley space of $D$, then $B^{\delta}\cong\mathcal{P}(X)$, $D^{\delta}\cong\mathcal{U}(X)$, and the closure operator $\overline{(\ )}:B^{\delta}\to D^{\delta}$ is, according to Proposition 4.1(a) simply the map that takes a $S\subseteq X$ to the upset ${\uparrow}S$. Furthermore, Proposition 4.1(b) tells us that if $K\subseteq X$ is closed then ${\uparrow}K$ is also closed. That is, it is the canonical extension incarnation of the second assertion in Proposition 2.5. We did not prove Proposition 4.1(c) in topological terms, but we could have. It says that if $\{W_{i}\}_{i\in I}$ is a down-directed family of closed subsets of a Priestley space, then

[TABLE]

A statement that is not true in general for down-directed families of subsets of a poset.

5 The difference hierarchy and directed families of

adjunctions

Let $B$ be a Boolean algebra, $I$ a directed partially ordered set, $\{S_{i}\}_{i\in I}$ an indexed family of meet-semilattices, and $\{f_{i}\colon B\leftrightarrows S_{i}\colon g_{i}\}_{i\in I}$ an indexed family of adjunctions satisfying:

1. (D.1)

${\rm Im}(g_{i})\subseteq{\rm Im}(g_{j})$ for all $i,j\in I$ with $i\leq j$; 2. (D.2)

$\bigcup_{i\in I}{\rm Im}(g_{i}):=D$ is a bounded sublattice of $B$.

For $b\in B$ and $i\in I$ we denote by $\overline{b}^{i}=g_{i}f_{i}(b)$. This is the closure operator on $B$ associated with the adjunction $f_{i}\colon B\leftrightarrows S_{i}\colon g_{i}$. We have the following relationship between these closure operators and the one given by $D$.

Proposition 5.1**.**

Let $B$, $D$ and $\{f_{i}\colon B\leftrightarrows S_{i}\colon g_{i}\}_{i\in I}$ be as specified above, then, for each $x\in B$, we have:

[TABLE]

where the meet is taken in $B^{\delta}$.

Proof.

For $x\in B$, since $\overline{(\ )}^{i}$ is a closure operator, we have $x\leq\overline{x}^{i}\in{\rm Im}(g_{i})\subseteq D\subseteq D^{\delta}$. Also $\overline{x}$ is the least element of $D^{\delta}$ above $x$. It follows that $\overline{x}\leq\overline{x}^{i}$ for each $i\in I$ and thus $\overline{x}\leq\bigwedge_{i\in I}\overline{x}^{i}$. On the other hand, by Proposition 4.1(b), since $x\in B\subseteq F(B^{\delta})$, we have $\overline{x}\in F(D^{\delta})$. That is, $\overline{x}=\bigwedge\{a\in D\mid x\leq a\}$. Now let $a\in D$ with $x\leq a$. Then, since $D=\bigcup_{i\in I}{\rm Im}(g_{i})$, there is $j\in I$ with $a\in{\rm Im}(g_{j})$. Now using the fact that $x\leq a$ and the monotonicity of $\overline{(\ )}^{j}$ we obtain

[TABLE]

We thus have

[TABLE]

Now fix $b\in B$ and define sequences $\{k_{n}\}_{n\geq 1}$ and $\{c_{n,i}\}_{n\geq 1}$, for each $i\in I$, as follows:

[TABLE]

[TABLE]

Remark 5.2**.**

Note that if $B=D^{-}$, then the sequence $\{k_{n}\}_{n\in\mathbb{N}}$ is exactly the canonical extension incarnation of the sequence $\{K_{n}\}_{n\in\mathbb{N}}$ given by $V=\widehat{b}$ of Theorem 3.7. We can say even more: By the universal property of the Booleanization $D^{-}$ of $D$, any embedding $D\hookrightarrow B$ factors through $D^{-}$. Thus it is the composition of two embeddings $D\hookrightarrow D^{-}$ and $D^{-}\hookrightarrow B$. The arguments given above thus apply to both of these and the canonical extensions of both of these embeddings have lower adjoints whose composition is the lower adjoint of the composition of the two embeddings. But if $b\in D^{-}$, then applying the lower adjoint of the embedding $D^{-}\hookrightarrow B$ to it, leaves it fixed. Accordingly, for any embedding $D\hookrightarrow B$, if $b\in B$ is in fact in $D^{-}$, then the sequence $\{k_{n}\}_{n\in\mathbb{N}}$ is exactly the canonical extension incarnation of the sequence $\{K_{n}\}_{n\in\mathbb{N}}$ given by $V=\widehat{b}$ of Theorem 3.7.

Lemma 5.3**.**

The following properties hold for the sequences as defined above:

1. (a)

$i\leq j$* implies $k_{n}\leq c_{n,j}\leq c_{n,i}$ for all $n\in\mathbb{N}$ and $i,j\in I$;* 2. (b)

$k_{n}=\bigwedge_{i\in I}c_{n,i}$* for all $n\in\mathbb{N}$.*

Proof.

Define $k_{0}=c_{0,i}=1$ for all $i\in I$. Also, define $b_{n}=b$ for $n$ odd and $b_{n}=\neg b$ for $n$ even then we have, for all $n\geq 1$, $k_{n+1}=\overline{k_{n}\wedge b_{n}}$ and similarly for the $c_{n,i}$. Proceeding by induction on $n$, we suppose (a) holds for $n\in\mathbb{N}$ and that $i\leq j$. Note that since ${\rm Im}(g_{i})\subseteq{\rm Im}(g_{j})\subseteq D$, we have $\overline{x}\leq\overline{x}^{j}\leq\overline{x}^{i}$ for all $x\in B$. Also, by the induction hypothesis $k_{n}\leq c_{n,j}\leq c_{n,i}$, and thus we have

[TABLE]

That is, $k_{n+1}\leq c_{n+1,j}\leq c_{n+1,i}$ as required.

For (b), again the case $n=0$ is clear by definition and we suppose $k_{n}=\bigwedge_{i\in I}c_{n,i}$. Then we have

[TABLE]

Now applying Proposition 4.1(c) and then Proposition 5.1, we obtain

[TABLE]

Now given $i,j\in I$, since $I$ is directed, there is $k\in I$ with $i,j\leq k$. By Lemma 5.3(a) we have $c_{n,k}\leq c_{n,i}$. Combining this with the fact that ${\rm Im}(g_{j})\subseteq{\rm Im}(g_{k})$ we obtain

[TABLE]

and thus

[TABLE]

We are now ready to state and prove our main theorem.

Theorem 5.4**.**

Let $B$ be a Boolean algebra, $I$ a directed partially ordered set, $\{S_{i}\}_{i\in I}$ an indexed family of meet-semilattices, and $\{f_{i}\colon B\leftrightarrows S_{i}\colon g_{i}\}_{i\in I}$ an indexed family of adjunctions satisfying (D.1) and (D.2), that is:

1. (D.1)

${\rm Im}(g_{i})\subseteq{\rm Im}(g_{j})$* for all $i,j\in I$ with $i\leq j$;* 2. (D.2)

$\bigcup_{i\in I}{\rm Im}(g_{i}):=D$* is a bounded sublattice of $B$.*

For each $b\in B$, define

[TABLE]

[TABLE]

If $b\in D^{-}\subseteq B$, then, there is $m\in\mathbb{N}$ and an $i\in I$ so that, for each $j\in I$ with $i\leq j$ we have

[TABLE]

Note that for $b\in D^{-}$ the fact that the first line of the conclusion holds is precisely the canonical extension reformulation of Theorem 3.7. See also Remark 5.2. The fact that the second line holds is the content of the following lemma.

Lemma 5.5**.**

Let $b,b^{\prime}\in B$ and $v\in B^{\delta}$ be such that $v\wedge k_{2l+1}=0$ and $b^{\prime}\leq k_{2l+2}$. Suppose $b=v\vee(k_{2l+1}-k_{2l+2})\vee b^{\prime}$. Then there is an $i\in I$ so that, for each $j\in I$ with $i\leq j$ we have $b=v\vee(c_{2l+1,j}-c_{2l+2,j})\vee b^{\prime}$.

Proof.

Since both $v$ and $k_{2l+1}-k_{2l+2}$ are below $\neg k_{2l+2}$ we have $b\leq\neg k_{2l+2}\vee b^{\prime}$, or equivalently, $b\wedge k_{2l+2}\leq b^{\prime}$. Now by Lemma 5.3(b) we have $b\wedge\bigwedge_{i\in I}c_{2l+2,i}\leq b^{\prime}$ and by compactness there is an $i_{1}\in I$ so that for all $j\in I$ with $i_{1}\leq j$ we have $b\wedge c_{2l+2,j}\leq b^{\prime}$, or equivalently, $b\leq\neg c_{2l+2,j}\vee b^{\prime}$. Now, for each $j\in I$ with $i_{1}\leq j$

[TABLE]

Consequently, for each $j\in I$ with $i_{1}\leq j$ we have $b=v\vee(k_{2l+1}-c_{2l+2,j})\vee b^{\prime}$. Now, since $c_{2l+2,j}=\overline{c_{2l+1,j}-b}^{j}\geq c_{2l+1,j}-b$, and thus, $b\geq c_{2l+1,j}-c_{2l+2,j}$, using also the inequality $c_{2l+1,j}\geq k_{2l+1}$ given by Lemma 5.3(a), we may deduce

[TABLE]

It then follows that for all $j\in I$ with $j\geq i_{1}$ we have

[TABLE]

Remark 5.6**.**

Notice that Corollary 3.10 could also be seen as a consequence of Theorem 5.4. Let $D$ be any bounded distributive lattice, $B$ its Booleanization. For each finite bounded sublattice $D^{\prime}$ of $D$, the embedding $f^{\prime}:D^{\prime}\hookrightarrow D\hookrightarrow B$ has an upper adjoint $g^{\prime}:B\to D^{\prime}$ given by $g^{\prime}(b)=\bigwedge\{a\in D^{\prime}\mid b\leq a\}=\min\{a\in D^{\prime}\mid b\leq a\}$ and this is a directed collection of adjunctions to which Theorem 5.4 applies thus yielding Corollary 3.10. In fact, in this way, we obtain more information as we see that the minimum length chain in $D$ is equal to the minimum length chain in $F(D^{\delta})$, or equivalently, in the lattice of closed upsets of the dual space of $D$. In turn, this is the same as the maximum length of difference chains in the dual with respect to the clopen corresponding to the given element.

In Section 7 we will give an application of the following consequence of Theorem 5.4, which needs its full generality.

Corollary 5.7**.**

Let $B$ be a Boolean algebra, $I$ a directed partially ordered set, $\{S_{i}\}_{i\in I}$ a family of meet-semilattices and $\{f_{i}\colon B\leftrightarrows S_{i}\colon g_{i}\}_{i\in I}$ an indexed family of adjunctions satisfying conditions (D.1) and (D.2). We denote by $D$ the bounded distributive lattice $\bigcup_{i\in I}{\rm Im}(g_{i})$. Let $B^{\prime}\leq B$ be a Boolean subalgebra closed under each of the closure operators $\overline{(\ )}^{i}=g_{i}f_{i}$ for $i\in I$. Then,

[TABLE]

where we view the Booleanization of any sublattice of $B$ as the Boolean subalgebra of $B$ that it generates.

Proof.

Since $D\cap B^{\prime}$ is contained in both of the Boolean algebras $D^{-}$ (also viewed as a subalgebra of $B$) and $B^{\prime}$, the Booleanization of $D\cap B^{\prime}$ is contained in their intersection.

For the converse, let $b\in D^{-}\cap B^{\prime}$. By Theorem 5.4, there exists an index $j$ so that $b$ can be written as a difference chain

[TABLE]

where $c_{1,j}=\overline{b}^{j}$, $c_{2n,j}=\overline{c_{2n-1,j}-b}^{j}$ and $c_{2n+1,j}=\overline{c_{2n,j}\wedge b}^{j}$, for $n\geq 1$. But then, by hypothesis it follows that $c_{1,j}\geq\dots\geq c_{2m,j}$ is a chain in $g_{j}f_{j}[B^{\prime}]\subseteq D\cap B^{\prime}$. Thus, $b$ belongs to $(D\cap B^{\prime})^{-}$. ∎

Remark 5.8**.**

We remark that the closure of $B^{\prime}$ under the closure operators $\overline{(\ )}^{i}$ for $i\in I$ implies that there is a family of adjunctions $\{f^{\prime}_{i}\colon B^{\prime}\leftrightarrows S^{\prime}_{i}=g_{i}^{-1}(B^{\prime})\colon g^{\prime}_{i}\}_{i\in I}$ obtained by considering the restrictions $f_{i}^{\prime}$ and $g_{i}^{\prime}$ of $f_{i}$ and $g_{i}$, respectively. Notice that the closure of $B^{\prime}$ under $g_{i}f_{i}$ implies that $f_{i}$ maps $B^{\prime}$ into $S^{\prime}_{i}$ as it is defined. Also, since upper adjoints preserve meets, $S_{i}^{\prime}$ is indeed a meet-semilattice. Finally, $D^{\prime}=\bigcup_{i\in I}{\rm Im}(g_{i}^{\prime})$ is the bounded distributive lattice $D\cap B^{\prime}$. Indeed, by definition of $S^{\prime}_{i}$ we have ${\rm Im}(g_{i}^{\prime})={\rm Im}(g_{i})\cap B^{\prime}$ and thus

[TABLE]

Notice that we also have ${\rm Im}(g_{i})\cap B^{\prime}=g_{i}f_{i}[B^{\prime}]$. The right-to-left inclusion is trivial. Conversely, let $b\in{\rm Im}(g_{i})\cap B^{\prime}$, say $b=g_{i}(s)$ for some $s\in S_{i}$. Then, we have $g_{i}f_{i}(b)=g_{i}f_{i}g_{i}(s)=g_{i}(s)=b$, where the second equality is well-known to hold for every adjoint pair $(f_{i},g_{i})$. Therefore, it follows that

[TABLE]

We give an example to show that the conclusion of Corollary 5.7 is by no means true in general.

Example 5.9**.**

Let $B=\mathcal{P}(\{a,b,c\})$ be the eight element Boolean algebra. Further, let $D$ be the bounded sublattice generated by $\{a\}$ and $\{a,b\}$ and let $B^{\prime}$ be the Boolean subalgebra generated by $\{b\}$. Then $B$ is, up to isomorphism, the Booleanization of $D$, and thus $D^{-}\cap B^{\prime}=B^{\prime}$, whereas $D\cap B^{\prime}=(D\cap B^{\prime})^{-}$ is the two-element Boolean subalgebra of $B$.

In order to formulate the application to the theory of formal languages, we will need some concepts from logic on words.

6 Preliminaries on formal languages and logic on

words

Formal languages

An alphabet is a finite set $A$, a word over $A$ is an element of the free $A$-generated monoid $A^{*}$, and a language is a set of words over some alphabet. For a word $w\in A^{*}$, we use $\left\lvert w\right\rvert$ to denote the length of $w$, that is, if $w=a_{1}\dots a_{n}$ with each $a_{i}\in A$, then we have $\left\lvert w\right\rvert=n$. Given a homomorphism $f:A^{*}\to M$ into a finite monoid $M$, we say that a language $L\subseteq A^{*}$ is recognized by $f$ if and only if there is a subset $P\subseteq M$ such that $L=f^{-1}(P)$, or equivalently, if $L=f^{-1}(f[L])$. The language $L$ is recognized by a finite monoid $M$ provided there is a homomorphism into $M$ recognizing $L$. Finally, a language is said to be regular if it is recognized by some finite monoid. Notice that the set of all regular languages forms a Boolean algebra. Indeed, if a language is recognized by a given finite monoid then so is its complement, and if $L_{1}$ and $L_{2}$ are recognized, respectively, by $M_{1}$ and $M_{2}$, then $L_{1}\cap L_{2}$ and $L_{1}\cup L_{2}$ are both recognized by the Cartesian product $M_{1}\times M_{2}$.

We present a technical result that will be used in Section 7. Given a monoid $M$, its powerset $\mathcal{P}(M)$ is a monoid when equipped with pointwise multiplication, that is, for subsets $P_{1},P_{2}\subseteq M$, we define

[TABLE]

It is easy to see that the preimage of a language recognized by $M$ under a homomorphism between free monoids is again recognized by $M$. This is not the case for direct images. However, the forward image under a length-preserving homomorphism between free monoids (i.e., a homomorphism mapping letters to letters) of a language recognized by $M$ is recognized by $\mathcal{P}(M)$ [21, Theorem 2.2]. This is an instance of the fact that modal algebras are dual to co-algebras for the Vietoris monad, enriched in the category of monoids, see [1]. However, since this finitary instance is quite simple to derive, for completeness, we include a proof. For a subset $P\subseteq M$, we denote

[TABLE]

Lemma 6.1**.**

Let $f:A^{*}\to B^{*}$ and $g:A^{*}\to M$ be homomorphisms, with $f$ length-preserving. Then, the map $h:B^{*}\to\mathcal{P}(M)$ defined by $h(w)=g[f^{-1}(w)]$ is also a homomorphism. Moreover, for every $P\subseteq M$, the equality $h^{-1}(\diamond P)=f[g^{-1}(P)]$ holds. In particular, if $L\subseteq A^{*}$ is a language recognized by $M$, then $f[L]\subseteq B^{*}$ is recognized by $\mathcal{P}(M)$.

Proof.

We first show that $h$ is a homomorphism. Let $v,w\in B^{*}$. Then, by definition, an element $m$ belongs to $h(vw)$ if and only if there exists $u\in f^{-1}(vw)$ such that $m=g(u)$. Since $f$ is length-preserving, there is a unique factorization of $u$, say $u=u_{1}u_{2}$, satisfying $u_{1}\in f^{-1}(v)$ and $u_{2}\in f^{-1}(w)$. Therefore, $h(uv)\subseteq h(u)h(v)$. The converse inclusion is trivial (in fact, it holds for every homomorphism $f$ between any two monoids).

Now, let $P\subseteq M$ and $w\in B^{*}$. We may deduce the following:

[TABLE]

Thus, $h^{-1}(\diamond P)=g[f^{-1}(P)]$ as claimed. ∎

Corollary 6.2**.**

Let $A$ and $B$ be alphabets and $f:A^{*}\to B^{*}$ a length-preserving homomorphism. Then the forward image under $f$ of a regular language over $A$ is a regular language over $B$.

We are mostly interested in languages that are defined by first-order formulas of logic on words. Accordingly, we now introduce this logic.

Syntax of first-order logic on words

Fix an alphabet $A$. We denote first-order variables by $x,y,z,x_{1},x_{2},\dots$. First-order formulas are inductively built as follows. For each letter $a\in A$, we consider a letter predicate, also denoted by $a$, which is unary. Thus, for any variable $x$, $a(x)$ is an (atomic) formula. A $k$-ary numerical predicate is a function $R:\mathbb{N}\to\mathcal{P}(\mathbb{N}^{k})$ satisfying $R(n)\subseteq\{1,\ldots,n\}^{k}$ for every $n\in\mathbb{N}$. That is, $R$ is an element of the Boolean algebra $\Pi_{n\in\mathbb{N}}\mathcal{P}(\{1,\ldots,n\}^{k})$. When we fix a set $\mathcal{R}$ of numerical predicates, we will assume it forms a Boolean subalgebra of $\Pi_{n\in\mathbb{N}}\mathcal{P}(\{1,\ldots,n\}^{k})$. Each $k$-ary numerical predicate $R$ and any sequence $x_{1},\dots,x_{k}$ of first-order variables define an (atomic) formula $R(x_{1},\ldots,x_{k})$. Finally, Boolean combinations of formulas are formulas, and if $\varphi$ is a formula and $x_{1},\ldots,x_{k}$ are distinct variables, then $\forall x_{1},\ldots,x_{k}\ \varphi$ is a formula. In order to simplify the notation, we usually also consider the quantifier $\exists$: the formula $\exists x_{1},\dots,x_{k}\ \varphi$ is an abbreviation for $\neg\forall x_{1},\dots,x_{k}\ \neg\varphi$. As usual in logic, we say that a variable occurs freely in a formula provided it is not in the scope of a quantifier, and a formula is said to be a sentence provided it has no free variables. Quantifier-free formulas are those that are Boolean combinations of atomic formulas.

Semantics of first-order logic on words

Let us fix an alphabet $A$ and a set of numerical predicates $\mathcal{R}$. To each non-empty word $w=a_{1}\dots a_{n}\in A^{*}$ with $a_{i}\in A$, we associate the relational structure $\mathcal{M}_{w}=(\{1,\dots,n\},A\cup\mathcal{R})$, given by the interpretation $a^{w}=\{i\in\{1,\dots,n\}\mid a_{i}=a\}$, for each $a\in A$, and $R^{w}=R(n)$, for each $R\in\mathcal{R}$. Models of first-order sentences are words, while models of formulas with free variables are the so-called structures. For a list of distinct variables ${\bf x}=(x_{1},\dots,x_{k})$, an ${\bf x}$-structure is a map $\{{\bf x}\}\to\mathcal{M}_{w}$ for some word $w\in A^{*}$, where $\{{{\bf x}}\}$ denotes the underlying set of ${\bf x}$. We identify maps from $\{{\bf x}\}$ to $\{1,\dots,\left\lvert w\right\rvert\}$ with $k$-tuples ${\bf i}=(i_{1},\dots,i_{k})\in\{1,\dots,\left\lvert w\right\rvert\}^{k}$. With a slight abuse of notation we write $\{1,\dots,\left\lvert w\right\rvert\}^{\bf x}$ for the set of all such maps. Given a word $w\in A^{*}$ and a vector ${\bf i}=(i_{1},\dots,i_{k})\in\{1,\dots,\left\lvert w\right\rvert\}^{\bf x}$, we denote by $w_{{\bf x}={\bf i}}$ the ${\bf x}$-structure mapping $x_{j}$ to $i_{j}$, for $j=1,\dots,k$. Moreover, if ${\bf x}=(x_{1},\dots,x_{k})$ and ${\bf y}=(y_{1},\dots,y_{\ell})$ are disjoint lists of variables, ${\bf i}=(i_{1},\dots,i_{k})\in\{1,\dots,\left\lvert w\right\rvert\}^{\bf x}$ and ${\bf j}=(j_{1},\dots,j_{\ell})\in\{1,\dots,\left\lvert w\right\rvert\}^{{\bf y}}$, then $w_{{\bf x}={\bf i},{\bf y}={\bf j}}$ denotes the ${\bf z}$-structure $w_{{\bf z}={\bf k}}$, where ${\bf z}=(x_{1},\dots,x_{k},y_{1},\dots,y_{\ell})$ and ${\bf k}=(i_{1},\dots,i_{k},j_{1},\dots,j_{\ell})$. The set of all ${\bf x}$-structures is denoted by $A^{*}\otimes{\bf x}$. Finally, the set of models $L_{\varphi({\bf x})}$ of a formula $\varphi({\bf x})$ having free variables in $\{{\bf x}\}$ is defined classically. For further details concerning logic on words, see [22, Chapter II]. The next example should help in understanding the semantics of first-order logic on an intuitive level.

Example 6.3**.**

The sentence $\varphi=\exists x,y\ (x<y\wedge a(x)\wedge b(y))$ is read: “there are positions $x$ and $y$ such that $x<y$, there is an $a$ at position $x$, and there is a $b$ at position $y$”. Thus, $\varphi$ defines the language $A^{*}aA^{*}bA^{*}$.

Formulas will always be considered up to semantic equivalence, even if not explicitly said. We denote by ${\bf FO}[\mathcal{N}]$ the set of all first-order sentences with arbitrary numerical predicates (up to semantic equivalence). For a set $\mathcal{R}$ of numerical predicates, ${\bf FO}[\mathcal{R}]$ denotes the set of first-order sentences using numerical predicates from $\mathcal{R}$. Notice that, as a Boolean algebra, ${\bf FO}[\mathcal{N}]$ is naturally equipped with a partial order, which in turn may be characterized in terms of semantic containment: $\varphi\leq\psi$ if and only if $L_{\varphi}\subseteq L_{\psi}$. Finally, for a subset of sentences $S\subseteq{\bf FO}[\mathcal{N}]$, we use $\mathcal{L}(S)$ to denote the set of languages $\{L_{\varphi}\mid\varphi\in S\}$. In particular, this yields an embedding of Boolean algebras ${\bf FO}[\mathcal{N}]\cong\mathcal{L}({\bf FO}[\mathcal{N}])\subseteq\mathcal{P}(A^{*})$.

Universal quantifiers as adjoints

Again, we fix a finite alphabet $A$ and a set of variables ${\bf x}$. We consider the projection map given by

[TABLE]

This gives rise, via the duality between sets and complete and atomic Boolean algebras, to the complete embedding of Boolean algebras

[TABLE]

This embedding, being a complete homomorphism between complete lattices, has an upper adjoint which we may call $\forall$ (and a lower adjoint $\exists$). These are given by

[TABLE]

and similarly

[TABLE]

As is well-known in categorical logic, $\forall$ and $\exists$ are the semantic incarnations of the classical universal and existential quantifiers, respectively. Explicitly, in the case of the universal quantifier, when $K=L_{\varphi({\bf x})}$ is definable by a formula $\varphi({\bf x})$, we have

[TABLE]

Recognition of languages of structures

We fix a list of distinct variables ${\bf x}=(x_{1},\dots,x_{k})$. Then, $2^{\bf x}$ is isomorphic to the powerset $\mathcal{P}({\bf x})$. There is a natural embedding of the set of all ${\bf x}$-structures into the free monoid $(A\times 2^{\bf x})^{*}$. Indeed, to an ${\bf x}$-structure $w_{{\bf x}={\bf i}}$, where ${\bf i}=(i_{1},\dots,i_{k})$, we may assign the word $(a_{1},S_{1})\dots(a_{n},S_{n})$, where $w=a_{1}\dots a_{n}$ with each $a_{i}\in A$ and, for $\ell\in\{1,\dots,n\}$, $S_{\ell}=\{x_{j}\in\{{\bf x}\}\mid i_{j}=\ell\}$. It is not hard to see that this mapping defines an injection $A^{*}\otimes{\bf x}\hookrightarrow(A\times 2^{\bf x})^{*}$. Moreover, an element $(a_{1},S_{1})\dots(a_{n},S_{n})$ of $(A\times 2^{\bf x})^{*}$ represents an ${\bf x}$-structure under this embedding precisely when $\{S_{\ell}\mid\ell\in\{1,\dots,n\},\ S_{\ell}\neq\emptyset\}$ forms a partition of ${\bf x}$. From hereon, we view $A^{*}\otimes{\bf x}$ as a subset of $(A\times 2^{\bf x})^{*}$ without further mention.

The identification $A^{*}\otimes{\bf x}\subseteq(A\times 2^{\bf x})^{*}$ enables us to extend the notion of recognition to languages of structures as follows. Given $L\subseteq A^{*}\otimes{\bf x}$, we say that $L$ is recognized by a homomorphism $f:(A\times 2^{\bf x})^{*}\to M$ if and only if there is $P\subseteq M$ such that $L=f^{-1}(P)$, and we say that $L$ is recognized by a monoid $M$ if there is a homomorphism $(A\times 2^{\bf x})^{*}\to M$ recognizing $L$. Accordingly, a language of structures is regular provided it is recognized by a finite monoid.

An important observation and a well-known fact in logic on words is that, as a language over the alphabet $A\times 2^{\bf x}$, the set of all ${\bf x}$-structures, $A^{*}\otimes{\bf x}$, is a regular language. To simplify the notation, take ${\bf x}=(x)$ and let $N=\{0,1,n\}$ be the three-element monoid satisfying $n^{2}=0$. Then, the unique homomorphism $f:(A\times 2^{\bf x})^{*}\to N$ satisfying $f(a,\emptyset)=1$ and $f(a,\{x\})=n$ recognizes $A^{*}\otimes{\bf x}$ via $\{n\}$. The general case is handled in a similar manner. This provides a shortcut for proving regularity for languages of structures.

Proposition 6.4**.**

Let $L\subseteq A^{*}\otimes{\bf x}$ be a language of structures. Then $L$ is regular if and only if there exists a monoid homomorphism $f\colon(A\times 2^{\bf x})^{*}\to M$ into a finite monoid $M$ and a subset $P\subseteq M$ with $L=f^{-1}(P)\cap(A^{*}\otimes{\bf x})$.

Proof.

If $L$ is regular, then there exists a monoid homomorphism $f\colon(A\times 2^{\bf x})^{*}\to M$ onto a finite monoid $M$ and a subset $P\subseteq M$ with $L=f^{-1}(P)$. Thus, in particular, $L=f^{-1}(P)\cap(A^{*}\otimes{\bf x})$. Conversely, suppose there exists a monoid homomorphism $f\colon(A\times 2^{\bf x})^{*}\to M$ onto a finite monoid $M$ and a subset $P\subseteq M$ with $L=f^{-1}(P)\cap(A^{*}\otimes{\bf x})$. Now, let $g\colon(A\times 2^{\bf x})^{*}\to N$ be the monoid homomorphism onto a finite monoid, which recognizes the set of all structures, and let $Q\subseteq N$ be so that $A^{*}\otimes{\bf x}=g^{-1}(Q)$. Then the product map $f\times g\colon(A\times 2^{\bf x})^{*}\to M\times N$ recognizes $L$ using the subset $P\times Q$. And we conclude that $L$ is regular. ∎

Quantifier-free formulas

We now provide an algebraic characterization of languages definable by quantifier-free formulas. Consider a fixed list of distinct variables ${\bf x}=(x_{1},\dots,x_{k})$. We want a characterization of the languages of the form $L_{\varphi({\bf x})}$ for $\varphi({\bf x})$ a quantifier-free formula whose free variables are in $\{{\bf x}\}$. We first provide a set theoretic characterization of these languages. For this purpose, we say that $L\subseteq X$ is set theoretically recognized by $f\colon X\to Y$ provided there is a subset $P\subseteq Y$ with $L=f^{-1}(P)$.

For a vector of letters ${\bf a}=(a_{1},\dots,a_{k})\in A^{\bf x}$, we denote by ${{\bf a}}({\bf x})$ the conjunction $a_{1}(x_{1})\wedge\dots\wedge a_{k}(x_{k})$. If $w\in A^{*}$ is a word and ${\bf i}\in\{1,\dots,\left\lvert w\right\rvert\}^{\bf x}$, then we denote by $w({\bf i})$ the unique vector ${\bf a}$ for which $w_{{\bf x}={\bf i}}\models{\bf a}({\bf x})$. That is, if $w=a_{1}\dots a_{n}$ with each $a_{i}\in A$ and ${\bf i}=(i_{1},\dots,i_{k})$, then $w({\bf i})=(a_{i_{1}},\dots,a_{i_{k}})$.

Lemma 6.5**.**

Let $K\subseteq A^{*}\otimes{\bf x}$. Then, $K$ is given by a quantifier-free first-order formula over ${\bf x}$ if and only if is it set theoretically recognized by the map

[TABLE]

Proof.

Let $P\subseteq\mathbb{N}^{k+1}\times A^{k}$. For each ${\bf a}\in A^{k}$ and $n\in\mathbb{N}$, let

[TABLE]

Then $R^{\,{\bf a}}$ is a ($k$-ary) numerical predicate for each ${\bf a}\in A^{k}$ and it is not difficult to see that $c_{A}^{-1}(P)=L_{\varphi({\bf x})}$ for

[TABLE]

On the other hand, for $a\in A$ and $i\in\{1,\dots,k\}$

[TABLE]

and for $R\subseteq\mathbb{N}^{m+1}$ an $m$-ary numerical predicate and ${\bf y}$ a list of $m$ (not necessarily distinct) variables with $\{{\bf y}\}\subseteq\{{\bf x}\}$, we have

[TABLE]

where $R^{\prime}$ is the $k$-ary numerical predicate given by

[TABLE]

where $j_{s}=i_{t}$ if and only if $y_{s}=x_{t}$. ∎

Observe that the above proof implies that the quantifier-free formulas with free variables in ${\bf x}$ form a complete Boolean algebra (isomorphic to a powerset Boolean algebra) and that each of these quantifier-free formulas may be written as a finite disjunction of formulas of the very special form ${\bf a}({\bf x})\wedge R({\bf x})$.

Now, to obtain an algebraic characterization, let $\varepsilon\notin A$ be a new symbol and denote $A_{\varepsilon}=A\cup\{\varepsilon\}$. We consider the length-preserving homomorphism $\Theta_{\bf x}:(A\times 2^{\bf x})^{*}\to(A_{\varepsilon}\times 2^{\bf x})^{*}$ given by

[TABLE]

Notice that, given ${\bf x}$-structures $v_{{\bf x}={\bf i}}$ and $w_{{\bf x}={\bf j}}$, we have

[TABLE]

Using this observation, it is straightforward to show:

Lemma 6.6**.**

The following diagrams are well defined and commute

(A\times 2^{\bf x})^{*}$$(A_{\varepsilon}\times 2^{\bf x})^{*}$$A^{*}\otimes{\bf x}$$A_{\varepsilon}^{*}\otimes{\bf x}$$\mathbb{N}^{k+1}\times A^{k}$$\mathbb{N}^{k+1}\times A_{\varepsilon}^{k}$$\Theta_{\bf x}$$\Theta_{\bf x}|_{(\ )\otimes{\bf x}}$$e$$c_{A}$$c_{A_{\varepsilon}}

where $e$ is the natural inclusion obtained from the inclusion of $A$ in $A_{\varepsilon}$. Furthermore, $\Theta_{{\bf x}}^{-1}(A^{*}_{\varepsilon}\otimes{\bf x})=A^{*}\otimes{\bf x}$ and the restriction of $c_{A_{\varepsilon}}$ to $\Theta_{{\bf x}}[A^{*}\otimes{\bf x}]$ is a bijection onto $e[\mathbb{N}^{k+1}\times A^{k}]$.

As an immediate consequence, we have:

Corollary 6.7**.**

Let $L\subseteq A^{*}\otimes{\bf x}$ be a language. Then, the following are equivalent:

1. (a)

$L$* is definable by a quantifier-free formula;* 2. (b)

$L=\Theta_{{\bf x}}^{-1}(\Theta_{{\bf x}}[L])$; 3. (c)

there is a subset $P\subseteq A_{\varepsilon}^{*}\otimes{\bf x}$ such that $L=\Theta_{{\bf x}}^{-1}(P)$.

The alternation hierarchy

The so-called alternation hierarchy has been widely considered in the literature and its decidability (beyond $k=2$) remains an open problem (see [18]). The hierarchy classifies formulas according to the minimum number of alternations of quantifiers that is needed to express them in prenex-normal formula, that is, in the form

[TABLE]

where $\varphi$ is a quantifier-free formula, $Q_{1},\dots,Q_{k}\in\{\forall,\exists\}$ and $Q_{\ell}=\forall$ if and only if $Q_{\ell+1}=\exists$ for each $\ell=1,\dots,k-1$. It is a well-known fact that for every first-order formula there is a semantically equivalent one in prenex-normal form. For $k\geq 1$ and a set of numerical predicates $\mathcal{R}$, $\Pi_{k}[\mathcal{R}]$ consists of all the sentences of ${\bf FO}[\mathcal{R}]$ that are semantically equivalent to a sentence of the form (12) where $Q_{1}=\forall$. In particular, we have $\Pi_{k}[\mathcal{R}]\subseteq\Pi_{\ell}[\mathcal{R}]$ whenever $k\leq\ell$. Similarly, $\Sigma_{k}[\mathcal{R}]$ denotes the set of all sentences that are semantically equivalent to a sentence of the form (12) with $Q_{1}=\exists$, and $\Sigma_{k}[\mathcal{R}]\subseteq\Sigma_{\ell}[\mathcal{R}]$ whenever $k\leq\ell$. It is not hard to see that both $\Pi_{k}[\mathcal{R}]$ and $\Sigma_{k}[\mathcal{R}]$ are closed under disjunction and conjunction but not under negation. In other words, $\Pi_{k}[\mathcal{R}]$ and $\Sigma_{k}[\mathcal{R}]$ are lattices, but not Boolean algebras. We denote by $\mathcal{B}\Pi_{k}[\mathcal{R}]$ and by $\mathcal{B}\Sigma_{k}[\mathcal{R}]$ the Boolean algebras generated by $\Pi_{k}[\mathcal{R}]$ and by $\Sigma_{k}[\mathcal{R}]$, respectively, that is,

[TABLE]

Clearly, we have $\mathcal{B}\Pi_{k}[\mathcal{R}]=\mathcal{B}\Sigma_{k}[\mathcal{R}]\subseteq\Pi_{k+1}[\mathcal{R}],\Sigma_{k+1}[\mathcal{R}]$.

In this paper we are only concerned with the first level of this hierarchy. For notational convenience, we will work with the fragment $\Pi_{1}[\mathcal{N}]$, although everything we prove for $\Pi_{1}[\mathcal{N}]$ admits a dual statement for $\Sigma_{1}[\mathcal{N}]$. We use ${\sf Reg}$ to denote both the subset of ${\bf FO}[\mathcal{N}]$ defining regular languages, and the subset of numerical predicates for which the associated language of structures $L_{R({\bf x})}$, where ${\bf x}$ is a list of distinct variables of the same length as the arity of $R$, is regular over the alphabet $A\times 2^{\bf x}$ as discussed under recognition of languages of structures. For $k>1$, it is an open problem to determine whether the following equality holds:

[TABLE]

Using the results of Section 5, we provide an elementary proof of the case $k=1$, which was first proved in [14].

Every formula of $\Pi_{1}[\mathcal{R}]$ is of the form $\psi=\forall{{\bf x}}\ \varphi({\bf x})$, for some quantifier-free formula $\varphi({\bf x})$. Inside $\Pi_{1}[\mathcal{R}]$, we classify formulas according to the size of ${\bf x}$: we let $\Pi_{1}^{d}[\mathcal{R}]$ consist of all equivalence classes of such formulas for which there is a representative $\psi$ for which ${\bf x}$ has $d$ variables. We remark that, $\Pi_{1}^{d}[\mathcal{R}]$ is closed under conjunction, since the formulas $\forall{\bf x}\ \varphi({\bf x})\wedge\forall{\bf x}\ \psi({\bf x})$ and $\forall{\bf x}\ (\varphi({\bf x})\wedge\psi({\bf x}))$ are semantically equivalent. However, in general, $\Pi_{1}^{d}[\mathcal{R}]$ fails to be closed under disjunction.

Example 6.8**.**

Let $\varphi(x)=a(x)$ and $\psi(x)=b(x)$. Then, $\forall x\ \varphi(x)$ defines the language $a^{*}$, while $\forall x\ \psi(x)$ defines the language $b^{*}$ and thus these are both in $\Pi_{1}^{1}[\mathcal{N}]$. The disjunction $\forall x\ \varphi(x)\vee\forall x\ \psi(x)$ defines the language $a^{*}\cup b^{*}$, while $\forall x\ (\varphi(x)\vee\psi(x))$ defines the language $\{a,b\}^{*}$. Indeed, one can show that $\forall x\ \varphi(x)\vee\forall x\ \psi(x)$ is not in $\Pi_{1}^{1}[\mathcal{N}]$ while it is in $\Pi_{1}^{2}[\mathcal{N}]$ as witnessed by the sentence $\forall x_{1},x_{2}\ \left(\varphi(x_{1})\vee\psi(x_{2})\right)$.

7 An application to Logic on Words

In this section we combine Corollary 5.7 and Remark 5.8 to prove the equality

[TABLE]

The idea is the following. Combining the fact that universal quantification may be seen as an adjoint and our algebraic characterization of quantifier-free formulas we obtain a directed family of adjunctions on $\mathcal{P}(A^{*})$ with joint image equal to $\mathcal{L}(\Pi_{1}[\mathcal{N}])$ allowing us to fit into the setting of Theorem 5.4. Finally we show that these adjunction restrict correctly to the regular fragment thus allowing us to apply Corollary 5.7 and Remark 5.8, thereby concluding that (13) holds.

Universal quantification (as an adjoint) and recognition of quantifier-free formulas are based on the following two maps, respectively

[TABLE]

Dually this gives rise to

[TABLE]

In particular, we have a (correct) composition of adjunctions as follows

[TABLE]

That is,

[TABLE]

is an adjunction and, combining quantification as adjunction with the description of quantifier free formulas in Corollary 6.7(c), we have $L\subseteq A^{*}$ is in $\mathcal{L}(\Pi_{1}^{k}[\mathcal{N}])$ if and only if $L=\forall(\Theta_{\bf x}^{-1}(P))=g_{k}(P)$ for some $P\subseteq A_{\varepsilon}^{*}\otimes{\bf x}$. That is, $(f_{k},g_{k})$ is an adjunction with associated closure operator

[TABLE]

and ${\rm Im}(\,\lceil\ \ \rceil_{k})={\rm Im}(g_{k})=\mathcal{L}(\Pi_{1}^{k}[\mathcal{N}])$. Accordingly, we are in the situation of Theorem 5.4 with

[TABLE]

We now aim to apply Corollary 5.7 with $B^{\prime}=\mathcal{L}({\sf Reg})$, the Boolean algebra of all regular languages over the alphabet $A$. This is possible given the following fact.

Lemma 7.1**.**

For each $k\in\mathbb{N}$, if $L\subseteq A^{*}$ is regular, then so is $\lceil L\rceil_{k}$.

Proof.

Fix $k\in\mathbb{N}$ and suppose $L\subseteq A^{*}$ is regular. We proceed through the four maps whose composition defines $\lceil\ \ \rceil_{k}$.

Claim 1. $L\otimes{\bf x}$ is regular.

Note that if $\mu\colon A^{*}\to M$ is a finite monoid recognizing $L$, then $\pi^{*}\colon(A\times 2^{\bf x})^{*}\to A^{*}$ composed with $\mu$, where $\pi^{*}$ is the homomorphism extending the projection of $A\times 2^{\bf x}$ onto $A$, recognizes $L\otimes{\bf x}$ once we restrict to structures. By Proposition 6.4 it follows that $L\otimes{\bf x}$ is regular.

Claim 2. $\Theta_{\bf x}[L\otimes{\bf x}]$ is regular.

Here we note that $\Theta_{\bf x}\colon(A\times 2^{\bf x})^{*}\to(A_{\varepsilon}\times 2^{\bf x})^{*}$ is a length-preserving homomorphism so that $L\otimes{\bf x}$ regular implies $\Theta_{\bf x}[L\otimes{\bf x}]$ is regular by Corollary 6.2.

Claim 3. $\Theta_{\bf x}^{-1}(\Theta_{\bf x}[L\otimes{\bf x}])$ is regular.

This is immediate as the inverse image with respect to a homomorphism between free monoids of a regular language is always regular: If $\Theta_{\bf x}[L\otimes{\bf x}]$ is recognized by $f^{\prime}\colon(A_{\varepsilon}\times 2^{\bf x})^{*}\to M^{\prime}$ then the composition $f^{\prime}\circ\Theta_{\bf x}\colon(A\times 2^{\bf x})^{*}\to M^{\prime}$ recognizes $\Theta_{\bf x}^{-1}(\Theta_{\bf x}[L\otimes{\bf x}])$.

Claim 4. $\forall(\Theta_{\bf x}^{-1}(\Theta_{\bf x}[L\otimes{\bf x}]))$ is regular.

As observed in Section 6, the upper adjoint $\forall$ is given by $K\mapsto(\pi[K^{c}])^{c}$ where $\pi\colon A^{*}\otimes{\bf x}\to A^{*}$ is the restriction of $\pi^{*}\colon(A\times 2^{\bf x})^{*}\to A^{*}$ to structures. It follows that $\pi[K^{c}]=\pi^{*}[K^{c}\cap(A^{*}\otimes{\bf x})]$. Now, since $K=\Theta_{\bf x}^{-1}(\Theta_{\bf x}[L\otimes{\bf x}])$ is regular, $K^{c}$ is also regular and $K^{c}\cap(A^{*}\otimes{\bf x})$ is regular. Further, noting that $\pi^{*}$ is a length-preserving monoid homomorphism, it follows by Corollary 6.2 that $\pi^{*}[(\Theta_{\bf x}^{-1}(\Theta_{\bf x}[L\otimes{\bf x}]))^{c}\cap A\otimes{\bf x}]$ is regular. Finally, we conclude that its complement $\forall(\Theta_{\bf x}^{-1}(\Theta_{\bf x}[L\otimes{\bf x}]))=(\pi^{*}[(\Theta_{\bf x}^{-1}(\Theta_{\bf x}[L\otimes{\bf x}]))^{c}\cap A\otimes{\bf x}])^{c}$ is regular as required. ∎

As a consequence, Corollary 5.7 applies and we obtain:

Corollary 7.2**.**

Considering each of the following Booleanizations as subalgebras of $\mathcal{P}(A^{*})$, we have

[TABLE]

Finally, applying Remark 5.8 in this particular case, we see that

[TABLE]

The languages in $g_{k}f_{k}[\mathcal{L}({\sf Reg})]$ are exactly the languages $\lceil L\rceil_{k}$ for $L$ regular. By the proof of Lemma 7.1, we have that $\lceil L\rceil_{k}=\forall(\Theta_{\bf x}^{-1}(P))$ where $P=\Theta_{\bf x}[L\otimes{\bf x}]\subseteq(A_{\varepsilon}\times 2^{\bf x})^{*}$, and, by Claim 2 in particular, we have that $P$ is regular. That is, $\lceil L\rceil_{k}=L_{\forall{\bf x}\,\varphi({\bf x})}$ where the atomic formula $\varphi({\bf x})$ is regular, or equivalently, $\lceil L\rceil_{k}\in\mathcal{L}(\Pi_{1}[{\sf Reg}])$.

On the other hand if $\varphi({\bf x})$ is an atomic formula that is regular, then the arguments in Claims 3 and 4 show that $L_{\forall{\bf x}\,\varphi({\bf x})}$ is regular. Thus

[TABLE]

and we have

[TABLE]

Since we consider logical formulas up to semantic equivalence, we obtain the desired result:

Theorem 7.3**.**

The following equality holds:

[TABLE]