Topo-Canonical Completions of Closure Algebras and Heyting Algebras

Home , Complete Heyting algebra, Heyting algebra, Spectral space

Algebra univers. Online First DOI 10.1007/s00012-007-2032-2 c Birkh¨auser Verlag, Basel, 2007 Algebra Universalis

Topo-canonical completions of closure algebras and Heyting algebras

Guram Bezhanishvili, Ray Mines, and Patrick J. Morandi

Abstract. We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactiﬁcations, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal completions preserve no identities of Heyting algebras. We also characterize deﬁnable classes of topological spaces.

1. Introduction

Closure algebras were introduced by McKinsey and Tarski [34] to give an algebraic treatment of point-set topology. They showed that every closure algebra can be represented as a subalgebra of the closure algebra over a topological space. We refer to this as the McKinsey-Tarski representation of closure algebras. One of their main results states that the variety of closure algebras is generated by the closure algebra over any dense-in-itself metric separable space. In particular, the variety of closure algebras is generated by the closure algebra over the real line, the rational line, or the Cantor space. A relational representation of closure algebras was given by Jónsson and Tarski [31], where it was shown that every closure algebra is isomorphic to a subalgebra of the closure algebra over a quasi-ordered set. We refer to this as the Jónsson-Tarski representation of closure algebras. In [13] Esakia combined the Jónsson-Tarski representation of closure algebras with the Stone duality for Boolean algebras to develop the duality theory between the category of closure algebras and the category of special quasi-ordered Stone spaces, to which we refer as Esakia spaces. The Jónsson-Tarski representation gives rise to the concept of the canonical completion (also known as the perfect extension or the canonical extension) of a closure algebra, or more generally, of a Boolean algebra with operators, whose

Presented by I. Hodkinson. Received January 20, 2006; accepted in final form September 12, 2006. 2000 Mathematics Subject Classification: Primary: 06E25; Secondary: 06D20, 06E15, 54D35, 03B45. Key words and phrases: Closure algebras, Heyting algebras, topo-canonical completions, compactifications.

1 2 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers. theory was extensively developed in [31] and in the subsequent papers [32, 30, 21, 22, 23, 12, 24]. The generalization of the concept of canonical completions to lattices with operators can be found in [21, 18, 19, 17, 20]. The McKinsey-Tarski representation gives rise to another completion of a closure algebra, which we call the topo-canonical completion. One of the aims of this paper is to give a detailed account of topo-canonical completions of closure algebras and their relationship with canonical completions of closure algebras. We give the algebraic and categorical characterizations of topo-canonical completions of closure algebras. We link topo-canonical completions to canonical completions by showing that the McKinsey-Tarski topology, which gives rise to the topo- canonical completion, is the intersection of the Stone topology and the Alexandroff topology, the latter of which gives rise to the canonical completion. We combine the McKinsey-Tarski representation of closure algebras with the Stone duality for Boolean algebras to develop the duality theory between the category of closure algebras and the category of spaces equipped with both the Stone topology and the McKinsey-Tarski topology. We show that the latter category is isomorphic to the category of Esakia spaces, thus providing an alternative to the Esakia duality. We characterize the dual of the topo-canonical completion of a closure algebra in terms of the Salbany compactification. We generalize the notion of a spectral space to that of a quasi-spectral space and show that the Salbany compactification is quasi-spectral. We introduce the notion of topo-canonical varieties and show that the variety of all closure algebras is the only nontrivial topo-canonical variety of closure algebras. We also introduce the notion of topo-complete varieties and give several examples of topo-complete varieties whose algebras have interesting topological representations. Furthermore, we introduce the notion of definable classes of topological spaces and give a necessary and sufficient condition for a class of topological spaces to be definable. The notions of topo-canonical and topo-complete varieties, as well as definable classes of topological spaces were motivated by the corresponding notions of canonical and complete varieties and definable classes of relational structures that have their origin in modal logic (see, e.g., [8]). Finally, since there is a close connection between closure algebras and Heyting algebras [35, 37, 10, 9, 13, 14], all our results transfer to the Heyting algebra setting. Moreover, some of them can even be improved. For example, we show that the topo- canonical completion of a Heyting algebra is isomorphic to its ideal completion. Consequently, since the variety of all Heyting algebras is the only nontrivial topo- canonical variety of Heyting algebras, we obtain that ideal completions do not preserve any identities in the language of Heyting algebras. This is in contrast to the well-known result in lattice theory that ideal completions of lattices preserve all lattice identities (see, e.g., [26]). We also characterize the dual of the topo-canonical Topo-canonical completions of closure algebras and Heyting algebras 3 completion of a Heyting algebra H as the Banaschewski compactification, which is a spectral space, and show that it is the T0-reflection of the Salbany compactification.

2. Topo-canonical completions

2.1. Closure algebras.

Deﬁnition 2.1. [34] A closure algebra is a pair A = (B, C) such that B is a Boolean algebra and C is a unary operation on B satisfying the Kuratowski identities: (1) a ≤ Ca, (2) CCa ≤ Ca, (3) C(a ∨ b)= Ca ∨ Cb, (4) C0=0.

Suppose A = (B, C) is a closure algebra. We call C the closure operator on B. The interior operator I on B is deﬁned by Ia = −(C(−a)). There are two natural examples of closure algebras, one coming from topology and one from order. For a topological space X, let Cl denote the closure operator and Int denote the interior operator on the power set P(X). Then (P(X), Cl) is a closure algebra. Similarly, if (X, R) is a quasi-ordered set, we deﬁne R−1 [A]= {x ∈ X : ∃a ∈ A with xRa}. Then (P(X), R−1) is a closure algebra.

Deﬁnition 2.2. Suppose A = (B, C) and A′ = (B′, C′) are closure algebras and h: B → B′ is a Boolean algebra homomorphism. We call h a closure algebra homomorphism if C′ha = hCa for all a ∈ B, and a closure algebra semi-homomorphism if C′ha ≤ hCa for all a ∈ B. We denote by CA the category of closure algebras and closure algebra homomorphisms, and by CA the category of closure algebras and closure algebra semi-homomorphisms.

We point out that if f : X → Y is a continuous map (resp. open continuous map), then f −1 : (P(Y ), Cl) → (P(X), Cl) is a closure algebra semi-homomorphism (resp. closure algebra homomorphism). Similarly, for quasi-ordered sets (X, R) and (Y,S), if f : X → Y is order preserving (resp. p-morphism), then f −1 : (P(Y ),S−1) → (P(X), R−1) is a closure algebra semi-homomorphism (resp. closure algebra homomorphism), where we recall that an order-preserving map f : X → Y is a p- morphism if f(x)Sy implies there exists z ∈ X with xRz and f(z)= y. There are two well-known representations of closure algebras, due to McKinsey and Tarski [34] and Jónsson and Tarski [31], respectively. One is topological and the other is order-theoretic. Let A = (B, C) be a closure algebra and let uf(B) 4 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers. denote the set of ultrafilters of B. We recall that the Stone map φ: B →P(uf(B)) is given by φ(a)= {x ∈ uf(B) : a ∈ x} , and the McKinsey-Tarski topology τ on uf(B) is defined by letting {φ(Ca) : a ∈ B} be a basis of closed sets. Then At := (P(uf(B)), Cl) is a closure algebra. The Jónsson-Tarski quasi-order R on uf(B) is defined by xRy if and only if a ∈ y implies Ca ∈ x for each a ∈ B. Then Aσ := (P(uf(B)), R−1) is a closure algebra as well. Moreover, we have the following representation theorem for closure algebras. Theorem 2.3. Let A be a closure algebra. (1) [34, Thm. 2.4] φ: A → At is a closure algebra embedding. (2) [31, Thms. 3.10, 3.14] φ: A → Aσ is a closure algebra embedding. An abstract characterization of Aσ was given in [31], which we now describe. Suppose B and B′ are Boolean algebras with B′ complete, and suppose e: B → B′ is a Boolean algebra embedding. We recall that B′ is a dense completion of B if each element of B′ is a join of meets and also a meet of joins of elements of e[B]. Also, B′ is a compact completion of B if, whenever E and F are subsets of B with V e[E] ≤ W e[F ], then there exist finite subsets E0 ⊆ E and F0 ⊆ F such that ′ ′ V e[E0] ≤ W e[F0]. If B is a dense and compact completion of B, then we call B a canonical completion of B. It is known [31, pp. 908-910] that every Boolean algebra B has up to isomorphism a unique canonical completion Bσ, which is isomorphic to P(uf(B)). We will often think of B as a Boolean subalgebra of Bσ by identifying B with e[B]. If A = (B, C) is a closure algebra, then we can extend C to Bσ by the formula σ C (x)= W{V{Ca : a ∈ B and k ≤ a} : k is closed and k ≤ x}, where an element is closed if it is a meet of elements from B. Then it follows from [31, pp. 933-935] that Aσ is isomorphic to (Bσ, Cσ). We will refer to Aσ as the canonical completion of A. We wish to give an algebraic characterization of At. Definition 2.4. Let A = (B, C) be a closure algebra. We define Ct on Bσ by the formula t C (x)= V{Ca : a ∈ B and x ≤ Ca}. We call the pair (Bσ, Ct) the topo-canonical completion of A. The name topo-canonical completion is motivated by the following proposition. Topo-canonical completions of closure algebras and Heyting algebras 5

Proposition 2.5. At is isomorphic to (Bσ, Ct).

Proof. We already observed that there is a Boolean isomorphism h from Bσ to P(uf(B)). Then it follows from the deﬁnition of Ct on Bσ and the McKinsey- Tarski topology on uf(B) that

t h(C x)= h(V{Ca : a ∈ B and x ≤ Ca}) = T{hCa : a ∈ B and x ≤ Ca} = Cl(hx).

Remark 2.6. Comparing Aσ and At we observe that although the Boolean carrier of both Aσ and At is Bσ, the closure operators Cσ and Ct are diﬀerent. In fact, one can think of the topo-canonical completion as a mixture of the canonical completion and the upper MacNeille completion. Indeed, the Boolean carrier of the topo- canonical completion At = (Bσ, Ct) of A = (B, C) is the canonical completion Bσ of B, while the closure operator Ct is deﬁned as the closure operator of the upper MacNeille completion of A [27].

We now give a category-theoretic characterization of the canonical and topo- canonical completions of closure algebras. We recall that a closure algebra A = (B, C) is complete if B is complete as a Boolean algebra, and that it is completely additive if Wa∈I Ca = C Wa∈I a for any I ⊆ B. Also, a semi-homomorphism between two closure algebras is complete if it is complete as a Boolean algebra homomorphism.

Theorem 2.7. Let A = (B, C) be a closure algebra. (1) Suppose A′ = (B′, C′) is a complete and completely additive closure algebra, and h: A → A′ is a closure algebra semi-homomorphism. Then there is a unique complete closure algebra semi-homomorphism hσ : Aσ → A′ such that σ h |A = h. (2) Suppose A′ = (B′, C′) is a complete closure algebra, and h: A → A′ is a closure algebra semi-homomorphism. Then there is a unique complete closure algebra t t ′ t semi-homomorphism h : A → A such that h |A = h.

Proof. (1) It is known (see, e.g., [19, Sec. 3]) that hσ : Bσ → B′, deﬁned by

σ h x = W{V{ha : a ∈ B and k ≤ a} : k is closed and k ≤ x}, σ is the unique complete Boolean algebra homomorphism such that h |B = h. It suﬃces to show that hσ is a closure algebra semi-homomorphism. Indeed, since A′ is complete and completely additive and h is a closure algebra semi-homomorphism, 6 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers. we have

′ σ ′ C (h (x)) = C (W{V{ha : a ∈ B and k ≤ a} : k is closed and k ≤ x}) ′ = W{C V{ha : a ∈ B and k ≤ a} : k is closed and k ≤ x} ′ ≤ W{V{C ha : a ∈ B and k ≤ a} : k is closed and k ≤ x} ≤ W{V{hCa : a ∈ B and k ≤ a} : k is closed and k ≤ x} = hσ(Cσ(x)).

(2) We deﬁne ht to be hσ from (1) and show that ht is a closure algebra semi- homomorphism. Let x ∈ Bσ and a ∈ B. Note that if x ≤ Ca, then htx ≤ hCa. Therefore, since ht is complete and h is a closure algebra semi-homomorphism, we have

′ t ′ C h x ≤ C V{hCa : a ∈ B and x ≤ Ca} ′ ≤ V{C hCa : a ∈ B and x ≤ Ca} ≤ V{hCa : a ∈ B and x ≤ Ca} t = h V{Ca : a ∈ B and x ≤ Ca} = htCtx.

Let CAσ denote the category of complete and completely additive closure algebras and complete closure algebra semi-homomorphisms. Also let CAt denote the category of complete closure algebras and complete closure algebra semi-homomorphisms. It follows from Theorem 2.7.1 that Aσ is freely generated by A in the category CAσ, and so is determined uniquely up to a unique isomorphism commuting with the embedding from A. Similarly, by Theorem 2.7.2, At is freely generated by A in the category CAt, and so is determined uniquely up to a unique isomorphism commuting with the embedding from A. We show that the universal mapping property of Theorem 2.7 does not hold for CA-homomorphisms.

Example 2.8. Consider the ordinals ω and ω+1 with the usual order and (interval) topology. Note that as topological spaces, ω is discrete and ω + 1 is a Stone space. We denote by CO(ω +1) the Boolean algebra of clopen subsets of ω + 1 and deﬁne e: CO(ω + 1) →P(ω) by e(A)= A ∩ ω. Then it is easy to see that e is a Boolean algebra embedding. For A ⊆ ω + 1, let ↓A = {x ∈ ω + 1 : ∃a ∈ A with x ≤ a} denote the downset of A. Again, it is easy to verify that e(↓A) =↓e(A) for each A ∈ CO(ω + 1). Therefore, e is a closure algebra embedding of A = (CO(ω + 1), ↓) into B = (P(ω), ↓). Also, it is obvious that B is complete and completely additive. The unique extension eσ : Aσ → B (resp. et : At → B) of e to Aσ = (P(ω + 1), ↓) (resp. At = (P(ω + 1), Cl)) is given by eσ(A) = A ∩ ω (resp. et(A) = A ∩ ω). To see that this is not a closure algebra homomorphism it is enough to note that Topo-canonical completions of closure algebras and Heyting algebras 7

↓eσ({ω}) =↓({ω} ∩ ω) =↓∅ = ∅ while eσ(↓{ω}) = eσ(ω +1)= (ω + 1) ∩ ω = ω (resp. Cl et({ω}) = Cl({ω} ∩ ω)=Cl ∅ = ∅ while et(↓{ω})= ω). As a consequence of Theorem 2.7 we obtain that both CAσ and CAt are reflective subcategories of CA. Corollary 2.9. (1) The inclusion functor CAσ ֒→ CA has a left adjoint functor (−)σ : CA → CAσ. Consequently, CAσ is a reflective subcategory of CA. .The inclusion functor CAt ֒→ CA has a left adjoint functor (−)t : CA → CAt (2) Consequently, CAt is a reflective subcategory of CA. Proof. (1) That (−)σ is a well-defined functor follows from [31, Thm. 1.22] and [19, Sec. 3]. For A ∈ CA and B ∈ CAσ, define σ ϕ: homCA(A, B) → homCAσ (A , B) by ϕ(h) = hσ. It follows from Theorem 2.7.1 that ϕ is well-defined. To see that ϕ is a bijection, define ψ in the opposite direction by ψ(h) = h|A. Clearly ψ is well-defined, and it follows from Theorem 2.7.1 that ψ = ϕ−1. So ϕ is a bijection. The naturality of ϕ is straightforward and will be left to the reader. Therefore, σ : CA → CAσ is a left adjoint to the inclusion functor CAσ ֒→ CA, and so CAσ(−) is a reflective subcategory of CA. (2) That (−)t is a well-defined functor follows from [31, Thm. 1.22] and Theorem 2.7.2. For A ∈ CA and B ∈ CAt, define t ϕ: homCA(A, B) → homCAt (A , B) by ϕ(h)= ht. It follows from Theorem 2.7.2 that ϕ is a well-defined bijection. The naturality of ϕ is straightforward. Therefore, (−)t : CA → CAt is a left adjoint to .the inclusion functor CAt ֒→ CA, and so CAt is a reflective subcategory of CA Remark 2.10. Let CAσ denote the category of complete and completely additive closure algebras and complete closure algebra homomorphisms and let CAt denote the category of complete closure algebras and complete closure algebra homomorphisms. The analogue of Corollary 2.9 holds for neither CAσ nor CAt. Indeed, it follows from Example 2.8 that (−)σ : CA → CAσ is not a reflection since the appropriate universal mapping property fails, while Remark 3.5 below implies that (−)t : CA → CAt is not even a functor. This provides one of the main reasons for considering semi-homomorphisms rather than just closure algebra homomorphisms. The other reason for considering semi-homomorphisms is that they correspond to continuous maps, while closure algebra homomorphisms correspond to open continuous maps (see the paragraph after Definition 2.2 and Section 3 below). Let A = (B, C) be a closure algebra, τ be the McKinsey-Tarski topology, and R be the Jónsson-Tarski quasi-order on uf(B). We recall that the usual Stone topology 8 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

τS on uf(B) has {φ(a) : a ∈ B} as a basis of clopen sets, and that the specialization order Rτ of τ is given by xRτ y if x ∈ Cl(y). Lemma 2.11. The J´onsson-Tarski quasi-order R is the specialization order of the McKinsey-Tarski topology τ. Proof. We observe that xRy ⇔ (∀a ∈ B)(a ∈ y → Ca ∈ x) ⇔ (∀a ∈ B)(a ∈ y → x ∈ φ(Ca)) ⇔ x ∈ T{φ(Ca) : a ∈ y} ⇔ x ∈ T{φ(Ca) : Ca ∈ y} ⇔ x ∈ Cl(y)

⇔ xRτ y.

Let (X, R) be a quasi-ordered set. We recall that a subset U of X is an upset if x ∈ U and xRy imply y ∈ U. Let the Alexandroﬀ topology τR on X be the set of all upsets of X. We show that the McKinsey-Tarski topology is the intersection of the Stone and Alexandroﬀ topologies. This is a generalization of [2, Fact 3.6].

Theorem 2.12. τ = τS ∩ τR.

Proof. By definition, τ ⊆ τS and, by Lemma 2.11, R is the specialization order of τ. Therefore, τ ⊆ τR. Thus, τ ⊆ τS ∩ τR. For the reverse inclusion, let A be closed in both the Stone and Alexandroff topologies. Then A = T{φ(a) : A ⊆ φ(a)} and A = R−1[A]. The collection {φ(a) : A ⊆ φ(a)} is clearly a downward directed family under inclusion. Therefore, by the Esakia Lemma (see [13, Lem. 3] or [11, Lem. 10.27]) we have −1 −1 R T{φ(a) : A ⊆ φ(a)} = T{R [φ(a)] : A ⊆ φ(a)}. Thus, by [13, Lem. 2] we obtain −1 −1 −1 A = R [A]= R [T{φ(a) : A ⊆ φ(a)}]= T{R [φ(a)] : A ⊆ φ(a)} = T{φ(Ca) : A ⊆ φ(a)} = T {φ(Ca) : A ⊆ φ(Ca)} = Cl(A), which shows that A is closed in the McKinsey-Tarski topology. Example 2.13. We give a simple example demonstrating the interaction of the three topologies. Let X denote the ordinal ω +1 with its usual order ≤ and interval topology τS. Then ≤ is a partial order and τS is a Stone topology on X. We can think of X as the set of ultrafilters of the closure algebra A = (B, C), where B is the Boolean algebra of finite and cofinite subsets of ω, and C(Y )=↓Y for each Y ⊆ ω. Topo-canonical completions of closure algebras and Heyting algebras 9

Moreover, τ≤ = {↑x : x ∈ X} is the Alexandroﬀ topology, and τ = {↑x : x 6= ω} is the McKinsey-Tarski topology. Clearly all three topologies are diﬀerent, but we have τ = τ≤ ∩ τS.

2.2. Heyting algebras. We recall that a Heyting algebra is a bounded distributive lattice (H, ∧, ∨, 0, 1) with an extra binary operation → satisfying a ∧ b ≤ c if and only if a ≤ b → c. We denote by HA the category of Heyting algebras and Heyting algebra homomorphisms, and by HA the category of Heyting algebras and bounded distributive lattice homomorphisms. There is a close correspondence between closure algebras and Heyting algebras [35, 37, 10, 9, 13, 14]. For a closure algebra A = (B, C), we have that

HA := {−Ca : a ∈ B} = {Ia : a ∈ B} is a Heyting algebra, where a → b = −C(a − b)= I(−a ∨ b) for a,b ∈ H. Moreover, if h: A → A′ is a closure algebra homomorphism (resp. closure algebra semi-homomorphism), then the restriction of h to HA is a Heyting algebra homomorphism (resp. bounded distributive lattice homomorphism). Con- versely, if H is a Heyting algebra, then

AH =: (B(H), CH ) is a closure algebra, where B(H) is the free Boolean extension of H (see, e.g., [3, Sec. V.4]), and for a ∈ B(H) with n a = W (ai − bi) i=1 for some ai,bi ∈ H, CH is deﬁned by n CH (a)= − V (ai → bi). i=1 Moreover, if h: H → E is a Heyting algebra homomorphism (resp. bounded distributive lattice homomorphism), then the extension of h to B(H) is a closure algebra homomorphism (resp. closure algebra semi-homomorphism). This correspondence deﬁnes functors F: CA → HA and G: HA → CA, and F: CA → HA and G: HA → CA. Theorem 2.14. (1) G is left adjoint to F. (2) G is left adjoint to F. Proof. For (1) see [9, P. 27] or [13, Thm. 2]; (2) is similar to (1). 10 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

Similar to closure algebras, we have two representations of Heyting algebras, one topological and one order-theoretic. The topological representation goes back to Stone [39] and Tarski [40], while the order-theoretic representation goes back to Kripke [33]. Let X be a topological space. We denote by O(X) the Heyting algebra of open subsets of X with operations of ∪, ∩, and implication defined by U → V = Int(−U ∪ V )= −Cl(U − V ). Similarly, if (X, R) is a quasi-ordered set, we denote by U(X) the Heyting algebra of upsets of (X, R), with operations of ∪, ∩, and implication defined by U → V = −R−1(U − V ). For a Heyting algebra H we denote the set of prime filters of H by pf(H). We let ≤ be set inclusion and τ be the spectral topology on pf(H) whose basis of open sets is {φ(a) : a ∈ H}, where φ is the usual Stone map. We define Hσ := U(pf(H)) and Ht := O(pf(H)). Then the following theorem gives a topological and order-theoretic representation of Heyting algebras. Theorem 2.15. Let H be a Heyting algebra. (1) (see, e.g., [37, P. 140, Thm. 9.2]) φ: H → Ht is a Heyting algebra embedding. (2) (see, e.g., [11, Cor. 8.19]) φ: H → Hσ is a Heyting algebra embedding. Suppose H and C are Heyting algebras with C complete as a lattice. We call C a canonical completion of H if C is a dense and compact completion of H. It follows from [18, Sec. 2] that every Heyting algebra H has a unique, up to isomorphism, canonical completion, which is isomorphic to Hσ. Definition 2.16. For a Heyting algebra H, we call Ht the topo-canonical completion of H. The next theorem gives a lattice-theoretic characterization of topo-canonical completions. Theorem 2.17. The topo-canonical completion of a Heyting algebra is isomorphic to its ideal completion. Proof. Let H be a Heyting algebra and τ be the spectral topology on pf(H). It is sufficient to observe that the lattice of ideals of H is isomorphic to the lattice of open sets of (pf(H), τ), by, e.g., [26, P. 100, Lemma 3]. A category-theoretic characterization of Hσ and of Ht follows easily from the results of [19] and [29]. We recall that a frame is a complete Heyting algebra, and that a frame homomorphism is a bounded lattice homomorphism preserving Topo-canonical completions of closure algebras and Heyting algebras 11 arbitrary joins. We also recall that for a complete Heyting algebra C, an element p ∈ C is completely join-prime if for each E ⊆ C, whenever p ≤ W E, there exists e ∈ E with p ≤ e; and that C is completely join-prime generated if every element of C is a join of completely join-prime elements underneath it. Corollary 2.18. Let H be a Heyting algebra. (1) Suppose h: H → C is a bounded lattice homomorphism into a complete and completely join-prime generated Heyting algebra C. Then there is a unique σ σ σ complete lattice homomorphism h : H → C such that h |H = h. (2) Suppose h: H → F is a bounded lattice homomorphism into a frame F . Then t t t there is a unique frame homomorphism h : H → F such that h |H = h. Proof. (1) It is known from [19, Sec. 3] that hσ : Hσ → C defined by σ h x = W{V{ha : a ∈ H and k ≤ a} : k is closed and k ≤ x} is the desired complete lattice homomorphism. (2) follows from Theorem 2.17 and the universal mapping property for the ideal completion of H [29, p. 59]. (In fact, by identifying Ht with the ideal lattice of H, we can define ht explicitly by ht(I)= W {h(a) : a ∈ I} for any ideal I of H.) Similar to the case of closure algebras, we have that the universal mapping properties above do not hold for Heyting algebra homomorphisms (as can be seen by modifying Example 2.8 appropriately). Let HAσ denote the category of complete and completely join-prime generated Heyting algebras and complete lattice homomorphisms. Let also FRM denote the category of frames and frame homomorphisms. As an immediate consequence of Corollary 2.18 we obtain the following statement. Corollary 2.19. (1) The inclusion functor HAσ ֒→ HA has a left adjoint functor (−)σ : HA → HAσ. Consequently, HAσ is a reflective subcategory of HA. → The inclusion functor FRM ֒→ HA has a left adjoint functor (−)t : HA (2) FRM. Consequently, FRM is a reflective subcategory of HA. Again, similar to the case of closure algebras, we have that the analogue of Corollary 2.19 is not true if we replace lattice homomorphisms by Heyting algebra homomorphisms. There is a close correspondence between canonical and topo-canonical completions of closure algebras and Heyting algebras. Indeed, for a closure algebra A = (B, C) we will see in Theorem 3.15 below that F(Aσ) is isomorphic to F(A)σ, and that F(At) is isomorphic to F(A)t. Since the functors F and F are the same on the objects, we obtain that the functor F has the same property. On the other hand, the functors G and G do not have an analogous property (see Example 3.16 below). 12 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

Let H be a Heyting algebra, let τ≤ denote the Alexandroﬀ topology associated with ≤ on pf(H), and let τP denote the Priestley topology [36] with {φ(a), −φ(a) : a ∈ H} as a subbasis for clopen sets. Then, similar to Lemma 2.11 and Theorem 2.12, we have that ≤ is the specialization order of τ, and that τ = τP ∩ τ≤.

3. Dualities for closure algebras and Heyting algebras

3.1. Dualities for closure algebras. Let TOP denote the category of topological spaces and continuous maps, and let QOS denote the category of quasi-ordered sets and order-preserving maps. We can think of QOS as a full subcategory of TOP by considering a quasi-ordered set to be a topological space with the Alexandroﬀ topology. We will also consider the category TOP of topological spaces and open continuous maps, and the category QOS of quasi-ordered sets and p-morphisms. Again we can think of QOS as a full subcategory of TOP. The two representations of closure algebras allow us to deﬁne the following contravariant functors.

Definition 3.1. (1) The functor (−)+ : CA → QOS is defined as follows. If A = (B, C) is a closure algebra, then A+ = (uf(B), R), where R is the Jónsson- Tarski quasi-order; and if h: A → A′ is a closure algebra semi-homomorphism, −1 then h+ = h . (2) The functor (−)+ : QOS → CA is defined as follows. If (Y,S) is a quasi-ordered set, then (Y,S)+ = (P(Y ),S−1); and if f : Y → Y ′ is an order-preserving map, then f + = f −1. (3) The functor (−)∗ : CA → TOP is defined as follows. If A = (B, C) is a closure algebra, then A∗ = (uf(B), τ), where τ is the McKinsey-Tarski topology; and ′ −1 if h: A → A is a closure algebra semi-homomorphism, then h∗ = h . (4) The functor (−)∗ : TOP → CA is defined as follows. If X is a topological space, then X∗ = (P(X), Cl); and if f : X → X′ is a continuous map, then f ∗ = f −1.

+ Theorem 3.2. (1) (−) is left adjoint to (−)+. ∗ (2) (−) is left adjoint to (−)∗. Proof. (1) If A = (B, C) is a closure algebra and (Y,S) is a quasi-ordered set, define + ϕ: homQOS((Y,S), A+) → homCA(A, (Y,S) ) by ϕ(f)(a)= f −1φ(a). Clearly ϕ(f) is a Boolean algebra homomorphism. Since f is order-preserving, we have that S−1ϕ(f)(a)= S−1f −1φ(a) ⊆ f −1R−1φ(a)= f −1φ(Ca)= ϕ(f)(Ca). Thus, ϕ is well-defined. To see that ϕ is a bijection, define ψ in the opposite direction by ψ(h)(y)= {a ∈ B : y ∈ h(a)}. Suppose that xSy, and let a ∈ ψ(h)(y). Then Topo-canonical completions of closure algebras and Heyting algebras 13 y ∈ h(a), and so x ∈ S−1(h(a)). Since h is a closure algebra semi-homomorphism, we have S−1(h(a)) ⊆ h(Ca). Therefore, x ∈ h(Ca), and hence Ca ∈ ψ(h)(x). + Thus, ψ(h) is order-preserving, and so ψ is well-defined. If h ∈ homCA(A, (Y,S) ) and a ∈ B, then ϕ(ψ(h))(a)= ψ(h)−1φ(a)= {y ∈ Y : ψ(h)(y) ∈ φ(a)} = {y ∈ Y : a ∈ ψ(h)(y)} = {y ∈ Y : y ∈ h(a)} = h(a) and if f ∈ homQOS((Y,S), (B, C)+) and y ∈ Y , then ψ(ϕ(f))(y)= {a ∈ B : y ∈ ϕ(f)(a)} = a ∈ B : y ∈ f −1φ(a) = {a ∈ B : f(y) ∈ φ(a)} = {a ∈ B : a ∈ f(y)} = f(y). Therefore, ψ = ϕ−1, so ϕ is a bijection. The naturality of ϕ is straightforward and will be left to the reader. (2) If A = (B, C) is a closure algebra and Y is a topological space, define ∗ ϕ: homTOP(Y, A∗) → homCA(A, Y ) by ϕ(f)(a)= f −1φ(a). Clearly ϕ(f) is a Boolean algebra homomorphism. Since f is continuous, we have that Cl ϕ(f)(a)=Cl f −1φ(a) ⊆ f −1 Cl φ(a)= f −1φ(Ca)= ϕ(f)(Ca). Thus, ϕ(f) is a closure algebra semi-homomorphism. We claim that ϕ is a bijection. ∗ Define ψ : homCA(A, Y ) → homTOP(Y, A∗) by ψ(h)(y) = {a ∈ B : y ∈ h(a)}. For ∗ each h ∈ homCA(A, Y ), ϕ(ψ(h))(a)= {y ∈ Y : a ∈ ψ(h)(y)} = {y ∈ Y : y ∈ h(a)} = h(a), and for each f ∈ homTOP(Y, A∗), ψ(ϕ(f))(y)= {a ∈ B : y ∈ ϕ(f)(a)} = {a ∈ B : a ∈ f(y)} = f(y). Therefore, ψ = ϕ−1, and ϕ is a bijection. The naturality of ϕ is straightforward.

+ Remark 3.3. The analogues of (−)+ and (−) for CA and QOS are also an adjoint pair of functors; the proof is the same as the proof of Theorem 3.2.1, with the A A′ A′ A addition that if h: → is a closure algebra homomorphism, then h+ : + → + is a p-morphism, and if f : X → X′ is a p-morphism, then f + : (X′)+ → X+ is a + closure algebra homomorphism. For a proof that h+ is a p-morphism and f is a closure algebra homomorphism see, e.g., [13, P. 150]. Similarly, the analogue of (−)∗ is a functor from TOP to CA since if f : X → X′ is continuous and open, then f ∗ : (X′)∗ → X∗ is a closure algebra homomorphism. However, as the next example shows, the analogue of (−)∗ is not necessarily a functor because it can happen that h: A → A′ is a closure algebra homomorphism, but h−1 : uf(A′) → uf(A) is not open. 14 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

Example 3.4. Let X = ω ∪ {∞1, ∞2} be the two-point compactification of the discrete space ω, where ∞1 (resp. ∞2) is the unique limit point of the odd integers (resp. even integers). Define f : X → ω +1 by f(n)= n if n ∈ ω, and f(∞i)= ω. It is obvious that f is continuous. To see that f is not open, let E denote the set of even integers. Then E ∪ {∞2} is open (even clopen) in X, E ∪{ω} is closed but not open in ω + 1, and f(E ∪ {∞2}) = E ∪{ω}. Let CO(X) be the Boolean algebra of clopen subsets of X, and let CO(ω +1) be the Boolean algebra of clopen subsets of ω + 1. We consider each of CO(X) and CO(ω + 1) as a closure algebra with the identity function as the closure operator. Then f induces a closure algebra embedding f −1 : CO(ω+1) → CO(X). Moreover, since the closure operators are the identity functions, we obtain, by Stone duality, that CO(ω + 1)∗ is homeomorphic −1 to ω + 1 and that CO(X)∗ is homeomorphic to X. Now, (f )∗ is not an open map since it can be identified with f, and f is not open, as we have already observed. σ + Remark 3.5. Since we can think of (−) as the composition of (−)+ and (−) , t ∗ and of (−) as the composition of (−)∗ and (−) , it follows from Example 3.4 that (−)t : CA → CAt is not a functor. Esakia [13] combined the Stone duality for Boolean algebras with the Jónsson- Tarski representation of closure algebras to obtain duality between closure algebras and certain quasi-ordered Stone spaces. We recall that a triple (X,τ,R) is an Esakia space if (i) (X, τ) is a Stone space (that is, it is compact Hausdorff 0-dimensional), (ii) R is a quasi-order on X, (iii) the set R[x] = {y ∈ X : xRy} is closed for each x ∈ X, and (iv) if A ⊆ X is clopen, then R−1[A] is clopen. We denote by ES the category of Esakia spaces and continuous order-preserving maps, and by ES the category of Esakia spaces and continuous p-morphisms. The following theorem was established in [13, Thm. 3]. Theorem 3.6. (1) CA is dually equivalent to ES. (2) CA is dually equivalent to ES. We now present an alternative duality for closure algebras by combining the Stone duality for Boolean algebras with the McKinsey-Tarski representation of closure algebras.

Deﬁnition 3.7. Let X be a set with two topologies τS and τ. For A ⊆ X we denote the closure of A in τS and τ by ClS(A) and Cl(A), respectively. The category T has as objects triples (X, τS, τ) such that (i) (X, τS) is a Stone space, (ii) compact open sets in τ are closed under ﬁnite intersections and form a basis for τ, (iii) compact open sets in τ are clopen in τS, and (iv) A clopen in τS implies that Cl(A) is clopen in τS. Morphisms in T are continuous maps with respect to both topologies. We also let T denote the category of such triples whose morphisms in addition satisfy the condition f −1(Cl(x)) = Cl(f −1(x)). Topo-canonical completions of closure algebras and Heyting algebras 15

Theorem 3.8. (1) ES is isomorphic to T. (2) ES is isomorphic to T.

Proof. (1) We recall that if R is a quasi-order on a set X, then τR denotes the Alexandroff topology associated to R, and if τ is a topology, then Rτ denotes the specialization order for τ. We define functors Γ: ES → T and ∆: T → ES as follows. If (X, τS, R) is an object of ES, then Γ(X, τS, R) = (X, τS, τ), where τ = τS ∩ τR, and if f is a morphism of ES then Γ(f) = f. Also, if (X, τS, τ) is an object of T, then ∆(X, τS , τ) = (X, τS, Rτ ), and if f is a morphism of T then ∆(f)= f. First we show that Γ is well-defined. If (X, τS, R) is an Esakia space, then obviously τS is a Stone topology. Moreover, opens in τ are exactly the open upsets of (X, τS, R), and compact opens in τ are exactly the clopen upsets of (X, τS , R). (To see this, let U be τS -clopen and an R-upset. Then U is τ-open by definition of τ. Since U is closed, and hence compact, in τS, it is compact in the weaker topology τ. Thus, U is τ-compact open. Conversely, let U be τ-compact open. Then U is τS -open and an R-upset by definition. Therefore, U is a union of τS -clopen R-upsets, which are τ-open by the previous argument. Since U is τ-compact, this union is finite, so U is a finite union of τS -clopen R-upsets, so U is a τS-clopen R-upset.) Thus, compact opens of τ are closed under finite intersections and form a basis for τ, and every compact open of τ is clopen in τS. Furthermore, the closure of a set A in τ is the −1 least closed downset of (X, τS, R) containing A. Therefore, Cl(A) = R [ClS(A)]. −1 Thus, if A is closed in τS, then Cl(A) = R [A], and so A clopen in τS implies −1 Cl(A) = R [A] is clopen in τS. It follows that (X, τS, τ) ∈ T. Now suppose ′ ′ ′ that f : (X, τS , R) → (X , τS, R ) is continuous and order-preserving. Then the ′ ′ ′ pullback of an open upset is an open upset. Therefore, f : (X, τS, τ) → (X , τS, τ ) is continuous with respect to both τS and τ. Thus, Γ is well-defined. Now we show that ∆ is well-defined. If (X, τS, τ) ∈ T, then τS is a Stone topology by Part (i) of Definition 3.7, and clearly Rτ is a quasi-order. To see that Rτ [x] is closed in τS, let y∈ / Rτ [x]. Then, by Part (ii) of Definition 3.7, there exists a compact open neighborhood U ∈ τ of x such that y∈ / U. Let V = −U. Since U is clopen in τS by Part (iii) of the definition, so is V . Moreover, y ∈ V and Rτ [x] ∩ V = ∅. Thus, there exists an open neighborhood V ∈ τS of y that misses Rτ [x], implying that Rτ [x] is closed in τS . Next we show that if A is closed in τS, −1 −1 −1 then Rτ [A] = Cl(A). Indeed, to see Rτ [A] ⊆ Cl(A) for any A, if x ∈ Rτ [A], then there exists y ∈ A with xRτ y. Thus, there exists y ∈ A with x ∈ Cl(y). This shows x ∈ Cl(A). On the other hand, if A is closed in τS and x ∈ Cl(A), then U ∩ A 6= ∅ for each compact open neighborhood U ∈ τ of x. Let {Ui : i ∈ I} be the set of these neighborhoods. Since the Ui are clopen in τS, A is closed in

τS, and (X, τS ) is compact, we obtain that Ti Ui ∩ A 6= ∅. Let y ∈ Ti Ui ∩ A. −1 Then xRτ y and y ∈ A. Thus, x ∈ Rτ [A]. But then A clopen in τS implies that 16 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

−1 Rτ [A] = Cl(A) is clopen in τS by Part (iv) of Deﬁnition 3.7. Therefore, (X, τS , Rτ ) ′ ′ ′ is an Esakia space. Now suppose that f : (X, τS, τ) → (X , τS, τ ) is continuous with respect to both τS and τ. Clearly f preserves the specialization order. Therefore, ′ ′ ′ f : (X, τS, Rτ ) → (X , τS, Rτ ) is continuous and order-preserving. Thus, ∆ is well- deﬁned. Finally

∆Γ(X, τS, R) = ∆(X, τS, τ) = (X, τS, Rτ ) = (X,τ,R) −1 because Rτ = R by the equality Cl(x)= R [x] for each x ∈ X. Also,

Γ∆(X, τS , τ)=Γ(X, τS , Rτ ) = (X, τS, τS ∩ τRτ ) = (X, τS , τ) since τ-opens are exactly τS-open Rτ -upsets, as we show. It is clear that a τ-open set is τS -open and an Rτ -upset. For the converse, if U is a τS-open set and an Rτ -upset, then X − U is a τS-closed Rτ -downset. We have shown earlier that −1 −1 Cl(X − U)= Rτ (X − U) since X − U is τ-closed. However, Rτ (X − U)= X − U since X − U is a Rτ -downset. Consequently, X − U = Cl(X − U). Therefore, U is τ-open. Therefore, Γ and ∆ establish an isomorphism between ES and T. (2) The proof is similar to (1). In addition we need to observe that a morphism f of ES is a p-morphism if and only if f −1 Cl(x) =Cl f −1(x), and a morphism g of T satisﬁes g−1 Cl(x) = Cl g−1(x) if and only if g is a p-morphism with respect to Rτ .

As an immediate consequence we obtain the following corollary.

Corollary 3.9. (1) CA is dually equivalent to T. (2) CA is dually equivalent to T.

Thus, with each closure algebra A = (B, C) we can associate the quadruple (uf(B), τS ,τ,R), where τS is the Stone topology, τ is the McKinsey-Tarski topology, and R is the Jónsson-Tarski quasi-order on uf(B). We can develop duality for closure algebras in the Esakia style by working with triples (X, τS, R) and recover the McKinsey-Tarski topology τ by τ = τS ∩ τR. Or else we can develop duality for closure algebras by working with triples (X, τS , τ) and recover the Jónsson-Tarski quasi-order R as the specialization order of τ. It thus is only a matter of taste to choose which dual category to work with. In fact, we find it convenient to work with the quadruples (X, τS ,τ, R) rather than either of the triples.

3.2. Dualities for Heyting algebras. We now discuss how to transfer the results above to the case of Heyting algebras. The functors described in Deﬁnition 3.1 have their analogues for Heyting algebras. Let T0 denote the category of T0-spaces and continuous maps, and let POS denote the category of posets and order-preserving maps. Also let T0 denote the category of T0-spaces and open continuous maps, and Topo-canonical completions of closure algebras and Heyting algebras 17

POS denote the category of posets and p-morphisms. The two representations of Heyting algebras then allow us to deﬁne the following contravariant functors.

Definition 3.10. (1) The functor (−)+ : HA → POS is defined as follows. If H ′ is a Heyting algebra, then H+ = (pf(H), ≤); and if h: H → H is a bounded −1 distributive lattice homomorphism, then h+ = h . (2) The functor (−)+ : POS → HA is defined as follows. If (X, ≤) is a poset, then (X, ≤)+ = U(X); and if f : X → X′ is an order-preserving map, then f + = f −1. (3) The functor (−)∗ : HA → T0 is defined as follows. If H is a Heyting algebra, ′ then H∗ = (pf(H), τ), where τ is the spectral topology; and if h: H → H is a −1 bounded distributive lattice homomorphism, then h∗ = h . ∗ (4) The functor (−) : T0 → HA is defined as follows. If X is a T0-space, then X∗ = O(X); and if f : X → X′ is a continuous map, then f ∗ = f −1. Then similar to Theorem 3.2 we have:

+ Theorem 3.11. (1) (−) is left adjoint to (−)+. ∗ (2) (−) is left adjoint to (−)∗.

+ The analogues of (−)+ and (−) for HA and POS are also an adjoint pair ∗ of functors. Similarly, the analogue of (−) is a functor from T0 to HA since if f : X → X′ is continuous and open, then f ∗ : (X′)∗ → X∗ is a Heyting algebra homomorphism. However, similar to the case of closure algebras, the analogue of (−)∗ is not necessarily a functor because it can happen, as can be seen by modifying Example 2.8 appropriately, that h: H → H′ is a Heyting algebra homomorphism, but h−1 : pf(H′) → pf(H) is not open. The Esakia duals of Heyting algebras are Esakia spaces (X,τ, ≤) where ≤ is a partial order. Thus, we deﬁne a Heyting space to be an Esakia space (X,τ, ≤) where ≤ is a partial order. We denote by HS the category of Heyting spaces and continuous order-preserving maps, and by HS the category of Heyting spaces and continuous p-morphisms. The following analogue of Theorem 3.6 was established in [13, Thm. 3]. Theorem 3.12. (1) HA is dually equivalent to HS. (2) HA is dually equivalent to HS.

The Esakia dual (pf(H), τP , ≤) of a Heyting algebra H is constructed by taking ≤ to be set inclusion, and τP to be the Priestley topology on pf(H) with subbasis {φ(a), −φ(a) : a ∈ H}. Instead we can work with the triple (pf(H), τP , τ), where τ is the spectral topology with basis {φ(a) : a ∈ H}. Again τ can be recovered from (pf(H), τP , ≤) by τ = τP ∩ τ≤, while ≤ can be recovered from (pf(H), τP , τ) as the specialization order of τ. In addition, it follows from the definition of τP 18 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers. that it is nothing more than the patch topology of τ [28, P. 45]. Therefore, we can simply work with pairs (pf(H), τ), where τ is the spectral topology on pf(H), and recover ≤ as the specialization order and τP as the patch topology of τ. The only additional condition on (pf(H), τ) is that if A is clopen in the patch topology, then so is Cl(A). To give an abstract definition of these spaces, we recall some terminology. Let X be a topological space. A nonempty subset A of X is irreducible if A is not the union of two proper subsets, each of which is closed in A. The space X is sober if every closed irreducible set is the closure of a point. Let E(X) denote the collection of compact open subsets of X. Then X is a spectral space (see, e.g., [28]) if X is T0, compact, sober, E(X) is closed under finite intersections, and is a basis for the topology. A continuous map f : X → Y between spectral spaces is a spectral map if the pullback of each compact open subset of Y is compact open in X.

Deﬁnition 3.13. Let (X, τ) be a spectral space. We call (X, τ) a spectral Heyting space provided that, for each subset A of X, if A is clopen in the patch topology of τ, then so is Cl(A). Let HSPEC denote the category of spectral Heyting spaces and spectral maps. Also let HSPEC denote the category of spectral Heyting spaces and spectral maps that satisfy the additional condition f −1 Cl(x)=Cl f −1(x).

Similar to Theorem 3.8 and Corollary 3.9, we have the following theorem.

Theorem 3.14. (1) HS is isomorphic to HSPEC. (2) HS is isomorphic to HSPEC. (3) HA is dually equivalent to HSPEC. (4) HA is dually equivalent to HSPEC.

We conclude this section by describing dually the correspondence between closure algebras and Heyting algebras. The functor F: CA → HA (resp. F: CA → HA) dually corresponds to the functor ES → HS (resp. ES → HS) that identifies the clusters C[x] = {y ∈ X : xRy and yRx} of Esakia spaces, while the functor G: HA → CA (resp. G: HA → CA) dually corresponds to the inclusion functor HS ֒→ ES (resp. HS ֒→ ES). Alternatively, F (resp. F) corresponds to the functor T → HSPEC (resp. T → HSPEC) that takes the T0-reflections, and G (resp. G) corresponds to the inclusion functor HSPEC ֒→ T (resp. HSPEC ֒→ T). We recall that the T0-reflection X0 of a topological space X is the quotient space X/E, where the equivalence relation E on X is defined by xEy if Cl(x) = Cl(y).

Theorem 3.15. Suppose A = (B, C) is a closure algebra. Then: (1) F(Aσ) =∼ F(A)σ. (2) F(At) =∼ F(A)t. Topo-canonical completions of closure algebras and Heyting algebras 19

Proof. (1) Let A = (B, C) be a closure algebra and let (uf(B), τS ,τ,R) be the dual of A. We denote by CO(uf(B)) the Boolean algebra of τS -clopen sets and by CU(uf(B)) the Heyting algebra of τS-clopen upsets of (uf(B), τS ,τ,R). Then Aσ =∼ (P(uf(B)), R−1) and so F(Aσ) =∼ U(uf(B)). On the other hand, A =∼ (CO(uf(B)), R−1), so F(A) =∼ CU(uf(B)), and so F(A)σ =∼ U(uf(B)). Therefore, F(Aσ) =∼ F(A)σ. (2) We have At =∼ (P(uf(B)), Cl) and so F(At) =∼ O(uf(B)), which is isomorphic to the Heyting algebra of τS -open upsets of (uf(B), τS ,τ,R). On the other hand, A =∼ (CO(uf(B)), R−1), so F(A) =∼ CU(uf(B)), and so F(A)t is isomorphic to t the Heyting algebra of τS -open upsets of (uf(B), τS ,τ,R). Therefore, F(A ) =∼ F(A)t.

Since the functors F and F are the same on the objects, we obtain that Theorem 3.15 also holds for F. On the other hand, the functors G and G do not have an analogous property as the following example shows.

Example 3.16. Let H be the Heyting algebra of clopen upsets of the ordinal ω + 1 (with its usual order and topology). Then G(H) =∼ (CO(ω + 1), ↓). Thus, G(H)σ =∼ (P(ω + 1), ↓) is uncountable. On the other hand, Hσ = U(ω + 1), so G(Hσ)—the free Boolean extension of U(ω + 1)—is countable, because U(ω +1) is countable, and hence is not isomorphic to G(H)σ. Similarly, G(H)t ≇ G(Ht).

4. The Salbany and Banaschewski compactiﬁcations

4.1. The Salbany compactification. Let B be a Boolean algebra. Then it is known (see, e.g., [6, Sec. 3]) that the dual space of Bσ is the Stone-Cechˇ compactification β(uf(B)) of uf(B). This result extends to canonical completions of closure algebras in that the dual space of (Bσ, Cσ) is (β(uf(B)), R), where R is the Jónsson-Tarski quasi-order. As for topo-canonical completions, we recall the ∗ adjoint functors (−)∗ : CA → TOP and (−) : TOP → CA described in the previous section. For a closure algebra A = (B, C), we can view the topo-canonical comple- t ∗ tion A of A as (A∗) . (Note that the canonical completion of A can respectively + t be viewed as (A+) ). The dual of A is then uf(P(uf(B))) with the Stone topology τS, the McKinsey-Tarski topology τ, and the Jónsson-Tarski quasi-order R. We will concentrate on (uf(P(uf(B))), τ). This is a special case of the compactification Salbany defined in [38, Sec. 2]. It can be described as the composition of (−)∗ and (−)∗.

Deﬁnition 4.1. The Salbany compactiﬁcation s(X) of a topological space X is the ∗ topological space (X )∗ = (uf(P(X)), τ), where τ is the McKinsey-Tarski topology. 20 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

In order to summarize the main properties of s(X) we recall some terminology. A topological space X is said to be locally compact if for every x ∈ X and every open neighborhood U of x, there is an open set V and a compact set C with x ∈ V ⊆ C ⊆ U. A space X is said to be supersober if for each ultrafilter P on X, the set T{Cl(A) : A ∈ P } is the closure of a singleton. It is not hard to see that a supersober space is compact: if {Ci : i ∈ I} is a collection of closed subsets of a supersober space X having the finite intersection property, then {Ci : i ∈ I} is contained in an ultrafilter P , and since T{Cl(A) : A ∈ P } is nonempty, so is T{Ci : i ∈ I}. Let Px denote the principal ultrafilter on X containing x. We define i: X → s(X) by i(x) = Px. The following properties of s(X) were proved in [38, Sec. 2]. Proposition 4.2. Let X be a topological space, and let s(X) be the Salbany compactification of X. (1) The natural map i: X → s(X) is an embedding. Moreover, i(X) is dense in s(X). (2) The space s(X) is locally compact and supersober. (3) The collection E(X) of compact open subsets of s(X) is equal to {φ(U) : U ∈ τ}, is closed under finite intersections, and is a basis for the topology. (4) If f : X → Y is a continuous map into a locally compact supersober space Y , then there is a continuous map g : s(X) → Y such that g ◦ i = f. We point out that the map g in Statement 4 of the proposition need not be unique. Salbany defines g on free ultrafilters by letting g(P ) = y for any point satisfying Cl(y)= T{Cl(A) : f −1(A) ∈ P }. The point y need not be unique if Y is not T0. We now discuss relations between supersober and sober spaces. Lemma 4.3. A supersober space is sober. Proof. Let C be a closed irreducible subset of X. We need to find an ultrafilter P on X with T{Cl(A) : A ∈ P } = C to conclude that C is the closure of a point. We use a Zorn’s lemma argument. Let S be the collection of all filters F on X with C ∈ F and satisfying the property that if A ∈ F , then C ⊆ Cl(A). Then ↑C ∈ S, so S is nonempty. By partially ordering S by inclusion, it is clear that Zorn’s lemma applies. Let P be a maximal element of S. We wish to show that P is an ultrafilter. Suppose that A, B are subsets of X with A ∪ B ∈ P . If both A/∈ P and B/∈ P , then the filters generated by P ∪{A} and P ∪{B} are properly larger than P , so do not lie in S. Therefore, there are S,T ∈ P with C 6⊆ Cl(S ∩ A) and C 6⊆ Cl(T ∩ B). However, since S ∩ T ∩ (A ∪ B) ∈ P , we have C ⊆ Cl(S ∩ T ∩ (A ∪ B)) = Cl((S ∩ T ∩ A) ∪ (S ∩ T ∩ B)) = Cl(S ∩ T ∩ A) ∪ Cl(S ∩ T ∩ B) ⊆ Cl(S ∩ A) ∪ Cl(S ∩ B). Topo-canonical completions of closure algebras and Heyting algebras 21

Because C is irreducible, we have C ⊆ Cl(S ∩ A) or C ⊆ Cl(T ∩ B). This contradiction shows that A ∈ P or B ∈ P , so P is an ultraﬁlter. Now, C ⊆ Cl(A) for each A ∈ P . Therefore, C ⊆ T{Cl(A) : A ∈ P }. The reverse inclusion is obvious because C ∈ P and C is closed. Thus, C = T{Cl(A) : A ∈ P }. Therefore, since X is supersober, C = Cl(x) for some x ∈ X. This proves that X is sober.

The converse of Lemma 4.3 is not true in general as any noncompact Hausdorﬀ space is sober but not supersober. Nevertheless, we do have that every spectral space is supersober as the next lemma shows.

Lemma 4.4. If X is a spectral space, then X is supersober.

Proof. Let Q be an ultrafilter on X, and let P = Q ∩E(X). Then P is a prime filter in E(X), so, by Priestley duality [36], P = ↑x ∩E(X) for some x ∈ X. We claim that T{Cl(A) : A ∈ Q} = Cl(x). For one inclusion, if U is a compact open neighborhood of x, then U ∈ P , so U ∈ Q. Therefore, for any A ∈ Q, we have U ∩ A 6= ∅, so x ∈ Cl(A). Conversely, if y ∈ T{Cl(A) : A ∈ Q}, then for every compact open neighborhood U of y we have U ∩ A 6= ∅. This forces U ∈ Q since Q is an ultrafilter. So, U ∈ P , which means x ∈ U. This proves that y ∈ Cl(x). Thus, Cl(x)= T{Cl(A) : A ∈ Q}. Therefore, X is supersober.

We generalize the notion of a spectral space to that of a quasi-spectral space by removing the requirement that X be T0 and replacing the notion of sober by that of supersober. This deﬁnition, motivated by Proposition 4.2, abstracts the properties of the McKinsey-Tarski topology.

Deﬁnition 4.5. We say that a topological space X is quasi-spectral if X is supersober, E(X) is closed under ﬁnite intersections, and is a basis for the topology.

Thus, by Proposition 4.2, the Salbany compactiﬁcation of a topological space is a quasi-spectral space. More generally, if (B, C) is a closure algebra, then the proof of Proposition 4.2 actually shows that uf(B) with the McKinsey-Tarski topology is quasi-spectral. There is a close connection between quasi-spectral spaces and spectral spaces, as we now see. Let X be a topological space and let X0 be the T0-reﬂection of X. −1 Let also π denote the canonical map X → X0. We point out that π (π(U)) = U for each open set U ⊆ X. Therefore, if U is open in X, then π(U) is open in −1 X0. Similarly, π (π(C)) = C for each closed set C. As a consequence, Cl(πA)= π(Cl A) for each A ⊆ X.

Theorem 4.6. Let X be a topological space. Then X is quasi-spectral if and only if the T0-reﬂection X0 is a spectral space. 22 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

Proof. Suppose that X is quasi-spectral. Then X0 = π(X) is compact since X is compact. Also, if U is compact open in X, then π(U) is compact open in X0. The set of all such U forms a basis for X, so the set of all the π(U) forms a basis for X0. Thus, the compact opens in X0 form a basis for X0. This collection is closed under finite intersections since the same is true for X. Finally, let C be a closed −1 −1 irreducible subset of X0. Then π (C) is closed in X. We claim that π (C) is irreducible. Let A, B be closed sets with π−1(C)= A ∪ B. Then C = π(A) ∪ π(B). Since π is a closed map and C is irreducible, we may assume that C = π(A). Therefore, π−1(C) = π−1(π(A)) = A since A is closed in X. Thus, π−1(C) is irreducible. By Lemma 4.3, X is sober, so there is x ∈ X with π−1(C) = Cl(x). Thus, C = π(Cl x) = Cl(π(x)). This proves that X0 is sober. Therefore, X0 is a spectral space. Conversely, let X0 be spectral. We prove that X is quasi-spectral. Since E(X) −1 = π (U) : U ∈E(X0) and E(X0) is a basis for X0, we see that E(X) is a basis for X. It is closed under finite intersections because E(X0) is closed under finite intersections and π−1(U) ∩ π−1(V ) = π−1(U ∩ V ). Finally, suppose that P is an ultrafilter on X. Let Q = B : π−1(B) ∈ P . It is easy to see that Q is an ultrafilter on X0. Then T{Cl(B) : B ∈ Q} = Cl(π(x)) for some x ∈ X by Lemma 4.4. We claim that T{Cl(A) : A ∈ P } = Cl(x). For one inclusion, let U be a compact open neighborhood of x. If there exists A ∈ P with U ∩ A = ∅, then U ∩ π−1(π(A)) = ∅ since U = π−1(π(U)). Thus, π(U) ∩ π(A) = ∅, which is false since π(U) is a neighborhood of π(x). Thus, x ∈ T {Cl(A) : A ∈ P }, and so Cl(x) ⊆{Cl(A) : A ∈ P }. Conversely, suppose that y ∈ T {Cl(A) : A ∈ P }. Let V be a compact open neighborhood of y. If π(V ) ∩ B = ∅ for some B ∈ Q, then V ∩ π−1(B) = ∅, which is false since π−1(B) ∈ P . Thus, π(V ) ∩ B 6= ∅ for each B ∈ Q. Therefore, {π(V ) ∩ Cl(B) : B ∈ Q} is a set of closed subsets of π(V ) with the finite intersection property, as Q is a filter. Because π(V ) is compact,

π(V ) ∩ Cl(π(x)) = π(V ) ∩ T {Cl(B) : B ∈ Q} = T {π(V ) ∩ Cl(B) : B ∈ Q} 6= ∅. Since π(V ) is open in π(X), we conclude that π(x) ∈ π(V ). This yields x ∈ V , and so y ∈ Cl(x). Hence, we have proved that T{Cl(A) : A ∈ P } = Cl(x), which shows that X is supersober. Therefore, X is quasi-spectral.

As we already pointed out, a continuous map f : X → Y into a locally compact supersober space does not extend uniquely to a continuous map g : s(X) → Y . Now we show that the situation improves if the target space Y is spectral.

Lemma 4.7. Let f : X → Y and g : s(X) → Y be continuous maps into a spectral space Y satisfying g ◦ i = f. Then g is spectral if and only if, for each compact Topo-canonical completions of closure algebras and Heyting algebras 23 open U ⊆ Y we have g−1(U)= φ(f −1(U)). Moreover, if g is a spectral map, then Cl(g(P )) = T Cl(A) : f −1(A) ∈ P . Proof. First suppose that g is spectral, and let U be compact open in Y . Then g−1(U) is compact open in s(X). By Proposition 4.2, there is an open set V in X with g−1(U)= φ(V ). Since g ◦ i = f and i−1(φ(V )) = V , we see that V = f −1(U), and so g−1(U) = φ(f −1(U)). Conversely, the equation g−1(U) = φ(f −1(U)) for each compact open U ⊆ Y says that g is spectral. Suppose that g is a spectral map. Let f −1(A) ∈ P . If U is a compact open neighborhood of g(P ), then P ∈ g−1(U)= φ(f −1(U)), and so f −1(U) ∈ P . Conse- quently, f −1(U ∩ A)= f −1(U) ∩ f −1(A) ∈ P . This implies U ∩ A 6= ∅. Therefore, g(P ) ∈ Cl(A), and so Cl(g(P )) ⊆ Cl(A) : f −1(A) ∈ P . Conversely, suppose that x ∈ Cl(A) for each A with f −1(A) ∈ P . If x∈ / Cl(g(P )), then there is a compact open V with x ∈ V but g(P ) ∈/ V . Then, P/∈ g−1(V ) = φ(f −1(V )), and so f −1(V ) ∈/ P . Set A = −V . Then f −1(A) ∈ P , so x ∈ Cl(A), and so V ∩ A 6= ∅. This is a contradiction. Thus, x ∈ Cl(g(P )). Proposition 4.8. Suppose f : X → Y is a continuous map into a spectral space Y . Then there is a unique spectral map g : s(X) → Y such that g ◦ i = f. Proof. It follows from [38, Sec. 2] that there is a continuous map g : s(X) → Y with g ◦ i = f. We recall the definition of g. If Px is a principal ultrafilter, then g(Px) = f(x), and if P is a free ultrafilter, then g(P ) is any point satisfying Cl(g(P )) = T Cl(A) : f −1(A) ∈ P (such a point always exists because a spectral space is supersober). We prove that g is a spectral map by showing that g−1(V )= φ(f −1(V )) for each compact open subset V of Y and applying Lemma 4.7. First, suppose that P ∈ g−1(V ). Then g(P ) ∈ V . If P/∈ φ(f −1(V )), then f −1(V ) ∈/ P , so f −1(−V ) = −f −1(V ) ∈ P . Thus, by the definition of g, we see that g(P ) ∈ Cl(−V ) = −V , a contradiction. Conversely, suppose that P ∈ φ(f −1(V )). Then f −1(V ) ∈ P . The collection Cl(A) ∩ V : f −1(A) ∈ P is easily seen to have the finite intersection property since P is a proper filter and f −1(V ) ∈ P . Thus, since V is compact, T Cl(A) ∩ V : f −1(A) ∈ P is nonempty. By Lemma 4.7 this intersection is equal to Cl(g(P )) ∩ V . Since V is open, we see that g(P ) ∈ V . This proves g−1(V ) = φ(f −1(V )). For uniqueness, if g′ : s(X) → Y is another spectral map satisfying g′ ◦i = f, then, by Lemma 4.7, Cl(g′(P )) = T{Cl(A) : f −1(A) ∈ P } for each ultrafilter P ∈ s(X). This equation also holds for g. Thus, Cl(g′(P )) = ′ ′ Cl(g(P )), and so g (P )= g(P ) since Y is a T0-space. We conclude that g = g. Note that, unlike the Stone-Cechˇ compactification, the Salbany compactification seldom satisfies strong separation axioms. In fact, it follows from [38, Ex. 8] that s(X) is a T1-space if and only if X is discrete (if and only if s(X) = β(X)). The next result characterizes when s(X)is a T0-space. 24 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

Proposition 4.9. Let (X, τ) be a topological space. Then s(X) is T0 if and only if the Boolean subalgebra B(τ) of P(X) generated by τ is P(X).

Proof. First suppose that B(τ) = P(X). Let P,Q ∈ s(X) be diﬀerent. Without loss of generality we may assume that there is an A ∈ P − Q. By assumption, we may write A = B1 ∪···∪ Bn with each Bi = Ui ∩ Ci with Ui open and Ci closed. Since P is an ultraﬁlter, Bi ∈ P for some i. Note that Bi ∈/ Q since Bi ⊆ A and A/∈ Q. Therefore, either Ui ∈/ Q or Ci ∈/ Q. However, Ui ∈ P and Ci ∈ P since they both contain Bi. If Ui ∈/ Q then φ(Ui) is an open neighborhood of P that misses Q, while if Ci ∈/ Q, then φ(−Ci) is an open neighborhood of Q missing P . Thus, s(X) is T0. Conversely, suppose that s(X) is T0. If B(τ) 6= P(X), then the proper inclusion B(τ) → P(X) yields a surjection uf(P(X)) → uf(B(τ)) which is not a bijection. So, there are P 6= Q in s(X) with P ∩ B(τ) = Q ∩ B(τ). Then P ∩ τ = Q ∩ τ. This means that there is no open set U such that φ(U) contains one of P and Q but not the other. Thus, s(X) is not T0. This contradiction shows that B(τ)= P(X).

Although it is uncommon for a topology τ on a set X to generate P(X) as a Boolean algebra, there are some interesting examples when this happens. Recall that a subset A of X is locally closed if it is an intersection of an open set and a closed set. It is clear that B(τ)= P(X) if and only if every subset of X is a finite union of locally closed subsets of X. To see when the latter condition is satisfied, we recall that the Hausdorff residue ρ(A) of a subset A of X is ρ(A) = A ∩ Cl(Cl(A) − A). Then, by [14, Cor. 5.26], a set A is a union of n locally closed sets if and only if ρn(A) = ∅. For specific examples, if X is a submaximal space (that is, each dense set is open), then ρ(A) = ∅ for each A ⊆ X [5, Cor. 3.3]. Thus, every submaximal space satisfies the condition of Proposition 4.9. Furthermore, if α is an ordinal space such that α<ωω, then for each β ⊆ α, there is an integer n with ρn(β) = ∅: for each β ⊆ α, let d(β) denote the set of limit points of β; then we have ρ(β) ⊆ d(β) ⊆ d(α), and since α ≤ ωn for some n ≥ 1, we have that dn(α) ⊆ dn(ωn) = ∅ so that ρn(β) = ∅. Therefore, every ordinal α<ωω satisfies the condition of Proposition 4.9. Finally, we claim that if X is finite, then X is T0 if and only if B(τ) = P(X). Let (X, τ) be a finite topological space. We note, since X is finite, that B(τ) = P(X) if and only if each point is locally closed. Suppose that X is T0. Let x ∈ X. For each y ∈ Cl(x) with y 6= x, Cl(y) is a proper subset of Cl(x). Let Uy = − Cl(y), an open neighborhood of x missing y. If U = S{Uy : y ∈ Cl(x),y 6= x}, then {x} = U ∩ Cl(x), so x is locally closed. Conversely, suppose that all points are locally closed. Let x, y ∈ X with Cl(x) = Cl(y). Write {x} = U ∩ Cl(x). If y 6= x, then as y ∈ Cl(x) and U is a neighborhood of x, we must have y ∈ U. This is impossible, so y = x, and X is T0. Topo-canonical completions of closure algebras and Heyting algebras 25

Thus, a ﬁnite space satisﬁes the condition of Proposition 4.9 if and only if it is a T0-space.

4.2. The Banaschewski compactification. The dual Heyting space of the canonical completion of a Heyting algebra was described in [6, Prop. 3.4] in terms of the Nachbin order compactification. We now describe the dual spectral space of the topo-canonical completion of a Heyting algebra. We recall the adjoint pair of ∗ functors (−)∗ : HA → T0 and (−) : T0 → HA described in Section 3.2. For a Heyt- t ∗ ing algebra H we can view the topo-canonical completion H of H as (H∗) . (Note + that the canonical completion of H can respectively be viewed as (H+) ). Then the dual spectral space of Ht is pf(O(pf(H))) with the spectral topology. This is a special case of the compactification Banaschewski defined in [4].

Deﬁnition 4.10. Let (X, τ) bea T0-space. Then the Banaschewski compactiﬁca- ∗ tion b(X) of X is the topological space (X )∗ = pf(O(pf(H))), with the spectral topology.

By the definition, b(X) is a spectral Heyting space. For x ∈ X let Px = {U ∈ τ : x ∈ U}. Then Px is a prime filter in O(X). We define the natural map i: X → b(X) by i(x) = Px. The following result was established in [4]. Alternatively, it follows from Propositions 4.2, 4.8, and Theorem 4.12 below.

Proposition 4.11. Let X be a T0-space. (1) The map i: X → b(X) is an embedding. (2) i(X) is dense in b(X). (3) If f : X → Y is a continuous map into a spectral space Y , then there is a unique spectral map g : b(X) → Y with g ◦ i = f.

The next theorem reveals a close connection between the Banaschewski and Salbany compactifications. In fact, it shows that we can view the Banaschewski compactification as the T0-reflection of the Salbany compactification.

Theorem 4.12. Let (X, τ) be a topological space. Then b(X) is homeomorphic to s(X)0.

Proof. If U is open in X, we recall the notation φ(U)= {Q ∈ s(X) : U ∈ Q}, and we set φb(U)= {P ∈ b(X) : U ∈ P }. Deﬁne f : s(X) → b(X) by f(Q)= Q ∩ τ. To see that f is continuous, let φb(U), for some U ∈ τ, be a compact open subset of b(X). Then

−1 f (φb(U)) = {Q ∈ s(X) : U ∈ Q ∩ τ} = {Q ∈ s(X) : U ∈ Q} = φ(U). 26 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

We next show that f is surjective. Let P ∈ b(X), and set

S = {F : F is a ﬁlter on X and F ∩ τ = P } .

Let F denote the filter on X generated by P . Then F ∈ S since F ∩τ = P . So, S is nonempty. Moreover, it is easy to see that Zorn’s lemma applies, by which S has a maximal element Q. A standard argument shows that Q is an ultrafilter. Moreover, f(Q)= P . Thus, f is surjective. This together with the proof of continuity shows that f(φ(U)) = φb(U) for each open set U of X. Consequently, f is an open map. Now, f(Q1)= f(Q2) if and only if Q1 ∩ τ = Q2 ∩ τ. This occurs if and only if for each open set U ⊆ X, we have Q1 ∈ φ(U) ⇔ Q2 ∈ φ(U). Finally, by the definition of the topology on s(X), this occurs if and only if Cl(Q1) = Cl(Q2). Therefore, f factors through s(X)0, and the induced map is open since f is open. Thus, s(X)0 and b(X) are homeomorphic.

Remark 4.13. The Banaschewski compactification and the Nachbin order compactification are closely related. Let (X, ≤) be a poset. We view X as a T0-space with the Alexandroff topology τ≤. Let b(X) be the Banaschewski compactification of (X, τ≤) and n(X) be the Nachbin order compactification of (X, ≤) (see [6, Sec. 3]). Then, by [6, Prop. 3.4] and Lemma 2.11, the topology on n(X) is the patch topology of b(X), and the order on n(X) is the specialization order of b(X).

5. Topo-canonicity, topo-completeness, and deﬁnability

5.1. Topo-canonicity and topo-completeness. We recall that a class of algebras is a variety if it is closed under homomorphic images, subalgebras, and direct products. By Birkhoff’s Theorem, a class of algebras is a variety if and only if it is equationally definable. A variety V of closure algebras (resp. Heyting algebras) is called canonical if A ∈ V implies that Aσ ∈ V (resp. H ∈ V implies that Hσ ∈ V). The theory of canonical varieties is relatively well-developed (see, e.g., [31, 32, 30, 21, 22, 23, 24, 12, 18, 19, 20, 17]). It is easy to see that both CA and HA are canonical, but there are many other canonical varieties of closure algebras and Heyting algebras. For instance, every variety that is generated by a finite closure algebra (resp. Heyting algebra) is canonical (see, e.g., [22, P. 227], [19, Cor. 22], or [6, Cor. 5.3]). The situation is different with topo-canonical varieties.

Deﬁnition 5.1. (1) Let V be a variety of closure algebras. We call V topo- canonical if A ∈ V implies that At ∈ V. (2) Let V be a variety of Heyting algebras. We call V topo-canonical if H ∈ V implies that Ht ∈ V. Topo-canonical completions of closure algebras and Heyting algebras 27

The next theorem shows that virtually no varieties of closure algebras (resp. Heyting algebras) are topo-canonical. Theorem 5.2. (1) The only topo-canonical varieties of closure algebras are CA and the trivial variety. (2) The only topo-canonical varieties of Heyting algebras are HA and the trivial variety. Proof. (1) It is clear that both CA and the trivial variety are topo-canonical. Now we show that these are the only topo-canonical varieties of closure algebras. Let V denote the variety of closure algebras satisfying the equation Ca = a. Then a closure algebra A = (B, C) belongs to V if and only if C is the identity function. Moreover, it is known that V is the least nontrivial variety of closure algebras (see, e.g., [11, Thm. 8.67]). Let 2 denote the two element Boolean algebra, and let B denote the closure algebra (2, id), where id: 2 → 2 is the identity function. Since 2 generates the variety of Boolean algebras, it is clear that B generates V. Now, let C be the closure algebra (CO(X), id) of clopen sets of the Cantor space X with the identity function. We have C ∈ V. Moreover, since B is a subalgebra of C and B generates V, then so does C. The topo-canonical completion Ct of C is (P(X), Cl). Thus, by [34, Thm. 5.10], Ct generates CA. Therefore, taking homomorphic images, subalgebras, and direct products of the topo-canonical completion Ct of C ∈ V yields the whole CA. This means that CA is the only nontrivial topo-canonical variety of closure algebras. (2) follows from (1), Theorem 3.15.2, and the well-known fact (see [10, Thm. 4.1] or [14, Cor. 2.6]) that if V is a variety of closure algebras, then F(V) is a variety of Heyting algebras. Corollary 5.3. Ideal completions of Heyting algebras preserve no HA identities (deﬁning a proper nontrivial subvariety of HA). Proof. This follows immediately from Theorems 2.17 and 5.2. This corollary is in sharp contrast to the well-known result that ideal completions of lattices preserve all lattice identities (see, e.g., [26, P. 28, Lemma 8]). Although CA (resp. HA) is the only nontrivial topo-canonical variety of closure algebras (resp. Heyting algebras), there are interesting varieties of closure algebras (resp. Heyting algebras) that come from topology. For a class K of topological ∗ spaces, we denote by VK the variety of closure algebras generated by {X : X ∈ K}. Deﬁnition 5.4. We say that a variety V of closure algebras (Heyting algebras) is topo-complete if there is a class K of topological spaces such that V = VK . We give several examples of topo-complete varieties of closure algebras and Heyt- ing algebras. The varieties of closure algebras that we consider are CA, CA.1, 28 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

CA.2, CA.1.2, and GRZ, which are algebraic counterparts of the well-known modal systems S4, S4.1, S4.2, S4.1.2, and S4.Grz, respectively. The varieties of Heyting algebras that we consider are HA and the subvariety KC of HA defined by the equation ¬a ∨ ¬¬a = 1, which correspond to the intuitionistic propositional logic and the logic of weak excluded middle (see, e.g., [11] for details). Example 5.5. Let CA.1 be the McKinsey variety of closure algebras defined by ICa ≤ CIa. For a topological space X, let D(X) be the set of dense subsets and Iso(X) be the set of isolated points of X. We call X a McKinsey space if D(X)isa filter. We also call X weakly scattered if Iso(X) ∈ D(X). Let K1 denote the class of McKinsey spaces and let K2 denote the class of weakly scattered spaces. It is clear that K2 ⊆ K1 since if X is weakly scattered, then D(X) is the filter generated by Iso(X). To see that K2 is a proper subclass of K1, let X be an infinite set and let P be a free ultrafilter on X. We define a topology on X by τ = P ∪{∅}. Then X has no isolated points, so is not weakly scattered. On the other hand, D(X)= P , implying that X is a McKinsey space. We claim that CA.1 = VK1 = VK2 . The inclusion VK2 ⊆ VK1 is clear. The inclusion VK1 ⊆ CA.1 follows from [7, Prop.

2.1]. For the inclusion CA.1 ⊆ VK2 , we note that CA.1 is generated by the collection of X∗ = (P(X), Cl), where X is weakly scattered [16, Thm. 1.3.8]. Example 5.6. Let CA.2 be the Church-Rosser variety of closure algebras defined by CIa ≤ ICa. We recall that a topological space X is extremally disconnected if the closure of every open set is clopen. Let K denote the class of extremally disconnected spaces. Then CA.2 = VK [16, Thm. 1.3.4]. Example 5.7. Let CA.1.2 be the variety of closure algebras defined by CIa = ICa. Recalling the previous two examples, we see that CA.1.2 = CA.1 ∩ CA.2. Now since CA.1.2 is generated by its finite members, which are algebras over finite extremally diconnected McKinsey spaces, and since the class of finite McKinsey spaces coincides with the class of finite weakly scattered spaces, we obtain that

CA.1.2 = VK1 = VK2 , where K1 is the class of extremally disconnected McKinsey spaces and K2 is the class of extremally disconnected weakly scattered spaces. Example 5.8. Let GRZ be the variety of closure algebras deﬁned by a ≤ C(a − C(Ca − a)). We recall that a topological space X is resolvable if it is the union of two disjoint dense subsets of X, and that it is irresolvable if it is not resolvable. Also, X is hereditarily irresolvable if every subspace of X is irresolvable, and it is scattered if every subspace of X has an isolated point. Let K1 be the class of hereditarily irresolvable spaces, K2 the class of scattered spaces, K3 the class of ordinals (viewed as topological spaces with the interval topology), and K4 = {α}, ω where α is an ordinal satisfying α ≥ ω . Then GRZ = VKi for i = 1, 2, 3, 4.

Indeed, since K4 ⊆ K3 ⊆ K2 ⊆ K1, it is obvious that VK4 ⊆ VK3 ⊆ VK2 ⊆ VK1 . Topo-canonical completions of closure algebras and Heyting algebras 29

By [7, Thm. 2.4], a topological space X is hereditarily irresolvable if and only if

A ⊆ Cl(A−Cl(Cl(A)−A)) for each A ⊆ X. Thus, VK1 ⊆ GRZ. Finally, it follows from [1] that VK4 = GRZ, which concludes the proof.

Thus, we obtain that each of the varieties CA, CA.1, CA.2, CA.1.2, and GRZ is topo-complete. We note that if K is a class of topological spaces and V is a variety of closure algebras, then V = VK implies that F(V)= VK , where the latter VK is viewed as a variety of Heyting algebras. Since HA = F(CA)= F(CA.1)= F(GRZ) (see, e.g., [11, Ex. 9.63]), we obtain that HA = VK for each of the following classes K: (i) all topological spaces, (ii) McKinsey spaces, (iii) weakly scattered spaces, (iv) hereditarily irresolvable spaces, (v) scattered spaces, (vi) ordinals, (vii) any ordinal α ≥ ωω. Moreover, since KC = F(CA.2)= F(CA.1.2)= F(GRZ.2) (see, e.g., [11, Ex. 9.63]), we obtain that KC = VK for each of the following classes K: (i) extremally disconnected spaces, (ii) extremally disconnected McK- insey spaces, (iii) extremally disconnected weakly scattered spaces, (iv) extremally disconnected hereditarily irresolvable spaces, (v) extremally disconnected scattered spaces. For more examples of topo-complete varieties we refer to [5].

5.2. Deﬁnability. We now investigate the classes of topological spaces that arise from varieties of closure algebras (resp. Heyting algebras). Let V be a variety of ∗ closure algebras (resp. Heyting algebras), and let KV = {X : X ∈ V}.

Deﬁnition 5.9. We call a class K of topological spaces deﬁnable if K = KV for some variety V of closure algebras (resp. Heyting algebras).

The notion of a definable class of topological spaces generalizes the notion of a definable or modal-axiomatic class of quasi-ordered sets (resp. posets) [25, 21]; that is, of a class K of the form {X : X+ ∈ V} for some variety V of closure algebras (resp. Heyting algebras). Some examples of definable classes of topological spaces by varieties of closure algebras are (i) all topological spaces (defined by CA), (ii) McKinsey spaces (defined by CA.1), (iii) extremally disconnected spaces (defined by CA.2), and (iv) hereditarily irresolvable spaces (defined by GRZ). On the other hand, such classes as weakly scattered spaces, scattered spaces, and ordinals are not definable. Indeed, it follows from the definition that a class K of topological spaces is definable if and only if K = KVK . Now, if K is the class of weakly scattered spaces, then KVK is the class of McKinsey spaces (see Example 5.5), which properly contains K, and if K is the class of scattered spaces or ordinals, then KVK is the class of hereditarily irresolvable spaces (see Example 5.8), which properly contains the classes of both scattered spaces and ordinals. Note that we have even fewer definable classes by varieties of Heyting algebras. For example, neither the class of McKinsey spaces nor the class of hereditarily irresolvable spaces is definable 30 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers. by varieties of Heyting algebras because if K denotes either of these classes, then

KVK = TOP, and so properly contains K. The next theorem gives a necessary and sufficient condition for a class of topological spaces to be definable. It generalizes [25, Thm. 3], which gives a necessary and sufficient condition for a class of quasi-ordered sets to be definable.

Theorem 5.10. Let K be a class of topological spaces. Then K is deﬁnable if and only if K satisﬁes the following conditions. (1) K is closed under topological sums; (2) If X, Y are topological spaces such that X ∈ K and Y ∗ is a homomorphic image of a subalgebra of X∗, then Y ∈ K.

Proof. Suppose K is definable. Then there is a variety V such that K = KV. Let ∗ ∼ ∗ {Xi}⊆ K. By [16, Prop. 2.2.8], (Li Xi) = Q(Xi) ∈ V. Thus, Li Xi ∈ K, so K is closed under topological sums. Let X, Y be topological spaces with X ∈ K and Y ∗ ∈ HS(X∗), where H and S denote the class operations of taking homomorphic images and subalgebras, respectively. Then X∗ ∈ V, implying that Y ∗ ∈ V. Therefore, Y ∈ K. Conversely, suppose K satisfies the two conditions of the theorem. We claim that K = KVK . It is clear that K ⊆ KVK . Suppose that Y ∈ KVK . Then ∗ ∗ ∗ Y ∈ VK . By definition of VK , there are Xi ∈ K such that Y ∈ HS(Q(Xi) ). ∗ ∼ ∗ ∗ ∗ But Q(Xi) = (Li Xi) and Li Xi ∈ K by (1). Therefore, Y ∈ HS(X ) for some X ∈ K, which, by (2), implies that Y ∈ K. Thus, KVK = K, and so K is definable.

While this theorem gives a necessary and sufficient condition for a class of topological spaces to be definable, the problem with it is that Condition (2) is hard to verify. Gabelaia [16, Thm. 2.3.4] established the following alternative to Theorem 5.10. Let X be a topological space. The Alexandroff extension a(X) of X is the topological space (uf(P(X)), τR), where R is the Jónsson-Tarski quasi-order; note that R is the specialization order of s(X) by Lemma 2.11 and the definition of ∗ s(X). Thus, a(X) is obtained by taking the Alexandroff topology of (X )+. We ∗ + point out that if X is a quasi-order, then (X )+ is equal to (X )+. Consequently, whenever X is a quasi-order, a(X) is nothing more than the ultrafilter extension of X [8, Def. 2.57]. We say that a class K of topological spaces preserves Alexandroff extensions if X ∈ K implies that a(X) ∈ K, and that K reflects Alexandroff extensions if a(X) ∈ K implies that X ∈ K. The natural map i: X → a(X) is defined by i(x) = Px. Then we have the following theorem, generalizing the well-known Goldblatt-Thomason theorem for quasi-ordered sets (see [25, Thm. 8] or [8, Thm. 3.19]). Topo-canonical completions of closure algebras and Heyting algebras 31

Theorem 5.11. [16, Thm. 2.3.4] Let K be a class of topological spaces which preserves Alexandroff extensions. Then K is definable if and only if K is closed under open subspaces, open continuous images, topological sums, and reflects Alexandroff extensions. It is definitely easier to verify the four conditions of Theorem 5.11 than Condition (2) of Theorem 5.10. Nevertheless, Gabelaia’s theorem has drawbacks of its own. For one, as seen below, there are definable classes of topological spaces which do not preserve Alexandroff extensions, so these are missed by the theorem. Example 5.12. Let K be the class of hereditarily irresolvable spaces. Then K is definable by GRZ (see Example 5.8). We claim that K does not preserve Alexan- droff extensions. First, if (X, R) is a quasi-ordered set topologized with the corresponding Alexandroff topology, then X is hereditarily irresolvable if and only if X has no infinite ascending chains [15]. Let X = {−1, −2,... } with the usual order ≤. Obviously (X, τ≤) is hereditarily irresolvable. We claim that a(X) is not hereditarily irresolvable. If Q is a free ultrafilter on X and x ∈ X, then QRPx because R−1[x] is cofinite, hence in Q. For the same reason, QRM for any two free ultrafilters Q and M. Thus, a(X) has infinite ascending chains: if Q,M are distinct free ultrafilters, we have the infinitely ascending chain Q,M,Q,M,... . Therefore, a(X) ∈/ K, and so K does not preserve Alexandroff extensions. But even more importantly, the Alexandroff extension a(X) of a topological space X has several topologically unpleasant properties (see Example 5.13 below): (i) the natural map i: X → a(X) need not be continuous, (ii) the image of X need not be dense in a(X), and (iii) a(X) need not be compact. Thus, a(X) is far from being a compactification of X. Example 5.13. Let X be an infinite set, let P be a free ultrafilter on X, and let τ be the topology P ∪{∅}. Because every element of P is dense in X, we have QRP for each ultrafilter Q on X. Let Q,M be distinct ultrafilters. We show that Q RM/ if M 6= P . Indeed, there is A ∈ Q − M and B ∈ P − M. Then A ∪ B ∈ Q ∩ P but A ∪ B/∈ M. Therefore, there is an open set in Q which is not in M. Thus, Q RM/ . Consequently, we can picture a(X) as follows: Pr ,, , , , ,q q q p p p

From this description of the quasi-order R on a(X), we see that i(X) has the discrete topology as a subspace of a(X). Consequently, i is not continuous. Moreover, since ↓ i(X) = i(X), we have that i(X) is not dense in a(X). Finally, observe 32 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers. that {↑ Q : Q ∈ a(X)} is an infinite cover of a(X) which has no finite subcover. Therefore, a(X) is not compact. It is a reasonable idea to replace a(X) by a compactification of X. One natural candidate is the Salbany compactification in the case of closure algebras (and the Banaschewski compactification in the case of Heyting algebras). However, the next example shows that a variant of Gabelaia’s theorem with s(X) (resp. b(X)) instead of a(X) is false. Example 5.14. We have seen that the class K of weakly scattered spaces is not definable. We show that it is closed under open subspaces, open continuous images, topological sums, and both preserves and reflects the Salbany compactification. Since the Banaschewski compactification is the T0-reflection of the Salbany compactification, the same result will also be true for the Banaschewski compactification. It is elementary to see that K is closed under open subspaces, open continuous images, and topological sums. Let X be weakly scattered. We recall that i: X → s(X) defined by i(x)= Px, where Px is the principal ultrafilter on X containing x, is an embedding. Therefore, if x ∈ X is an isolated point, then Px is an isolated point of s(X). Consequently, since i(X) is dense in s(X) by Proposition 4.2, {Px : x ∈ Iso(X)} is a dense subset of s(X) contained in Iso(s(X)). Thus, s(X) is weakly scattered. Therefore, K preserves the Salbany compactification. To see that K reflects the Salbany compactification, suppose that s(X) is weakly scattered. Then Iso(s(X)) is dense in s(X). Also, i(X) is dense in s(X) by Proposition 4.2. Therefore, Iso(s(X))∩i(X) is dense in s(X) because s(X) is a McKinsey space by Example 5.5. Thus, Iso(s(X))∩i(X) = Iso(i(X)) is dense in i(X), which implies that i(X) is weakly scattered. Moreover, by Proposition 4.2, i(X) is homeomorphic to X. Therefore, X is weakly scattered.

References

[1] M. Abashidze and L. Esakia, Cantor’s scattered spaces and the provability logic, Baku International Topological Conference, Volume of Abstracts Part I, 1987, p. 3 (Russian). [2] M. Aiello, J. van Benthem, and G. Bezhanishvili, Reasoning about space: The modal way, J. Logic Comput. 13 (2003), no. 6, 889–920. [3] R. Balbes and P. Dwinger, Distributive lattices, University of Missouri Press, Columbia, Mo., 1974. [4] B. Banaschewski, Coherent frames, Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices, Lecture Notes in Math., Vol. 871, Springer-Verlag, Berlin, 1981, pp. 1–11. [5] G. Bezhanishvili, L. Esakia, and D. Gabelaia, Some results on modal axiomatization and definability for topological spaces, Studia Logica 81 (2005), no. 3, 325–355. [6] G. Bezhanishvili, M. Gehrke, R. Mines, and P. J. Morandi, Profinite completions and canonical extensions of Heyting algebras, Order 23 (2006), pp. 143–161. [7] G. Bezhanishvili, R. Mines, and P. J. Morandi, Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces, Topology Appl. 132 (2003), no. 3, 291–306. Topo-canonical completions of closure algebras and Heyting algebras 33

[8] P. Blackburn, M. de Rijke, and Y. Venema, Modal logic, Cambridge University Press, 2001. [9] W. J. Blok, Varieties of interior algebras, Ph.D. thesis, University of Amsterdam, 1976. [10] W. J. Blok and P. Dwinger, Equational classes of closure algebras. I, Indag. Math. 37 (1975), 189–198. [11] A. Chagrov and M. Zakharyaschev, Modal logic, Oxford Logic Guides, vol. 35, The Clarendon Press Oxford University Press, New York, 1997. [12] M. de Rijke and Y. Venema, Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras, Studia Logica 54 (1995), no. 1, 61–78. [13] L. L. Esakia, Topological Kripke models, Soviet Math. Dokl. 15 (1974), 147–151. [14] , Heyting algebras I. Duality theory (Russian), “Metsniereba”, Tbilisi, 1985. [15] D. Gabelaia, Modal logics GL and Grz: semantical comparison, Proceedings of the ESSLLI Student Session, 1999, pp. 91–97. [16] , Modal definability in topology, Master’s Thesis, 2001. [17] M. Gehrke and J. Harding, Bounded lattice expansions, J. Algebra 238 (2001), no. 1, 345–371. [18] M. Gehrke and B. Jónsson, Bounded distributive lattices with operators, Math. Japon. 40 (1994), no. 2, 207–215. [19] , Monotone bounded distributive lattice expansions, Math. Japon. 52 (2000), no. 2, 197–213. [20] , Bounded distributive lattice expansions, Math. Scand. 94 (2004), no. 1, 13–45. [21] R. Goldblatt, Varieties of complex algebras, Ann. Pure Appl. Logic 44 (1989), no. 3, 173–242. [22] , On closure under canonical embedding algebras, Algebraic logic (Budapest, 1988), Colloq. Math. Soc. János Bolyai, vol. 54, North-Holland, Amsterdam, 1991, pp. 217–229. [23] , Elementary generation and canonicity for varieties of Boolean algebras with operators, Algebra Universalis 34 (1995), no. 4, 551–607. [24] R. Goldblatt, I. Hodkinson, and Y. Venema, Erd˝os graphs resolve Fine’s canonicity problem, Bull. Symbolic Logic 10 (2004), no. 2, 186–208. [25] R. I. Goldblatt and S. K. Thomason, Axiomatic classes in propositional modal logic, Algebra and logic (Fourteenth Summer Res. Inst., Austral. Math. Soc., Monash Univ., Clayton, 1974), Lecture Notes in Math., Vol. 450, Springer-Verlag, Berlin, 1975, pp. 163–173. [26] G. Grätzer, General lattice theory, Birkhäuser Verlag, Basel, 1978. [27] J. Harding and G. Bezhanishvili, Macneille completions of modal algebras, Houston Journal of Mathematics 33 (2007), pp. 355–384. [28] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. [29] P. T. Johnstone, Stone spaces, Cambridge University Press, Cambridge, 1982. [30] B. Jónsson, On the canonicity of Sahlqvist identities, Studia Logica 53 (1994), no. 4, 473–491. [31] B. Jónsson and A. Tarski, Boolean algebras with operators. I, Amer. J. Math. 73 (1951), 891–939. [32] , Boolean algebras with operators. II, Amer. J. Math. 74 (1952), 127–162. [33] S. A. Kripke, Semantical analysis of intuitionistic logic. I, Formal Systems and Recursive Functions (Proc. Eighth Logic Colloq., Oxford, 1963), North-Holland, Amsterdam, 1965, pp. 92–130. [34] J. C. C. McKinsey and A. Tarski, The algebra of topology, Annals of Mathematics 45 (1944), 141–191. [35] , On closed elements in closure algebras, Ann. of Math. (2) 47 (1946), 122–162. [36] H. A. Priestley, Representation of distributive lattices by means of ordered stone spaces, Bull. London Math. Soc. 2 (1970), 186–190. 34 G.Bezhanishvili,R.Mines,andP.J.Morandi Algebra univers.

[37] H. Rasiowa and R. Sikorski, The mathematics of metamathematics, Monografie Matematyczne, Tom 41, Państwowe Wydawnictwo Naukowe, Warsaw, 1963. [38] S. Salbany, Ultrafilter spaces and compactifications, Portugal. Math. 57 (2000), no. 4, 481–492. [39] M. Stone, Topological representation of distributive lattices and Brouwerian logics, Casopisˇ peˇst. mat. fys. 67 (1937), 1–25. [40] A. Tarski, Der aussagenkalkül und die topologie, Fund. Math. 31 (1938), 103–134.

Guram Bezhanishvili, Ray Mines, and Patrick J. Morandi Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, USA e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]