Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by
piecing together results from various authors. Specifically, we show that if
$\mathbb{A}$ is a residually finite, finitely generated modal algebra such that
$\operatorname{HSP}(\mathbb{A})$ has equationally definable principal
congruences, then the profinite completion of $\mathbb{A}$ is isomorphic to its
MacNeille completion, and $\Diamond$ is smooth. Specific examples of such modal
algebras are the free $\mathbf{K4}$-algebra and the free
$\mathbf{PDL}$-algebra.Comment: 5 page

Vosmaer, Jacob

English

arXiv

Following Bezhanishvili and Vosmaer, we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if A is a residually finite, finitely generated modal algebra such that HSP(A) has equationally definable principal congruences, then the profinite completion of A is isomorphic to its MacNeille completion, and ◊ is smooth. Specific examples of such modal algebras are the free K4-algebra and the free PDL-algebra

Vosmaer, J.

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)MacNeille completion and profinite completion can coincide on finitely generatedmodal algebrasVosmaer, J.DOI10.1007/s00012-009-0028-9Publication date2009Document VersionFinal published versionPublished inAlgebra UniversalisLink to publicationCitation for published version (APA):Vosmaer, J. (2009). MacNeille completion and profinite completion can coincide on finitelygenerated modal algebras. Algebra Universalis, 61(3-4), 449-453.https://doi.org/10.1007/s00012-009-0028-9General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:09 Mar 2023Algebra Univers. 61 (2009) 449–453 DOI 10.1007/s00012-009-0028-9Published online November 18, 2009© 2009 The Author(s)This article is published with open access at Springerlink.comAlgebra UniversalisMacNeille completion and profinite completion cancoincide on finitely generated modal algebrasJacob VosmaerAbstract. Following Bezhanishvili and Vosmaer, we confirm a conjecture of YdeVenema by piecing together results from various authors. Specifically, we show thatif A is a residually finite, finitely generated modal algebra such that HSP(A) hasequationally definable principal congruences, then the profinite completion of A isisomorphic to its MacNeille completion, and ♦ is smooth. Specific examples of suchmodal algebras are the free K4-algebra and the free PDL-algebra.1. IntroductionIn this paper we compare two mathematical constructions applied to modalalgebras. The first is the MacNeille completion, which is an order-theoreticgeneralization of the construction of the reals from the rationals using Dedekindcuts [12]. It has been applied in logic to prove, for example, the completenessof predicate calculi [14]. The second is the profinite completion, which is auniversal algebraic construction, transforming an algebra into a topological al-gebra endowed with a Stone (compact, Hausdorff, zero-dimensional) topology.This construction stems from Galois theory [15], but has more recently alsobeen connected with lattice completions [3, 10, 19, 4].This paper is a companion piece to [4]. In that paper, parallel versionsof our Theorems 3.1 and 4.2 arise in a study of the connections between dif-ferent completions of Heyting algebras, using Esakia duality. In light of thetopological character of the profinite completion, in the present paper we willpresent topological algebra proofs instead. This establishes a strong connec-tion with the body of work on canonicity [8] and MacNeille canonicity [17].Another advantage of our present perspective is that we can show how the twomain theorems pivot around an interaction between principal lattice filters andprincipal algebra congruences, which are in a one-to-one correspondence forHeyting algebras, but not for modal algebras. Finally, we will briefly mentionsome of the connections of our results to modal logic.Presented by I. Hodkinson.Received November 13, 2007; accepted in final form April 28, 2008.2000 Mathematics Subject Classification: Primary: 06E25; Secondary: 06B23, 03B45,22A30.Key words and phrases: modal algebra, MacNeille completion, profinite completion.This research was supported by VICI grant 639.073.501 of the Netherlands Organizationfor Scientific Research (NWO).450 J. Vosmaer Algebra Univers.2. Completions and topologiesLet B = 〈B;∧,∨,¬, 0, 1〉 be a Boolean algebra. Given b ∈ B we writeb↓ = {a ∈ B | a ≤ b} (b↑ is defined dually). We say S ⊆ B is join-dense in B iff,for every a ∈ B, a =∨(a↓∩S) (meet-density is defined dually). A completionof a lattice B is a pair (m, C), where m : B ↪→ C is a lattice embedding intoa complete lattice C. Completions (m, C) and (k, D) of B are isomorphic ifgm = k for some lattice isomorphism g : C → D. If (m, C) is a completion ofB, let ρB be the topology on C generated by basis {[m(a),m(b)] | a, b ∈ B}(where [x, y] = {z ∈ C | x ≤ z ≤ y}). By γ↓B, γ↑Band γB we denote the Scotttopology, the dual Scott topology, and the biScott topology on B respectively.Let At B be the (possibly empty) set of atoms of B, and let Atω B be the set ofall finite joins of atoms of B. Then ιB is the topology generated by the basis{[a,¬b] | a, b ∈ Atω B}. By [8, Section 2], ιB = γB if B is complete and atomic.The MacNeille completion [2] of a Boolean algebra B is defined up to iso-morphism as a completion (m, C) such that m[B] is join-dense in C (by [5,Theorem V-27] C is then also a Boolean algebra). We denote the MacNeillecompletion of B by B̄. Alternatively [17, Theorem 4.5], B̄ can be characterizedup to isomorphism as a completion (m, C) of B such that 〈C, ρB〉 is Haus-dorff. If f : B → C is an order-preserving map between Boolean algebras, thenf◦ : B̄ → C̄, defined by f◦ : x →∨{f(a) | mB(a) ≤ x}, is the lower extensionof f . The upper extension f• is defined dually. Alternatively [17, Section 5],f◦ is the (pointwise) largest (ρB, γ↓C̄)-continuous extension of f , and f• is thesmallest (ρB, γ↑C̄)-continuous extension of f . We say f is smooth if f◦ = f•.Given a modal algebra A = 〈A;♦〉, let ΦA := {θ ∈ Con A | A/θ is finite}.We say A is residually finite if, for all a, b ∈ A with a = b, there exists θ ∈ ΦAsuch that a/θ = b/θ. The inverse system 〈{A/θ}θ∈ΦA , fθψ〉, where fθψ : A/θ A/ψ (for all θ, ψ ∈ ΦA such that θ ⊆ ψ) is defined by fθψ : a/θ → a/ψ, has aprojective limitÂ ={α ∈∏ΦAA/θ | ∀θ, ψ ∈ ΦA with θ ⊆ ψ, if α(θ) = a/θ then α(ψ) = a/ψ}.The map µ : A → Â, defined by µ : a → (a/θ)θ∈ΦA , is a modal algebra homo-morphism which is injective iff A is residually finite. We call Â the profinitecompletion of A [15]. Since Â is a complete lattice [10], it follows that (µ, Â)is a completion of A iff A is residually finite. If we define the discrete topol-ogy on each A/θ, Â inherits a topology τÂas a closed subspace of the prod-uct∏ΦA A/θ. Now 〈Â, τÂ〉 is a Stone space [3, Section 2], and in particular♦̂ : Â → Â is (τÂ, τÂ)-continuous [1].Lemma 2.1. If A is a Boolean algebra expansion, then τÂ= ιÂ= γÂ.Proof. Since 〈Â, τÂ〉 is a compact Hausdorff topological lattice, it follows by [9,Corollary VII-2.3] that τÂ= γÂ. Since Â is also a complete, atomic Booleanalgebra [3, 19], we know that ιÂ= γÂ[8, Section 2].  MacNeille completion and profi nite completion of modal algebras 4513. Comparing profinite completion and MacNeille completionTheorem 3.1 (cf. [4, Theorem 4.12]). Let A be a modal algebra. The followingare equivalent:(1) the profinite completion (µ, Â) is the MacNeille completion of A, and ♦is smooth;(2) A is residually finite and, for every θ ∈ ΦA, 1/θ is a principal lattice filter.Proof. If (µ, Â) is the MacNeille completion of A, then µ : A → Â must beinjective, so that A is residually finite (see above). Let θ ∈ ΦA; then it followsfrom the definition of Â that the projection πθ : Â  A/θ commutes with µ andthe natural map a → a/θ; i.e., a/θ = πθµ(a). By [3, Lemma 2.7], π−1θ (1/θ) is aclosed principal filter of Â; say π−1θ (1/θ) = α↑. Because of the correspondencebetween modal filters and modal congruences [18, Theorem 29], α↑ completelycharacterizes A/θ in the following sense: A/θ ∼= [0, α]Âas a bounded lattice[7, Exercise 4.12]. This implies that α↓ is finite. Since (µ, Â) is the MacNeillecompletion of A, µ[A] is join-dense in Â, so by finiteness of α↓, there must exista ∈ A such that µ(a) = α. Now b/θ = 1/θ iff µ(b) ∈ π−1θ (1/θ) = α↑ = µ(a)↑iff b ≥ a, so 1/θ = a↑ is a principal lattice filter.Conversely, if A is residually finite, then µ : A → Â is injective, so (µ, Â) isa completion of A. To show that (µ, Â) is the MacNeille completion of A, wewill consider the different topologies on Â. We first show that At Â ⊆ µ[A].If α ∈ At Â, there must be some θ ∈ ΦA such that α(θ) ∈ At A/θ. Because1/θ is a principal lattice filter c↑, we know that A/θ ∼= [0, c]A as a boundedlattice, so there must be some a ≤ c with a ∈ At A and a/θ = α(θ). But thenµ(a) = α. It follows that At Â ⊆ µ[A], whence ι ⊆ ρ. Since ι is Hausdorff, sois ρ. Using [17, Theorem 4.5] we can thus conclude that, as far as the Booleansubstructure of A is concerned, (µ, Â) is the MacNeille completion of A. Nowto show that ♦ is smooth, remember that ♦̂ : Â → Â is (τ, τ)-continuous. Sinceτ = ι = γ by Lemma 2.1 and ι ⊆ ρ, it follows that ♦̂ is (ρ, γ)-continuous. Butthen since γ↓, γ↑ ⊆ γ, it follows by [17, Proposition 5.9] that ♦• ≤ ♦̂ ≤ ♦◦.Since also ♦◦ ≤ ♦• [17, Proposition 5.6], it follows that ♦ is smooth. Note that the theorem above also admits a third equivalent condition, char-acterizing the dual space of A. This perspective is further explored in [4].4. Finitely generated modal algebras with EDPCHaving equationally definable principal congruences (EDPC) is a strongmeta-logical property of varieties of algebras that coincides with, for example,the existence of a deduction theorem or of a master modality [6, 11] for themodal logic corresponding to a variety of modal algebras. Examples of suchlogics are logics of bounded depth, n-transitive logics such as K4, or regulartest-free PDL with finitely many basic programs.452 J. Vosmaer Algebra Univers.Lemma 4.1 (Proposition 3.4.3 of [11]). Let V be a variety of modal algebras.V has EDPC iff every principal modal filter of an algebra in V is a principallattice filter.Note that the hypotheses below strongly resemble those of [20, Theorem 4].Theorem 4.2 (cf. [4, Corollary 4.5.3]). If A is a residually finite, finitelygenerated modal algebra such that HSP(A) has equationally definable principalcongruences, then the profinite completion (µ, Â) is the MacNeille completionof A and ♦ is smooth.Proof. Since A is a finitely generated algebra, every θ ∈ ΦA is compact [16,Theorem 1]. As is remarked in [20], every compact congruence of a modalalgebra is principal, so under our hypotheses, every θ ∈ ΦA is principal. Nowsince HSP(A) has EDPC, Lemma 4.1 tells us that 1/θ is a principal latticefilter for all θ ∈ ΦA, so by Theorem 3.1, (µ, Â) is the MacNeille completion ofA and ♦ is smooth. Note that the EDPC clause in the theorem above is suppressed in the Heyt-ing algebra case [4], because every variety of Heyting algebras has EDPC.The conditions of Theorem 4.2 above are sufficient; what about necessity?From Theorem 3.1 we know that it is necessary that A is residually finite.Moreover, in light of [19, Section 3.3], we know that it is necessary for Theorem4.2 that A is atomic. This helps us to find counterexamples to the theoremif we remove the requirements of being finitely generated or having EDPC.For instance, let A be the free algebra on one generator for the modal logicT (the logic of reflexive Kripke frames). Then [20, Corollary 7: Example 1]tells us that A is residually finite and finitely generated but not atomic. Thisshows us that being residually finite and finitely generated is not sufficientfor the conclusion of Theorem 4.2. Alternatively, the free transitive modalalgebra on ω generators is an example of a residually finite modal algebra A,generating a variety with EDPC, such that A is not atomic. In summary,residual finiteness is necessary for Theorem 4.2, and if we remove either of theother two conditions, we can find non-atomic counterexamples.Corollary 4.3. Let A be a finitely generated free algebra for K4 or PDL.Then the MacNeille completion and the profinite completion of A are the sameand ♦ is smooth.Proof. Let L be either K4 or PDL and let V be the variety corresponding toL. Since L has the finite model property, V is generated by its finite members.By [13, Theorem IV-14.4], this implies that A, being a finitely generated freealgebra for V, is residually finite. Using the fact that L has a master modality,it follows that we can apply Theorem 4.2. Acknowledgements. The author would like to thank Yde Venema, who sug-gested that Theorem 4.2 might be true. Additionally, the author is grateful tothe Editor and the Referee for their criticisms and suggestions. MacNeille completion and profi nite completion of modal algebras 453References[1] Banaschewski, B.: On profinite universal algebras. In: Novák, J. (ed.) Proceedings ofthe Third Prague Topological Symposium, pp. 51–62, September 1971[2] Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeillecompletion. Arch. Math. (Basel) 18, 369–377 (1967)[3] Bezhanishvili, G., Gehrke, M., Mines, R., Morandi, P.J.: Profinite completions andcanonical extensions of Heyting algebras. Order 23, 143–161 (2006)[4] Bezhanishvili, G., Vosmaer, J.: Comparison of MacNeille, canonical and profinitecompletions. Order 25, 299–320 (2008)[5] Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society (1967)[6] Blok, W., Pigozzi, D.: On the structure of varieties with equationally definableprincipal congruences III. Algebra Universalis 32, 545–608 (1994)[7] Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. CambridgeUniversity Press (2002)[8] Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94,13–45 (2004)[9] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: ACompendium of Continuous Lattices. Springer (1980)[10] Harding, J.: On profinite completions and canonical extensions. Algebra Universalis55, 293–296 (2006)[11] Kracht, M.: Tools and Techniques in Modal Logic. Elsevier (1999)[12] MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42, 416–160 (1927)[13] Mal’cev, A.I.: Algebraic Systems. Springer (1973)[14] Rasiowa, H., Sikorski, R.: A proof of the completeness theorem of Gödel. Fund. Math.37, 193–200 (1950)[15] Ribes, L., Zalesskii, P.: Profinite Groups. Springer (2000)[16] Rival, I., Sands, B.: A note on the congruence lattice of a finitely generated algebra.Proc. Amer. Math. Soc. 72, 451–455 (1978)[17] Theunissen, M., Venema, Y.: MacNeille completions of lattice expansions. AlgebraUniversalis 57, 143–193 (2007)[18] Venema, Y.: Algebras and Coalgebras. In: Blackburn, P., et al. (eds.) Handbook ofModal Logic. Elsevier (2007)[19] Vosmaer, J.: Connecting the profinite completion and the canonical extension usingduality. Master’s thesis, University of Amsterdam (2006)[20] Wolter, F.: A note on atoms in polymodal algebras. Algebra Universalis 37, 334–341(1997)Jacob VosmaerPostbus 94242, 1090 GE Amsterdam, The Netherlandse-mail : jvosmaer@science.uva.nlOpen Access This article is distributed under the terms of the Creative Commons Attribution Noncom-mercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

MacNeille completion and profinite completion can coincide on finitely generated modal algebras

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)MacNeille completion and profinite completion can coincide on finitely generatedmodal algebrasVosmaer, J.DOI10.1007/s00012-009-0028-9Publication date2009Document VersionFinal published versionPublished inAlgebra UniversalisLink to publicationCitation for published version (APA):Vosmaer, J. (2009). MacNeille completion and profinite completion can coincide on finitelygenerated modal algebras. Algebra Universalis, 61(3-4), 449-453.https://doi.org/10.1007/s00012-009-0028-9General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:25 Jul 2022Algebra Univers. 61 (2009) 449–453 DOI 10.1007/s00012-009-0028-9Published online November 18, 2009© 2009 The Author(s)This article is published with open access at Springerlink.comAlgebra UniversalisMacNeille completion and profinite completion cancoincide on finitely generated modal algebrasJacob VosmaerAbstract. Following Bezhanishvili and Vosmaer, we confirm a conjecture of YdeVenema by piecing together results from various authors. Specifically, we show thatif A is a residually finite, finitely generated modal algebra such that HSP(A) hasequationally definable principal congruences, then the profinite completion of A isisomorphic to its MacNeille completion, and ♦ is smooth. Specific examples of suchmodal algebras are the free K4-algebra and the free PDL-algebra.1. IntroductionIn this paper we compare two mathematical constructions applied to modalalgebras. The first is the MacNeille completion, which is an order-theoreticgeneralization of the construction of the reals from the rationals using Dedekindcuts [12]. It has been applied in logic to prove, for example, the completenessof predicate calculi [14]. The second is the profinite completion, which is auniversal algebraic construction, transforming an algebra into a topological al-gebra endowed with a Stone (compact, Hausdorff, zero-dimensional) topology.This construction stems from Galois theory [15], but has more recently alsobeen connected with lattice completions [3, 10, 19, 4].This paper is a companion piece to [4]. In that paper, parallel versionsof our Theorems 3.1 and 4.2 arise in a study of the connections between dif-ferent completions of Heyting algebras, using Esakia duality. In light of thetopological character of the profinite completion, in the present paper we willpresent topological algebra proofs instead. This establishes a strong connec-tion with the body of work on canonicity [8] and MacNeille canonicity [17].Another advantage of our present perspective is that we can show how the twomain theorems pivot around an interaction between principal lattice filters andprincipal algebra congruences, which are in a one-to-one correspondence forHeyting algebras, but not for modal algebras. Finally, we will briefly mentionsome of the connections of our results to modal logic.Presented by I. Hodkinson.Received November 13, 2007; accepted in final form April 28, 2008.2000 Mathematics Subject Classification: Primary: 06E25; Secondary: 06B23, 03B45,22A30.Key words and phrases: modal algebra, MacNeille completion, profinite completion.This research was supported by VICI grant 639.073.501 of the Netherlands Organizationfor Scientific Research (NWO).450 J. Vosmaer Algebra Univers.2. Completions and topologiesLet B = 〈B;∧,∨,¬, 0, 1〉 be a Boolean algebra. Given b ∈ B we writeb↓ = {a ∈ B | a ≤ b} (b↑ is defined dually). We say S ⊆ B is join-dense in B iff,for every a ∈ B, a =∨(a↓∩S) (meet-density is defined dually). A completionof a lattice B is a pair (m, C), where m : B ↪→ C is a lattice embedding intoa complete lattice C. Completions (m, C) and (k, D) of B are isomorphic ifgm = k for some lattice isomorphism g : C → D. If (m, C) is a completion ofB, let ρB be the topology on C generated by basis {[m(a),m(b)] | a, b ∈ B}(where [x, y] = {z ∈ C | x ≤ z ≤ y}). By γ↓B, γ↑Band γB we denote the Scotttopology, the dual Scott topology, and the biScott topology on B respectively.Let At B be the (possibly empty) set of atoms of B, and let Atω B be the set ofall finite joins of atoms of B. Then ιB is the topology generated by the basis{[a,¬b] | a, b ∈ Atω B}. By [8, Section 2], ιB = γB if B is complete and atomic.The MacNeille completion [2] of a Boolean algebra B is defined up to iso-morphism as a completion (m, C) such that m[B] is join-dense in C (by [5,Theorem V-27] C is then also a Boolean algebra). We denote the MacNeillecompletion of B by B̄. Alternatively [17, Theorem 4.5], B̄ can be characterizedup to isomorphism as a completion (m, C) of B such that 〈C, ρB〉 is Haus-dorff. If f : B → C is an order-preserving map between Boolean algebras, thenf◦ : B̄ → C̄, defined by f◦ : x →∨{f(a) | mB(a) ≤ x}, is the lower extensionof f . The upper extension f• is defined dually. Alternatively [17, Section 5],f◦ is the (pointwise) largest (ρB, γ↓C̄)-continuous extension of f , and f• is thesmallest (ρB, γ↑C̄)-continuous extension of f . We say f is smooth if f◦ = f•.Given a modal algebra A = 〈A;♦〉, let ΦA := {θ ∈ Con A | A/θ is finite}.We say A is residually finite if, for all a, b ∈ A with a = b, there exists θ ∈ ΦAsuch that a/θ = b/θ. The inverse system 〈{A/θ}θ∈ΦA , fθψ〉, where fθψ : A/θ A/ψ (for all θ, ψ ∈ ΦA such that θ ⊆ ψ) is defined by fθψ : a/θ → a/ψ, has aprojective limitÂ ={α ∈∏ΦAA/θ | ∀θ, ψ ∈ ΦA with θ ⊆ ψ, if α(θ) = a/θ then α(ψ) = a/ψ}.The map µ : A → Â, defined by µ : a → (a/θ)θ∈ΦA , is a modal algebra homo-morphism which is injective iff A is residually finite. We call Â the profinitecompletion of A [15]. Since Â is a complete lattice [10], it follows that (µ, Â)is a completion of A iff A is residually finite. If we define the discrete topol-ogy on each A/θ, Â inherits a topology τÂas a closed subspace of the prod-uct∏ΦA A/θ. Now 〈Â, τÂ〉 is a Stone space [3, Section 2], and in particular♦̂ : Â → Â is (τÂ, τÂ)-continuous [1].Lemma 2.1. If A is a Boolean algebra expansion, then τÂ= ιÂ= γÂ.Proof. Since 〈Â, τÂ〉 is a compact Hausdorff topological lattice, it follows by [9,Corollary VII-2.3] that τÂ= γÂ. Since Â is also a complete, atomic Booleanalgebra [3, 19], we know that ιÂ= γÂ[8, Section 2].  MacNeille completion and profi nite completion of modal algebras 4513. Comparing profinite completion and MacNeille completionTheorem 3.1 (cf. [4, Theorem 4.12]). Let A be a modal algebra. The followingare equivalent:(1) the profinite completion (µ, Â) is the MacNeille completion of A, and ♦is smooth;(2) A is residually finite and, for every θ ∈ ΦA, 1/θ is a principal lattice filter.Proof. If (µ, Â) is the MacNeille completion of A, then µ : A → Â must beinjective, so that A is residually finite (see above). Let θ ∈ ΦA; then it followsfrom the definition of Â that the projection πθ : Â  A/θ commutes with µ andthe natural map a → a/θ; i.e., a/θ = πθµ(a). By [3, Lemma 2.7], π−1θ (1/θ) is aclosed principal filter of Â; say π−1θ (1/θ) = α↑. Because of the correspondencebetween modal filters and modal congruences [18, Theorem 29], α↑ completelycharacterizes A/θ in the following sense: A/θ ∼= [0, α]Âas a bounded lattice[7, Exercise 4.12]. This implies that α↓ is finite. Since (µ, Â) is the MacNeillecompletion of A, µ[A] is join-dense in Â, so by finiteness of α↓, there must exista ∈ A such that µ(a) = α. Now b/θ = 1/θ iff µ(b) ∈ π−1θ (1/θ) = α↑ = µ(a)↑iff b ≥ a, so 1/θ = a↑ is a principal lattice filter.Conversely, if A is residually finite, then µ : A → Â is injective, so (µ, Â) isa completion of A. To show that (µ, Â) is the MacNeille completion of A, wewill consider the different topologies on Â. We first show that At Â ⊆ µ[A].If α ∈ At Â, there must be some θ ∈ ΦA such that α(θ) ∈ At A/θ. Because1/θ is a principal lattice filter c↑, we know that A/θ ∼= [0, c]A as a boundedlattice, so there must be some a ≤ c with a ∈ At A and a/θ = α(θ). But thenµ(a) = α. It follows that At Â ⊆ µ[A], whence ι ⊆ ρ. Since ι is Hausdorff, sois ρ. Using [17, Theorem 4.5] we can thus conclude that, as far as the Booleansubstructure of A is concerned, (µ, Â) is the MacNeille completion of A. Nowto show that ♦ is smooth, remember that ♦̂ : Â → Â is (τ, τ)-continuous. Sinceτ = ι = γ by Lemma 2.1 and ι ⊆ ρ, it follows that ♦̂ is (ρ, γ)-continuous. Butthen since γ↓, γ↑ ⊆ γ, it follows by [17, Proposition 5.9] that ♦• ≤ ♦̂ ≤ ♦◦.Since also ♦◦ ≤ ♦• [17, Proposition 5.6], it follows that ♦ is smooth. Note that the theorem above also admits a third equivalent condition, char-acterizing the dual space of A. This perspective is further explored in [4].4. Finitely generated modal algebras with EDPCHaving equationally definable principal congruences (EDPC) is a strongmeta-logical property of varieties of algebras that coincides with, for example,the existence of a deduction theorem or of a master modality [6, 11] for themodal logic corresponding to a variety of modal algebras. Examples of suchlogics are logics of bounded depth, n-transitive logics such as K4, or regulartest-free PDL with finitely many basic programs.452 J. Vosmaer Algebra Univers.Lemma 4.1 (Proposition 3.4.3 of [11]). Let V be a variety of modal algebras.V has EDPC iff every principal modal filter of an algebra in V is a principallattice filter.Note that the hypotheses below strongly resemble those of [20, Theorem 4].Theorem 4.2 (cf. [4, Corollary 4.5.3]). If A is a residually finite, finitelygenerated modal algebra such that HSP(A) has equationally definable principalcongruences, then the profinite completion (µ, Â) is the MacNeille completionof A and ♦ is smooth.Proof. Since A is a finitely generated algebra, every θ ∈ ΦA is compact [16,Theorem 1]. As is remarked in [20], every compact congruence of a modalalgebra is principal, so under our hypotheses, every θ ∈ ΦA is principal. Nowsince HSP(A) has EDPC, Lemma 4.1 tells us that 1/θ is a principal latticefilter for all θ ∈ ΦA, so by Theorem 3.1, (µ, Â) is the MacNeille completion ofA and ♦ is smooth. Note that the EDPC clause in the theorem above is suppressed in the Heyt-ing algebra case [4], because every variety of Heyting algebras has EDPC.The conditions of Theorem 4.2 above are sufficient; what about necessity?From Theorem 3.1 we know that it is necessary that A is residually finite.Moreover, in light of [19, Section 3.3], we know that it is necessary for Theorem4.2 that A is atomic. This helps us to find counterexamples to the theoremif we remove the requirements of being finitely generated or having EDPC.For instance, let A be the free algebra on one generator for the modal logicT (the logic of reflexive Kripke frames). Then [20, Corollary 7: Example 1]tells us that A is residually finite and finitely generated but not atomic. Thisshows us that being residually finite and finitely generated is not sufficientfor the conclusion of Theorem 4.2. Alternatively, the free transitive modalalgebra on ω generators is an example of a residually finite modal algebra A,generating a variety with EDPC, such that A is not atomic. In summary,residual finiteness is necessary for Theorem 4.2, and if we remove either of theother two conditions, we can find non-atomic counterexamples.Corollary 4.3. Let A be a finitely generated free algebra for K4 or PDL.Then the MacNeille completion and the profinite completion of A are the sameand ♦ is smooth.Proof. Let L be either K4 or PDL and let V be the variety corresponding toL. Since L has the finite model property, V is generated by its finite members.By [13, Theorem IV-14.4], this implies that A, being a finitely generated freealgebra for V, is residually finite. Using the fact that L has a master modality,it follows that we can apply Theorem 4.2. Acknowledgements. The author would like to thank Yde Venema, who sug-gested that Theorem 4.2 might be true. Additionally, the author is grateful tothe Editor and the Referee for their criticisms and suggestions. MacNeille completion and profi nite completion of modal algebras 453References[1] Banaschewski, B.: On profinite universal algebras. In: Novák, J. (ed.) Proceedings ofthe Third Prague Topological Symposium, pp. 51–62, September 1971[2] Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeillecompletion. Arch. Math. (Basel) 18, 369–377 (1967)[3] Bezhanishvili, G., Gehrke, M., Mines, R., Morandi, P.J.: Profinite completions andcanonical extensions of Heyting algebras. Order 23, 143–161 (2006)[4] Bezhanishvili, G., Vosmaer, J.: Comparison of MacNeille, canonical and profinitecompletions. Order 25, 299–320 (2008)[5] Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society (1967)[6] Blok, W., Pigozzi, D.: On the structure of varieties with equationally definableprincipal congruences III. Algebra Universalis 32, 545–608 (1994)[7] Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. CambridgeUniversity Press (2002)[8] Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94,13–45 (2004)[9] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: ACompendium of Continuous Lattices. Springer (1980)[10] Harding, J.: On profinite completions and canonical extensions. Algebra Universalis55, 293–296 (2006)[11] Kracht, M.: Tools and Techniques in Modal Logic. Elsevier (1999)[12] MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42, 416–160 (1927)[13] Mal’cev, A.I.: Algebraic Systems. Springer (1973)[14] Rasiowa, H., Sikorski, R.: A proof of the completeness theorem of Gödel. Fund. Math.37, 193–200 (1950)[15] Ribes, L., Zalesskii, P.: Profinite Groups. Springer (2000)[16] Rival, I., Sands, B.: A note on the congruence lattice of a finitely generated algebra.Proc. Amer. Math. Soc. 72, 451–455 (1978)[17] Theunissen, M., Venema, Y.: MacNeille completions of lattice expansions. AlgebraUniversalis 57, 143–193 (2007)[18] Venema, Y.: Algebras and Coalgebras. In: Blackburn, P., et al. (eds.) Handbook ofModal Logic. Elsevier (2007)[19] Vosmaer, J.: Connecting the profinite completion and the canonical extension usingduality. Master’s thesis, University of Amsterdam (2006)[20] Wolter, F.: A note on atoms in polymodal algebras. Algebra Universalis 37, 334–341(1997)Jacob VosmaerPostbus 94242, 1090 GE Amsterdam, The Netherlandse-mail : jvosmaer@science.uva.nlOpen Access This article is distributed under the terms of the Creative Commons Attribution Noncom-mercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)MacNeille completion and profinite completion can coincide on finitely generatedmodal algebrasVosmaer, J.DOI10.1007/s00012-009-0028-9Publication date2009Document VersionFinal published versionPublished inAlgebra UniversalisLink to publicationCitation for published version (APA):Vosmaer, J. (2009). MacNeille completion and profinite completion can coincide on finitelygenerated modal algebras. Algebra Universalis, 61(3-4), 449-453.https://doi.org/10.1007/s00012-009-0028-9General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:06 Jan 2026Algebra Univers. 61 (2009) 449–453 DOI 10.1007/s00012-009-0028-9Published online November 18, 2009© 2009 The Author(s)This article is published with open access at Springerlink.comAlgebra UniversalisMacNeille completion and profinite completion cancoincide on finitely generated modal algebrasJacob VosmaerAbstract. Following Bezhanishvili and Vosmaer, we confirm a conjecture of YdeVenema by piecing together results from various authors. Specifically, we show thatif A is a residually finite, finitely generated modal algebra such that HSP(A) hasequationally definable principal congruences, then the profinite completion of A isisomorphic to its MacNeille completion, and ♦ is smooth. Specific examples of suchmodal algebras are the free K4-algebra and the free PDL-algebra.1. IntroductionIn this paper we compare two mathematical constructions applied to modalalgebras. The first is the MacNeille completion, which is an order-theoreticgeneralization of the construction of the reals from the rationals using Dedekindcuts [12]. It has been applied in logic to prove, for example, the completenessof predicate calculi [14]. The second is the profinite completion, which is auniversal algebraic construction, transforming an algebra into a topological al-gebra endowed with a Stone (compact, Hausdorff, zero-dimensional) topology.This construction stems from Galois theory [15], but has more recently alsobeen connected with lattice completions [3, 10, 19, 4].This paper is a companion piece to [4]. In that paper, parallel versionsof our Theorems 3.1 and 4.2 arise in a study of the connections between dif-ferent completions of Heyting algebras, using Esakia duality. In light of thetopological character of the profinite completion, in the present paper we willpresent topological algebra proofs instead. This establishes a strong connec-tion with the body of work on canonicity [8] and MacNeille canonicity [17].Another advantage of our present perspective is that we can show how the twomain theorems pivot around an interaction between principal lattice filters andprincipal algebra congruences, which are in a one-to-one correspondence forHeyting algebras, but not for modal algebras. Finally, we will briefly mentionsome of the connections of our results to modal logic.Presented by I. Hodkinson.Received November 13, 2007; accepted in final form April 28, 2008.2000 Mathematics Subject Classification: Primary: 06E25; Secondary: 06B23, 03B45,22A30.Key words and phrases: modal algebra, MacNeille completion, profinite completion.This research was supported by VICI grant 639.073.501 of the Netherlands Organizationfor Scientific Research (NWO).450 J. Vosmaer Algebra Univers.2. Completions and topologiesLet B = 〈B;∧,∨,¬, 0, 1〉 be a Boolean algebra. Given b ∈ B we writeb↓ = {a ∈ B | a ≤ b} (b↑ is defined dually). We say S ⊆ B is join-dense in B iff,for every a ∈ B, a =∨(a↓∩S) (meet-density is defined dually). A completionof a lattice B is a pair (m, C), where m : B ↪→ C is a lattice embedding intoa complete lattice C. Completions (m, C) and (k, D) of B are isomorphic ifgm = k for some lattice isomorphism g : C → D. If (m, C) is a completion ofB, let ρB be the topology on C generated by basis {[m(a),m(b)] | a, b ∈ B}(where [x, y] = {z ∈ C | x ≤ z ≤ y}). By γ↓B, γ↑Band γB we denote the Scotttopology, the dual Scott topology, and the biScott topology on B respectively.Let At B be the (possibly empty) set of atoms of B, and let Atω B be the set ofall finite joins of atoms of B. Then ιB is the topology generated by the basis{[a,¬b] | a, b ∈ Atω B}. By [8, Section 2], ιB = γB if B is complete and atomic.The MacNeille completion [2] of a Boolean algebra B is defined up to iso-morphism as a completion (m, C) such that m[B] is join-dense in C (by [5,Theorem V-27] C is then also a Boolean algebra). We denote the MacNeillecompletion of B by B̄. Alternatively [17, Theorem 4.5], B̄ can be characterizedup to isomorphism as a completion (m, C) of B such that 〈C, ρB〉 is Haus-dorff. If f : B → C is an order-preserving map between Boolean algebras, thenf◦ : B̄ → C̄, defined by f◦ : x →∨{f(a) | mB(a) ≤ x}, is the lower extensionof f . The upper extension f• is defined dually. Alternatively [17, Section 5],f◦ is the (pointwise) largest (ρB, γ↓C̄)-continuous extension of f , and f• is thesmallest (ρB, γ↑C̄)-continuous extension of f . We say f is smooth if f◦ = f•.Given a modal algebra A = 〈A;♦〉, let ΦA := {θ ∈ Con A | A/θ is finite}.We say A is residually finite if, for all a, b ∈ A with a = b, there exists θ ∈ ΦAsuch that a/θ = b/θ. The inverse system 〈{A/θ}θ∈ΦA , fθψ〉, where fθψ : A/θ A/ψ (for all θ, ψ ∈ ΦA such that θ ⊆ ψ) is defined by fθψ : a/θ → a/ψ, has aprojective limitÂ ={α ∈∏ΦAA/θ | ∀θ, ψ ∈ ΦA with θ ⊆ ψ, if α(θ) = a/θ then α(ψ) = a/ψ}.The map µ : A → Â, defined by µ : a → (a/θ)θ∈ΦA , is a modal algebra homo-morphism which is injective iff A is residually finite. We call Â the profinitecompletion of A [15]. Since Â is a complete lattice [10], it follows that (µ, Â)is a completion of A iff A is residually finite. If we define the discrete topol-ogy on each A/θ, Â inherits a topology τÂas a closed subspace of the prod-uct∏ΦA A/θ. Now 〈Â, τÂ〉 is a Stone space [3, Section 2], and in particular♦̂ : Â → Â is (τÂ, τÂ)-continuous [1].Lemma 2.1. If A is a Boolean algebra expansion, then τÂ= ιÂ= γÂ.Proof. Since 〈Â, τÂ〉 is a compact Hausdorff topological lattice, it follows by [9,Corollary VII-2.3] that τÂ= γÂ. Since Â is also a complete, atomic Booleanalgebra [3, 19], we know that ιÂ= γÂ[8, Section 2].  MacNeille completion and profi nite completion of modal algebras 4513. Comparing profinite completion and MacNeille completionTheorem 3.1 (cf. [4, Theorem 4.12]). Let A be a modal algebra. The followingare equivalent:(1) the profinite completion (µ, Â) is the MacNeille completion of A, and ♦is smooth;(2) A is residually finite and, for every θ ∈ ΦA, 1/θ is a principal lattice filter.Proof. If (µ, Â) is the MacNeille completion of A, then µ : A → Â must beinjective, so that A is residually finite (see above). Let θ ∈ ΦA; then it followsfrom the definition of Â that the projection πθ : Â  A/θ commutes with µ andthe natural map a → a/θ; i.e., a/θ = πθµ(a). By [3, Lemma 2.7], π−1θ (1/θ) is aclosed principal filter of Â; say π−1θ (1/θ) = α↑. Because of the correspondencebetween modal filters and modal congruences [18, Theorem 29], α↑ completelycharacterizes A/θ in the following sense: A/θ ∼= [0, α]Âas a bounded lattice[7, Exercise 4.12]. This implies that α↓ is finite. Since (µ, Â) is the MacNeillecompletion of A, µ[A] is join-dense in Â, so by finiteness of α↓, there must exista ∈ A such that µ(a) = α. Now b/θ = 1/θ iff µ(b) ∈ π−1θ (1/θ) = α↑ = µ(a)↑iff b ≥ a, so 1/θ = a↑ is a principal lattice filter.Conversely, if A is residually finite, then µ : A → Â is injective, so (µ, Â) isa completion of A. To show that (µ, Â) is the MacNeille completion of A, wewill consider the different topologies on Â. We first show that At Â ⊆ µ[A].If α ∈ At Â, there must be some θ ∈ ΦA such that α(θ) ∈ At A/θ. Because1/θ is a principal lattice filter c↑, we know that A/θ ∼= [0, c]A as a boundedlattice, so there must be some a ≤ c with a ∈ At A and a/θ = α(θ). But thenµ(a) = α. It follows that At Â ⊆ µ[A], whence ι ⊆ ρ. Since ι is Hausdorff, sois ρ. Using [17, Theorem 4.5] we can thus conclude that, as far as the Booleansubstructure of A is concerned, (µ, Â) is the MacNeille completion of A. Nowto show that ♦ is smooth, remember that ♦̂ : Â → Â is (τ, τ)-continuous. Sinceτ = ι = γ by Lemma 2.1 and ι ⊆ ρ, it follows that ♦̂ is (ρ, γ)-continuous. Butthen since γ↓, γ↑ ⊆ γ, it follows by [17, Proposition 5.9] that ♦• ≤ ♦̂ ≤ ♦◦.Since also ♦◦ ≤ ♦• [17, Proposition 5.6], it follows that ♦ is smooth. Note that the theorem above also admits a third equivalent condition, char-acterizing the dual space of A. This perspective is further explored in [4].4. Finitely generated modal algebras with EDPCHaving equationally definable principal congruences (EDPC) is a strongmeta-logical property of varieties of algebras that coincides with, for example,the existence of a deduction theorem or of a master modality [6, 11] for themodal logic corresponding to a variety of modal algebras. Examples of suchlogics are logics of bounded depth, n-transitive logics such as K4, or regulartest-free PDL with finitely many basic programs.452 J. Vosmaer Algebra Univers.Lemma 4.1 (Proposition 3.4.3 of [11]). Let V be a variety of modal algebras.V has EDPC iff every principal modal filter of an algebra in V is a principallattice filter.Note that the hypotheses below strongly resemble those of [20, Theorem 4].Theorem 4.2 (cf. [4, Corollary 4.5.3]). If A is a residually finite, finitelygenerated modal algebra such that HSP(A) has equationally definable principalcongruences, then the profinite completion (µ, Â) is the MacNeille completionof A and ♦ is smooth.Proof. Since A is a finitely generated algebra, every θ ∈ ΦA is compact [16,Theorem 1]. As is remarked in [20], every compact congruence of a modalalgebra is principal, so under our hypotheses, every θ ∈ ΦA is principal. Nowsince HSP(A) has EDPC, Lemma 4.1 tells us that 1/θ is a principal latticefilter for all θ ∈ ΦA, so by Theorem 3.1, (µ, Â) is the MacNeille completion ofA and ♦ is smooth. Note that the EDPC clause in the theorem above is suppressed in the Heyt-ing algebra case [4], because every variety of Heyting algebras has EDPC.The conditions of Theorem 4.2 above are sufficient; what about necessity?From Theorem 3.1 we know that it is necessary that A is residually finite.Moreover, in light of [19, Section 3.3], we know that it is necessary for Theorem4.2 that A is atomic. This helps us to find counterexamples to the theoremif we remove the requirements of being finitely generated or having EDPC.For instance, let A be the free algebra on one generator for the modal logicT (the logic of reflexive Kripke frames). Then [20, Corollary 7: Example 1]tells us that A is residually finite and finitely generated but not atomic. Thisshows us that being residually finite and finitely generated is not sufficientfor the conclusion of Theorem 4.2. Alternatively, the free transitive modalalgebra on ω generators is an example of a residually finite modal algebra A,generating a variety with EDPC, such that A is not atomic. In summary,residual finiteness is necessary for Theorem 4.2, and if we remove either of theother two conditions, we can find non-atomic counterexamples.Corollary 4.3. Let A be a finitely generated free algebra for K4 or PDL.Then the MacNeille completion and the profinite completion of A are the sameand ♦ is smooth.Proof. Let L be either K4 or PDL and let V be the variety corresponding toL. Since L has the finite model property, V is generated by its finite members.By [13, Theorem IV-14.4], this implies that A, being a finitely generated freealgebra for V, is residually finite. Using the fact that L has a master modality,it follows that we can apply Theorem 4.2. Acknowledgements. The author would like to thank Yde Venema, who sug-gested that Theorem 4.2 might be true. Additionally, the author is grateful tothe Editor and the Referee for their criticisms and suggestions. MacNeille completion and profi nite completion of modal algebras 453References[1] Banaschewski, B.: On profinite universal algebras. In: Novák, J. (ed.) Proceedings ofthe Third Prague Topological Symposium, pp. 51–62, September 1971[2] Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeillecompletion. Arch. Math. (Basel) 18, 369–377 (1967)[3] Bezhanishvili, G., Gehrke, M., Mines, R., Morandi, P.J.: Profinite completions andcanonical extensions of Heyting algebras. Order 23, 143–161 (2006)[4] Bezhanishvili, G., Vosmaer, J.: Comparison of MacNeille, canonical and profinitecompletions. Order 25, 299–320 (2008)[5] Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society (1967)[6] Blok, W., Pigozzi, D.: On the structure of varieties with equationally definableprincipal congruences III. Algebra Universalis 32, 545–608 (1994)[7] Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. CambridgeUniversity Press (2002)[8] Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94,13–45 (2004)[9] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: ACompendium of Continuous Lattices. Springer (1980)[10] Harding, J.: On profinite completions and canonical extensions. Algebra Universalis55, 293–296 (2006)[11] Kracht, M.: Tools and Techniques in Modal Logic. Elsevier (1999)[12] MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42, 416–160 (1927)[13] Mal’cev, A.I.: Algebraic Systems. Springer (1973)[14] Rasiowa, H., Sikorski, R.: A proof of the completeness theorem of Gödel. Fund. Math.37, 193–200 (1950)[15] Ribes, L., Zalesskii, P.: Profinite Groups. Springer (2000)[16] Rival, I., Sands, B.: A note on the congruence lattice of a finitely generated algebra.Proc. Amer. Math. Soc. 72, 451–455 (1978)[17] Theunissen, M., Venema, Y.: MacNeille completions of lattice expansions. AlgebraUniversalis 57, 143–193 (2007)[18] Venema, Y.: Algebras and Coalgebras. In: Blackburn, P., et al. (eds.) Handbook ofModal Logic. Elsevier (2007)[19] Vosmaer, J.: Connecting the profinite completion and the canonical extension usingduality. Master’s thesis, University of Amsterdam (2006)[20] Wolter, F.: A note on atoms in polymodal algebras. Algebra Universalis 37, 334–341(1997)Jacob VosmaerPostbus 94242, 1090 GE Amsterdam, The Netherlandse-mail : jvosmaer@science.uva.nlOpen Access This article is distributed under the terms of the Creative Commons Attribution Noncom-mercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

MacNeille completion and profinite completion can coincide on finitely generated modal algebras

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