scholarly journals Bitopological duality for distributive lattices and Heyting algebras

2010 ◽  
Vol 20 (3) ◽  
pp. 359-393 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
NICK BEZHANISHVILI ◽  
DAVID GABELAIA ◽  
ALEXANDER KURZ
Keyword(s):  
Boolean Algebras ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Priestley Spaces ◽  
Spectral Spaces ◽  
Algebraic Concepts ◽  
Stone Spaces

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

Profinite Heyting Algebras and Profinite Completions of Heyting Algebras

2009 ◽  
Vol 16 (1) ◽  
pp. 29-47
Author(s):  
Guram Bezhanishvili ◽  
Patrick J. Morandi
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Heyting Algebra ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Profinite Completion ◽  
Bounded Distributive Lattice ◽  
Recent Developments ◽  
Profinite Completions

Abstract This paper surveys recent developments in the theory of profinite Heyting algebras (resp. bounded distributive lattices, Boolean algebras) and profinite completions of Heyting algebras (resp. bounded distributive lattices, Boolean algebras). The new contributions include a necessary and sufficient condition for a profinite Heyting algebra (resp. bounded distributive lattice) to be isomorphic to the profinite completion of a Heyting algebra (resp. bounded distributive lattice). This results in simple examples of profinite bounded distributive lattices that are not isomorphic to the profinite completion of any bounded distributive lattice. We also show that each profinite Boolean algebra is isomorphic to the profinite completion of some Boolean algebra. It is still an open question whether each profinite Heyting algebra is isomorphic to the profinite completion of some Heyting algebra.

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

Frontal operators in distributive lattices with a generalized implication

2015 ◽  
Vol 08 (03) ◽  
pp. 1550039
Author(s):  
Sergio A. Celani ◽  
Hernán J. San Martín
Keyword(s):  
Distributive Lattice ◽  
Unary Operation ◽  
Duke Math ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Bounded Distributive Lattice ◽  
Modal Lattices ◽  
Weak Heyting Algebras ◽  
San Martín

We introduce a family of extensions of bounded distributive lattices. These extensions are obtained by adding two operations: an internal unary operation, and a function (called generalized implication) that maps pair of elements to ideals of the lattice. A bounded distributive lattice with a generalized implication is called gi-lattice in [J. E. Castro and S. A. Celani, Quasi-modal lattices, Order 21 (2004) 107–129]. The main goal of this paper is to introduce and study the category of frontal gi-lattices (and some subcategories of it). This category can be seen as a generalization of the category of frontal weak Heyting algebras (see [S. A. Celani and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100(1–2) (2012) 91–114]). In particular, we study the case of frontal gi-lattices where the generalized implication is defined as the annihilator (see [B. A. Davey, Some annihilator conditions on distributive lattices, Algebra Universalis 4(1) (1974) 316–322; M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377–386]). We give a Priestley’s style duality for each one of the new classes of structures considered.

On decidable varieties of Heyting algebras

1992 ◽  
Vol 57 (3) ◽  
pp. 988-991 ◽  
Author(s):  
Devdatt P. Dubhashi
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
Semantic Interpretation ◽  
Structure Theory ◽  
Heyting Algebras ◽  
Quantitative Classification ◽  
First Order ◽  
First Order Theory ◽  
Famous Result

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

Yap Hian Poh. Postulational study of an axiom system of Boolean algebra. Majallah Tahunan 'Ilmu Pasti—Shu Hsüeh Nien K'an—Bulletin of Mathematical Society of Nanyang University (1960), pp. 94–110. - R. M. Dicker. A set of independent axioms for Boolean algebra. Proceedings of the London Mathematical Society, ser. 3 vol. 13 (1963), pp. 20–30. - P. J. van Albada. A self-dual system of axioms for Boolean algebra. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, series A vol. 67 (1964), pp. 377–381; also Indagationes mathematicae, vol. 26 (1964), pp. 377–381. - Antonio Diego and Alberto Suárez. Two sets of axioms for Boolean algebras. Portugaliae mathematica, vol. 23 nos. 3–4 (for 1964, pub. 1965), pp. 139–145. (Reprinted from Notas de lógica matemática no. 16, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca 1964, 13 pp.) - P. J. van Albada. Axiomatique des algèbres de Boole. Bulletin de la Société Mathématique de Belgique, vol. 18 (1966), pp. 260–272. - Lawrence J. Dickson. A short axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 6 (1967), pp. 253–257. - Leroy J. Dickey. A shorter axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 8 (1968), p. 336. - Chinthayamma . Independent postulate sets for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 9 (1968), pp. 378–379. - Kiyoshi Iséki. A simple characterization of Boolean rings. Proceedings of the Japan Academy, vol. 44 (1968), pp. 923–924. - Sakiko Ôhashi. On definitions of Boolean rings and distributive lattices. Proceedings of the Japan Academy, vol. 44 (1968), pp. 1015–1017.

1973 ◽  
Vol 38 (4) ◽  
pp. 658-660
Author(s):  
Donald H. Potts
Keyword(s):  
Boolean Algebra ◽  
London Mathematical Society ◽  
Axiom System ◽  
Dual System ◽  
Boolean Algebras ◽  
Axiomatic System ◽  
Mathematical Society ◽  
Distributive Lattices ◽  
Boolean Rings

Decidability problem for finite Heyting algebras

1988 ◽  
Vol 53 (3) ◽  
pp. 729-735 ◽  
Author(s):  
Katarzyna Idziak ◽  
Pawel M. Idziak
Keyword(s):  
Boolean Algebras ◽  
Heyting Algebras ◽  
First Order ◽  
Decidable Theory ◽  
Decidability Problem ◽  
Ordered Algebras

AbstractThe aim of this paper is to characterize varieties of Heyting algebras with decidable theory of their finite members. Actually we prove that such varieties are exactly the varieties generated by linearly ordered algebras. It contrasts to the result of Burris [2] saying that in the case of whole varieties, only trivial variety and the variety of Boolean algebras have decidable first order theories.

scholarly journals Finitely Presented Heyting Algebras

1998 ◽  
Vol 5 (30) ◽  
Author(s):  
Carsten Butz
Keyword(s):  
Simple Proof ◽  
Heyting Algebra ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Model Property ◽  
Disjunction Property ◽  
Irreducible Elements ◽  
Finite Set ◽  
Finite Distributive Lattices ◽  
Finitely Presented

In this paper we study the structure of finitely presented Heyting<br />algebras. Using algebraic techniques (as opposed to techniques from proof-theory) we show that every such Heyting algebra is in fact co- Heyting, improving on a result of Ghilardi who showed that Heyting algebras free on a finite set of generators are co-Heyting. Along the way we give a new and simple proof of the finite model property. Our main technical tool is a representation of finitely presented Heyting algebras in terms of a colimit of finite distributive lattices. As applications we construct explicitly the minimal join-irreducible elements (the atoms) and the maximal join-irreducible elements of a finitely presented Heyting algebra in terms of a given presentation. This gives as well a new proof of the disjunction property for intuitionistic propositional logic.<br />Unfortunately not very much is known about the structure of Heyting algebras, although it is understood that implication causes the complex structure of Heyting algebras. Just to name an example, the free Boolean algebra on one generator has four elements, the free Heyting algebra on one generator is infinite.<br />Our research was motivated a simple application of Pitts' uniform interpolation theorem [11]. Combining it with the old analysis of Heyting algebras free on a finite set of generators by Urquhart [13] we get a faithful functor J : HAop<br />f:p: ! PoSet; sending a finitely presented Heyting algebra to the partially ordered set of its join-irreducible elements, and a map between Heyting algebras to its leftadjoint<br />restricted to join-irreducible elements. We will explore on the induced duality more detailed in [5]. Let us briefly browse through the contents of this paper: The first section<br />recapitulates the basic notions, mainly that of the implicational degree of an element in a Heyting algebra. This is a notion relative to a given set of generators. In the next section we study nite Heyting algebras. Our contribution is a simple proof of the nite model property which names in particular a canonical family of nite Heyting algebras into which we can<br />embed a given finitely presented one.<br />In Section 3 we recapitulate the standard duality between nite distributive lattices and nite posets. The `new' feature here is a strict categorical<br />formulation which helps simplifying some proofs and avoiding calculations. In the following section we recapitulate the description given by Ghilardi [8]<br />on how to adjoin implications to a nite distributive lattice, thereby not destroying a given set of implications. This construction will be our major technical ingredient in Section 5 where we show that every nitely presented<br />Heyting algebra is co-Heyting, i.e., that the operation (−) n (−) dual to implication is dened. This result improves on Ghilardi's [8] that this is true<br />for Heyting algebras free on a finite set of generators. Then we go on analysing the structure of finitely presented Heyting algebras<br />in Section 6. We show that every element can be expressed as a finite join of join-irreducibles, and calculate explicitly the maximal join-irreducible elements in such a Heyting algebra (in terms of a given presentation). As a consequence we give a new proof of the disjunction property for propositional intuitionistic logic. As well, we calculate the minimal join-irreducible elements, which are nothing but the atoms of the Heyting algebra. Finally, we show how all this material can be used to express the category of finitely presented Heyting algebras as a category of fractions of a certain category with objects morphism between finite distributive lattices.

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