Locally finite varieties of Heyting algebras

Bezhanishvili, Guram; Grigolia, Revaz

doi:10.1007/s00012-005-1958-5

Cited by 11 publications

(12 citation statements)

References 10 publications

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“…Consider, as an example, the variety of Heyting algebras. Drawing on results of L. Maximova [51] and Bezhanishvili and Grigolia [5,Theorem 4.1], we obtain that a variety V of Heyting algebras satisfies (the equivalent) conditions (1)-(4) of Theorem 5.1 if and only if V contains none of (a) the variety generated by all finite Heyting chains, that is, the variety L of relative Stone algebras; (b) the variety generated by algebras B ⊕ 1, where B is a Boolean algebra; (c) the variety generated by algebras 1⊕B⊕1, where B is a Boolean algebra.…”

Section: Varieties Of Lattices and Lattice-based Algebrasmentioning

confidence: 99%

Natural extensions and profinite completions of algebras

et al. 2011

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This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class A = ISP(M), where M is a set, not necessarily finite, of finite algebras, it is shown that each A ∈ A embeds as a topologically dense subalgebra of a topological algebra n A (A) (its natural extension), and that n A (A) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that M is finite and A possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.

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Section: Varieties Of Lattices and Lattice-based Algebrasmentioning

confidence: 99%

Natural extensions and profinite completions of algebras

et al. 2011

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“…V has only finitely many subvarieties. 4. The dual space of every algebra in V has finite depth and finite width.…”

Section: Finitely Generated Varieties Of Heyting Algebrasmentioning

confidence: 99%

“…Alternative characterizations were given in [4][5][6]. Let V 1 denote the variety of Heyting algebras generated by finite chains, let V 2 denote the variety of Heyting algebras generated by finite Boolean algebras with a new adjoined top, and let V 3 denote the variety of Heyting algebras generated by finite Boolean algebras with a new adjoined top and bottom.…”

Section: Finitely Generated Varieties Of Heyting Algebrasmentioning

confidence: 99%

Profinite Completions and Canonical Extensions of Heyting Algebras

Bezhanishvili

Gehrke

Mines

et al. 2006

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We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and of Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion H of a Heyting algebra H, and characterize the dual space of H. We also give a necessary and sufficient condition for the profinite completion of H to coincide with its canonical extension, and provide a new criterion for a variety V of Heyting algebras to be finitely generated by showing that V is finitely generated if and only if the profinite completion of every member of V coincides with its canonical extension. From this we obtain a new proof of a well-known theorem that every finitely generated variety of Heyting algebras is canonical. Mathematics Subject Classifications (2000)Primary 06D20 · Secondary 06D50 · 06B30, 03B55

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“…Mardaev [19] showed that, unlike extensions of S4, there are continuum pre-locally tabular intermediate logics. G. Bezhanishvili and Grigolia [3] gave a characterization of locally tabular intermediate logics using coproducts of three element Heyting algebra. They also conjectured that an intermediate logic L is locally tabular iff the 2-generated free algebra in the corresponding variety is finite.…”

Section: Locally Tabular and Tabular Intermediate Logicsmentioning

confidence: 99%

Frame Based Formulas for Intermediate Logics

Bezhanishvili

2008

Stud Logica

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In this paper we define the notion of frame based formulas. We show that the well-known examples of formulas arising from a finite frame, such as the Jankovde Jongh formulas, subframe formulas and cofinal subframe formulas, are all particular cases of the frame based formulas. We give a criterion for an intermediate logic to be axiomatizable by frame based formulas and use this criterion to obtain a simple proof that every locally tabular intermediate logic is axiomatizable by Jankov-de Jongh formulas. We also show that not every intermediate logic is axiomatizable by frame based formulas.

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Locally finite varieties of Heyting algebras

Cited by 11 publications

References 10 publications

Natural extensions and profinite completions of algebras

Natural extensions and profinite completions of algebras

Profinite Completions and Canonical Extensions of Heyting Algebras

Frame Based Formulas for Intermediate Logics

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