Bounded distributive lattices with strict implication

Abstract

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices.

Compatible operations in some subvarieties of the variety of weak Heyting algebras

“…A weak Heyting algebra, or WH-algebra ( [2], [5]), is an ordered algebraic structure A, ∧, ∨, →, 0, 1 , where the reduct algebra A, ∧, ∨, 0, 1 is a bounded distributive lattice and → : A × A → A is a map such that for all a, b, c ∈ A satisfies the following conditions: We write WH to indicate the variety of W Halgebras.…”

“…The variety WH is an arithmetical variety and has the congruence extension property, therefore all of its subvarieties, and in particular the five considered, have these properties too. Moreover, the varieties TWH and SRL have equationally definable principal congruences, but WH and RWH do not (see [5]). …”

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AbstractWeak Heyting algebras are a natural generalization of Heyting algebras (see [2], [5]). In this work we study certain subvarieties of the variety of weak Heyting algebras in order to extend some known results about compatible functions in Heyting algebras.

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n‐linear weakly Heyting algebras

“…In [3] was introduced the variety WH of weakly Heyting algebras (WH-algebras). These algebras corresponds to the strict implication fragments of the local and global consequence relations of the least normal modal logic K (also called subintuitionistic logics).…”

“…Each one of the varieties of WH-algebras studied in [3] is related with both propositional logics wK σ and sK σ defined in [4]. The logics wK σ and sK σ are the strict implication fragments of the local and global consequence relations defined by means of Kripke models (see [4]), respectively.…”