Generalizing relational structures and formal languages to structures whose relations are evaluated by elements of a lattice, we show that such structure classes form a Heyting algebra if and only if the evaluation lattice is a Heyting algebra. Hence various new and some older results obtained for Heyting algebras can be applied to such structure classes.

The theory of elementary toposes plays the fundamental role in the categorial analysis of the intuitionistic logic. The main theorem of this theory uses the fact that sets E(A,Ω) (for any object A of a topos E) are Heyting algebras with operations defined in categorial terms. More exactly, subobject classifier true: 1 → Ω permits us define truth-morphism on Ω and operations in E(A,Ω) are define...

In this paper, it is shown that the category of stratified $L$-generalized convergence spaces is monoidal closed if the underlying truth-value table $L$ is a complete residuated lattice. In particular, if the underlying truth-value table $L$ is a complete Heyting Algebra, the Cartesian closedness of the category is recaptured by our result.

some appropriate axioms (for details see [7]). These axioms imply that the relation ≈ on A defined by: a ≈ b if and only if a → b = 1 and b → a = 1, 1)/ ≈ is a Heyting algebra. For a given Heyting algebra B there always exists a Nelson algebra A such that A h is isomorphic to B: the Fidel-Vakarelov construction of the Nelson algebra N (B) (see e.g. [8]) yields an example of such an algebra. In ...

We prove that (1) for any complete lattice $L$, the set $\mathcal {D}(L)$ of all non-empty saturated compact subsets Scott space $L$ is a Heyting algebra (with reverse inclusion order); and (2) if latti

In this note we provide a topological description of the MacNeille completion of a Heyting algebra similar to the description of the MacNeille completion of a Boolean algebra in terms of regular open sets of its Stone space. We also show that the only varieties of Heyting algebras that are closed under MacNeille completions are the trivial variety, the variety of all Boolean algebras, and the v...

This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasi-De Morgan Stone semi-Heyting algebras of level 1 satisfies the strongly blended $lor$-De Morgan law introduced in cite{Sa12}. Then, using this result and the results of cite{Sa12}, we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) ...

The variety DQD of semi-Heyting algebras with a weak negation, called dually quasi-De Morgan operation, and several of its subvarieties were investigated in the series [31], [32], [33], and [34]. In this paper we define and investigate a new subvariety JID of DQD, called “JI-distributive, dually quasi-De Morgan semi-Heyting algebras”, defined by the identity: x ∨ (y → z) ≈ (x ∨ y) → (x ∨ z), as...