The variety of Heyting algebras has a nice property that HS = SH. Heyting algebras are the algebraic dual of intuitionistic descriptive frames. The goal of this paper is to define proper dual notions so as to formulate this algebraic properties in the frame language, and to give a frame-based proof of this property and some other duality theorems.

This paper aims to introduce fuzzy congruence relations over Heyting algebras (HA) and give constructions of quotient Heyting algebras induced by fuzzy congruence relations on HA. The Fuzzy First, Second and Third Isomorphism Theorems of HA are established. MSC: 06D20, 06D72, 06D75.

The paper presents general machinery for extending a duality between complete, cocomplete categories to a duality between corresponding categories of semilattice representations (i.e. sheaves over Alexandrov spaces). This enables known dualities to be regularized. Among the applications, regularized Lindenbaun-Tarski duality shows that the weak extension of Boolean logic (i.e. the semantics of ...

For a Heyting algebra A, we show that the following conditions are equivalent: (i) A is profinite; (ii) A is finitely approximable, complete, and completely joinprime generated; (iii) A is isomorphic to the Heyting algebra Up(X) of upsets of an image-finite poset X. We also show that A is isomorphic to its profinite completion iff A is finitely approximable, complete, and the kernel of every fi...

this paper is the second of a two part series. in this part, we prove, using the description of simples obtained in part i, that the variety $mathbf{rdqdstsh_1}$ of regular dually quasi-de morgan stone semi-heyting algebras of level 1 is the join of the variety generated by the twenty 3-element $mathbf{rdqdstsh_1}$-chains and the variety of dually quasi-de morgan boolean semi-heyting algebras--...

We show that for a variety V of Heyting algebras the following conditions are equivalent: (1) V is locally finite; (2) the V-coproduct of any two finite V-algebras is finite; (3) either V coincides with the variety of Boolean algebras or finite V-copowers of the three element chain 3 ∈ V are finite. We also show that a variety V of Heyting algebras is generated by its finite members if, and onl...