a heyting algebra is a distributive lattice with implication and a dual $bck$-algebra is an algebraic system having as models logical systems equipped with implication. the aim of this paper is to investigate the relation of heyting algebras between dual $bck$-algebras. we define notions of $i$-invariant and $m$-invariant on dual $bck$-semilattices and prove that a heyting semilattice is equiva...

A Heyting algebra is a distributive lattice with implication and a dual $BCK$-algebra is an algebraic system having as models logical systems equipped with implication. The aim of this paper is to investigate the relation of Heyting algebras between dual $BCK$-algebras. We define notions of $i$-invariant and $m$-invariant on dual $BCK$-semilattices and prove that a Heyting semilattice is equiva...

Any finite distributive lattice has a unique Heyting structure. Every finite Heyting semilattice has, of course, a unique lattice structure and it is necessarily distributive. ∗ This ms was born in 2002. The first appendix was added in 2005, the first footnote in 2013, the subscorings in 2015 and the addendum in 2017. 1 [ ] See 2nd appendix for subscorings. First x and y meet x↔ y in the same w...

. 1 Preliminaries Consider the variety Eω generated by the algebra ω := (N, ·), where i · j := max (i, j) for i 6= j, and i · j := 1 for i = j, i, j ∈ N1. These variety is a subvariety of the variety of equivalential algebras E . By an equivalential algebra we mean a grupoid A = (A,↔) that is a subreduct of a Brouwerian semilattice (or, equivalently, a Heyting algebra) with the operation ↔ give...

The determination of the injective and projective members of a category is usually a challenging problem and adds to knowledge of the category. In this paper we consider these questions for the category of Heyting algebras. There has been a lack of uniformity in terminology in recent years. In [6] Heyting algebras are referred to as pseudo-Boolean algebras, and in [1] they are called Brouwerian...

In this paper we study the structure of finitely presented Heyting algebras. Using algebraic techniques (as opposed to techniques from proof-theory) we show that every such Heyting algebra is in fact coHeyting, 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 pro...

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct. A residuated lattice is an algebra A = (A,∨,∧, ·, e, /, \) such that (A,∨,∧) is a lattice, (A, ·, e) is a monoid and for every a, b, c ∈ A ab ≤ c ⇔ a ≤ c/b ⇔ b ≤ a\c. The last condition is equivalent to the fact that (A,∨,∧, ·, e) is a lattice-ordered monoid and for every a, b ∈ A there is a great...

We investigate the class of reflexive graphs that admit semilattice polymorphisms, and in doing so, give an algebraic characterisation of chordal graphs. In particular, we show that a graph G is chordal if and only if it has a semilattice polymorphism such that G is a subgraph of the comparability graph of the semilattice. Further, we find a new characterisation of the leafage of a chordal grap...

In the paper we introduce logical calculi for representation of propositions with modal operators indexed by fuzzy values. There calculi are called Heyting-valued modal logics. We introduce the concept of a Heyting-valued Kripke model and consider a semantics of Heyting-valued modal logics at the class of Heyting-valued Kripke models.

A modular semilattice is a semilattice S in which w > a A ft implies that there exist i,jeS such that x > a. y > b and x A y = x A w. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical charac...