Let S, T be semilattices. Let us assume that if S is upper-bounded, then T is upper-bounded. A map from S into T is said to be a semilattice morphism from S into T if: (Def. 1) For every finite subset X of S holds it preserves inf of X . Let S, T be semilattices. Observe that every map from S into T which is meet-preserving is also monotone. Let S be a semilattice and let T be an upper-bounded ...

We prove that the topology of a compact Hausdorff topological Heyting algebra is a Stone topology. It then follows from known results that a Heyting algebra is profinite iff it admits a compact Hausdorff topology that makes it a compact Hausdorff topological Heyting algebra.

In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras are not generated by their complete members. It follows extensions the Heyting–Brouwer logic [Formula: see text] topologically incomplete. This result provides further insight into long-standing open problem Kuznetsov yielding a negative solution reformulation from to text].

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation τ(a) ≤ b ∨ (b → a), for all a, b ∈ A. These operators were studied from an algebraic, logical an...

In [3] we have claimed that finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor, and in particular all finite chains generate a proper subvariety of the latter. As Xavier Caicedo made us notice, this claim is not true. He proved, using techniques of Kripke models, that the intuitionistic calculus with S has finite model propert...

ing frames O (X ) coming from a topological space to general frames is a genuine generalization of the concept of a space, as plenty of frames exist tha t are not of the form O (X ). A simple example is the frame Oreg(R) of regular open subsets of R, i.e. of open subsets U with the property ——U = U, where —U is the interior of the complement of U . This may be contrasted with the situation for ...

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 techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality...

As Jonathan Leech has pointed out, many natural examples of inverse semigroups are semilattice ordered under the natural partial order. But there are many interesting examples of semilattice ordered inverse semigroups in which the imposed partial order is not the natural one. In this talk we shall explore the structure and properties of these examples and some other questions related to semilat...