Via the introduction of (infinitary) disjunctions on any complete lattice while inheriting the meet as a conjunction, we construct a bijective correspondence (up to isomorphism) between complete lattices L and complete Heyting algebras DI(L) equipped with a so called disjunctive join dense closure operator RL. If L is itself a complete Heyting algebra then DI(L) ∼= L and RL = idDI(L). Ortholatt...

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) ...

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--...

In this paper we study the logic Lωω, which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of Lωω with respect to models in Grothendieck toposes, which can be sharpened t...

We show that a locally finite variety which omits abelian types is self–rectangulating if and only if it has a compatible semilattice term operation. Such varieties must have type–set {5 }. These varieties are residually small and, when they are finitely generated, they have definable principal congruences. We show that idempotent varieties with a compatible semilattice term operation have the ...

A specialization semilattice is a join together with coarser preorder $ \sqsubseteq satisfying an appropriate compatibility condition. If $X$ topological space, then $(\mathcal P(X), \cup, )$ semilattice, where x y$ if $x \subseteq Ky$, for $x,y X$, and $K$ closure. Specialization semilattices posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology....

In this paper, we study structures such as distributive lattices, semilattices, and median graphs from an algorithmic point of view. Such are very useful in classification phylogeny for representing lineage relationships example. A lattice can be considered a graph while ∨-semilattice provided that some conditions holding on triple elements satisfied. Starting structure with different represent...

A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete archimedean Heyting …eld, a terminal object in the category of archimedean Heyting …elds.