Join-semilattices whose principal filters are pseudocomplemented lattices

TL;DR

This paper explores join-semilattices with principal filters as pseudocomplemented lattices, connecting them to non-classical implication structures and substructural logics, highlighting their algebraic properties and potential logical applications.

Contribution

It introduces a new class of join-semilattices with pseudocomplemented principal filters and relates them to non-classical implication semilattices and substructural logics.

Findings

Principal filters are pseudocomplemented lattices.

Connections established with non-classical implication semilattices.

I-algebras form a congruence distributive, 3-permutable variety.

Abstract

This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a\vee b in the section [b,1] is denoted by a\rightarrow b and can be considered as the connective implication in a certain kind of intuitionistic logic. Contrary to the case of Brouwerian semilattices, sections need not be distributive lattices. This essentially allows possible applications in non-classical logics. We present a connection of the semilattices mentioned in the beginning with so-called non-classical implication semilattices which can be converted into I-algebras having everywhere defined operations. Moreover, we relate our structures to sectionally and relatively residuated semilattices which means that our logical structures are closely connected with substructural logics. We show that I-algebras form a congruence distributive, 3-permutable and weakly regular variety.