Filters and congruences in sectionally pseudocomplemented lattices and posets

TL;DR

This paper explores the algebraic structures called sectionally pseudocomplemented lattices and posets, defining congruences and filters, and analyzing their properties and relationships, with potential applications in algebraic semantics of intuitionistic logics.

Contribution

It introduces the concepts of congruences and filters in these structures and studies their properties, extending the tools used for related algebraic systems.

Findings

Mutual relationships between congruences and filters are established.

Basic properties of congruences in strongly sectionally pseudocomplemented posets are described.

The machinery for filters applies even when ideal terms are not everywhere defined.

Abstract

In our previous papers, together with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that - similar to relatively pseudocomplemented lattices - these structures can serve as an algebraic semantics of certain intuitionistic logics. The aim of the present paper is to define congruences and filters in these structures, derive mutual relationships between them and describe basic properties of congruences in strongly sectionally pseudocomplemented posets. For the description of filters both in sectionally pseudocomplemented lattices and posets, we use the tools introduced by A. Ursini, i.e. ideal terms and the closedness with respect to them. It seems to be of some interest that a similar machinery can be applied also for strongly sectionally pseudocomplemented posets in spite of the fact that the corresponding ideal terms are not everywhere defined.