On implications in sectionally pseudocomplemented posets

TL;DR

This paper investigates the structure and properties of sectionally pseudocomplemented posets, focusing on an implication-like operation derived from sectional pseudocomplements, and explores related lattice structures and implications.

Contribution

It characterizes the implication-like operation in sectionally pseudocomplemented posets and studies properties of various lattice structures with this operation.

Findings

Characterization of the implication-like operation

Analysis of properties of upper and lower semilattices with this operation

Discussion of weaker versions of implication

Abstract

A sectionally pseudocomplemented poset P is one which has the top element and in which every principal order filter is a pseudocomplemented poset. The sectional pseudocomplements give rise to an implication-like operation on P which coincides with the relative pseudocomplementation if P is relatively psudocomplemented. We characterise this operation and study some elementary properties of upper semilattices, lower semilattices and lattices equipped with this kind of implication. We deal also with a few weaker versions of implication. Sectionally pseudocomplemented lattices have already been studied in the literature.