Extension of sectional pseudocomplementation in posets

TL;DR

This paper explores the extension of sectional pseudocomplementation in posets, introducing new classes and properties of extended operations, correcting previous approaches, and broadening the understanding of implication-like operations in poset structures.

Contribution

It introduces new classes of posets with extended sp-complementation, refines existing definitions, and corrects prior inaccuracies in the study of these operations.

Findings

Several classes of posets with extended sp-complementation are characterized.

Elementary properties of the extended operations are established.

Previous attempts to define particular classes are corrected and improved.

Abstract

Sectional pseudocomplementation (sp-complementation) on a poset is a partial operation $*$ which associates with every pair $(x,y)$ of elements, where $x \ge y$, the pseudocomplement $x*y$ of $x$ in the upper section $[y)$. Any total extension $\to$ of $*$ is said to be an extended sp-complementation and is considered as an implication-like operation. Extended sp-complementations have already be studied on semilattices and lattices. We describe several naturally arising classes of general posets with extended sp-complementation, present respective elementary properties of this operation, demonstrate that two other known attempts to isolate particular such classes are in fact not quite correct, and suggest suitable improvements.