c-ideals in complemented posets

TL;DR

This paper extends the concept of c-ideals to complemented posets with weakly defined complementation, exploring their properties, separation theorems, and providing illustrative examples.

Contribution

It introduces and analyzes c-ideals in complemented posets with non-involutive, non-antitone complementation, expanding the theoretical framework of pseudocomplementation.

Findings

Characterization of c-ideals and c-filters in complemented posets

Basic properties and separation theorems for c-ideals

Illustrative examples demonstrating the concepts

Abstract

In their recent paper on posets with a pseudocomplementation denoted by * the first and the third author introduced the concept of a *-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, respectively, and we prove basic properties of them. Finally, we prove so-called Separation Theorems for c-ideals. The text is illustrated by several examples.