Filters and ideals in pseudocomplemented posets

TL;DR

This paper explores the structure of ideals and filters in pseudocomplemented posets, extending existing concepts and proving a Separation Theorem, thereby advancing the theoretical understanding of these algebraic structures.

Contribution

It generalizes the concept of *-ideals to pseudocomplemented posets and establishes a Separation Theorem within this broader context.

Findings

Extended *-ideals concept to pseudocomplemented posets

Established connections between prime filters, ultrafilters, and dense elements

Proved a Separation Theorem for *-ideals

Abstract

We study ideals and filters of posets and of pseudocomplemented posets and show a version of the Separation Theorem, known for ideals and filters in lattices and semilattices, within this general setting. We extend the concept of a *-ideal already introduced by Rao for pseudocomplemented distributive lattices and by Talukder, Chakraborty and Begum for pseudocomplemented semilattices to pseudocomplemented posets. We derive several important properties of such ideals. Especially, we explain connections between prime filters, ultrafilters, filters satisfying the *-condition and dense elements. Finally, we prove a Separation Theorem for *-ideals.