On mesoprimary decomposition of monoid congruences

TL;DR

This paper advances the understanding of mesoprimary decomposition of monoid congruences by identifying universal associated prime congruences and characterizing the posets that can arise as sets of associated primes.

Contribution

It completes the theory of mesoprimary decomposition by determining which associated prime congruences appear universally and characterizes the finite posets that can be realized as associated prime sets.

Findings

Identifies associated prime congruences present in all mesoprimary decompositions.

Provides a characterization of finite posets that can be sets of associated prime congruences.

Enhances the analogy between mesoprimary and primary decomposition.

Abstract

We prove two main results concerning mesoprimary decomposition of monoid congruences, as introduced by Kahle and Miller. First, we identify which associated prime congruences appear in every mesoprimary decomposition, thereby completing the theory of mesoprimary decomposition of monoid congruences as a more faithful analog of primary decomposition. Second, we answer a question posed by Kahle and Miller by characterizing which finite posets arise as the set of associated prime congruences of monoid congruences.