Idempotent Varieties of Incidence Monoids and Bipartite Posets

TL;DR

This paper explores the algebraic structure of idempotent elements in incidence monoids, characterizing their irreducible components, and analyzing conjugacy and Green's relations in bipartite poset monoids.

Contribution

It determines the irreducible components of the variety of idempotents and characterizes conjugacy in antichain monoids of bipartite posets.

Findings

Irreducible components of the idempotent variety are explicitly determined.

Antichain monoids of bipartite posets are shown to be orthodox semigroups.

Conjugacy in the monoid is characterized via Green's relations and $ ext{J}$-classes.

Abstract

The algebraic variety defined by the idempotents of an incidence monoid is investigated. Its irreducible components are determined. The intersection with an antichain submonoid is shown to be the union of these irreducible components. The antichain monoids of bipartite posets are shown to be orthodox semigroups. The Green's relations are explicitly determined, and applications to conjugacy problems are described. In particular, it is shown that two elements in the antichain monoid are primarily conjugate in the monoid if and only if they belong to the same $\mathcal{J}$-class and their multiplication by an idempotent of the same $\mathcal{J}$-class gives conjugate elements in the group.