Idempotent Varieties of Incidence Monoids and Bipartite Posets

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.