Indecomposable Ideals in Incidence Algebras

Abstract

The elements of a finite partial order $P$ can be identified with the maximal indecomposable two-sided ideals of its incidence algebra $\A$, and then for two such ideals, $I\prec J \iff IJ \not=0$. This offers one way to recover a poset from its incidence algebra. In the course of proving the above, we classify all of the two-sided ideals of $\A$.