Representations of finite partially ordered sets over commutative artinian uniserial rings

Abstract

Categories of representations of finite partially ordered sets over commutative artinian uniserial rings arise naturally from categories of lattices over orders and abelian groups. By a series of functorial reductions and a combinatorial analysis, the representation type of a category of representations of a finite partially ordered set S over a commutative artinian uniserial ring R is characterized in terms of S and the index of nilpotency of the Jacobson radical of R. These reductions induce isomorphisms of Auslander–Reiten quivers and preserve and reflect Auslander–Reiten sequences. Included, as an application, is the completion of a partial characterization of representation type of a category of representations arising from pairs of finite rank completely decomposable abelian groups.