Representations of finite posets over the ring of integers modulo a prime power

Abstract

The classical category Rep(S, Zp) of representations of a finite poset S over the field Zp is extended to two categories, Rep(S, Z p m ) and uRep(S, Z p m ), of representations of S over the ring Z p m . A list of values of S and m for which Rep(S, Z p m ) or uRep(S, Z p m ) has infinite representation type is given for the case that S is a forest. Applications include a computation of the representation type for certain classes of abelian groups, as the category of sincere representations in (uRep(S, Z p m )) Rep(S, Z p m ) has the same representation type as (homocyclic) (S, p m )-groups, a class of almost completely decomposable groups of finite rank. On the other hand, numerous known lists of examples of indecomposable (S, p m )-groups give rise to lists of indecomposable representations.

Completely decomposable direct summands of torsion-free abelian groups of finite rank

“…Then T cr (X) is called the critical typeset of X. The problem has been largely solved when the critical typeset is an inverted forest in a number of papers by Arnold-Mader-Mutzbauer-Solak ( [2], [3], [4], [5], [6], [7], [8], [9], [10]) using representations of posets as a tool.…”