THE CANONICAL DECOMPOSITION OF THE POSET OF A HAMMOCK

In the Auslander-Reiten quiver of a representation-directed algebra several hammocks occur naturally; they begin at the projective cover of a simple module E and end in the corresponding injective hull. It is known that hammocks are Auslander-Reiten quivers of posets, so there is a poset corresponding to each simple module; it describes the set of modules having E as a composition factor. In this paper we show that this poset S decomposes canonically into a coideal S+ and an ideal S- which can easily be described by vectorspace-categories corresponding to a one-point extension or a one-point coextension, respectively. In addition, we describe the simple modules for which S+ and S- are not comparable, and also those for which S+ greater-than-or-equal-to S-. We also show how to use the results in order to prove for certain posets that they do not occur as posets corresponding to simple modules.