THE CANONICAL DECOMPOSITION OF THE POSET OF A HAMMOCK

In the Auslander-Reiten quiver of a representation-directed algebra se veral hammocks occur naturally; they begin at the projective cover of a simple module E and end in the corresponding injective hull. It is k nown 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 t his poset S decomposes canonically into a coideal S+ and an ideal S- w hich can easily be described by vectorspace-categories corresponding t o a one-point extension or a one-point coextension, respectively. In a ddition, we describe the simple modules for which S+ and S- are not co mparable, 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.