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The Canonical Decomposition of the Poset of a Hammock
Author(s) -
Scheuer Tobias
Publication year - 1994
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/49.2.232
Subject(s) - partially ordered set , simple (philosophy) , mathematics , quiver , injective function , combinatorics , set (abstract data type) , pure mathematics , order (exchange) , ideal (ethics) , algebra over a field , computer science , philosophy , epistemology , finance , economics , programming language
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 + ⩾ 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.