REPRESENTATIONS OF FINITE POSETS OVER THE RING OF INTEGERS MODULO A PRIME POWER

Date

2016-12-01

Author

Arnold, David
Mader, Adolf
Mutzbauer, Otto
Solak, Ebru

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The classical category Rep(S, Z(p)) of representations of a finite poset S over the field Z(P) 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.

Subject Keywords

Poset, Representation, Indecomposable, Representation type, Almost completely decomposable group

URI

https://hdl.handle.net/11511/38005

Journal

JOURNAL OF COMMUTATIVE ALGEBRA

DOI

https://doi.org/10.1216/jca-2016-8-4-461

Collections

Department of Mathematics, Article

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REPRESENTATIONS OF POSETS AND INDECOMPOSABLE TORSION-FREE ABELIAN GROUPS Arnold, David; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2014-03-04) Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsion-free abelian groups. For a special class of almost completely decomposable groups we determine the possible ranks of indecomposable groups and show that the possible ranks are realized by indecomposable groups in the class.
Classification of a class of torsion-free abelian groups Solak, Ebru (2016-01-01) The class of almost completely decomposable groups with a critical typeset of type (2, 2) and a regulator quotient of exponent <= p(2) is shown to have exactly 4 near-isomorphism classes of indecomposable groups. Every group of the class is up to near-isomorphism uniquely a direct sum of these four indecomposable groups.
Indecomposable (1,3)-Groups and a matrix problem Arnold, David M.; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2013-06-01) Almost completely decomposable groups with a critical typeset of type (1, 3) and a p-primary regulator quotient are studied. It is shown that there are, depending on the exponent of the regulator quotient p (k) , either no indecomposables if k a (c) 1/2 2; only six near isomorphism types of indecomposables if k = 3; and indecomposables of arbitrary large rank if k a (c) 3/4 4.
(1,4)-GROUPS WITH HOMOCYCLIC REGULATOR QUOTIENT OF EXPONENT p(3) Arnold, David M.; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2015-01-01) The class of almost completely decomposable groups with a critical typeset of type (1,4) and a homocyclic regulator quotient of exponent p(3) is shown to be of bounded representation type. There are precisely four near-isomorphism classes of indecomposables, all of rank 6.
Almost completely decomposable groups and unbounded representation type Arnold, David M.; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2012-01-01) Almost completely decomposable groups with a regulating regulator and a p-primary regulator quotient are studied. It is shown that there are indecomposable such groups of arbitrarily large rank provided that the critical typeset contains some basic configuration and the exponent of the regulator quotient is sufficiently large.

Citation Formats

D. Arnold, A. Mader, O. Mutzbauer, and E. Solak, “REPRESENTATIONS OF FINITE POSETS OVER THE RING OF INTEGERS MODULO A PRIME POWER,” JOURNAL OF COMMUTATIVE ALGEBRA, pp. 461–491, 2016, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/38005.