Functors and dyadic sets of finite type
type: Item
instance of: scholarly article
title: Functors and dyadic sets of finite type
zbMATH Open document ID: 0976.16013
DOI: 10.1007/BF02482433
author: L. A. Nazarova
author: A. V. Rojter
publication date: 2001-06-12T00:00:00Z
review text: A dyadic set is a finite set \(D\) with three binary relations: a partial order \(\leq\), an equivalence \(\approx\) and a relation \(\vartriangleleft\) such that \(s\vartriangleleft t\) implies \(s\leq t\), if \(s\vartriangleleft t\leq u\) or \(s\leq t\vartriangleleft u\) then \(s\vartriangleleft u\), \(s\vartriangleleft s\) if and only if the cardinality \(r(s)\) of the \(\approx\)-equivalence class of \(s\) equals 1. The following is also assumed in the definition of a dyadic set: \(r(s)\leq 2\) for every \(s\in D\). If \(r(s)=2\) then \(s^*\) denotes the element such that \(s\approx s^*\) and \(s\neq s^*\). If \(s<t\) but not \(s\vartriangleleft t\) then the same holds with respect to \(s^*\) and \(t^*\). Finally, \(a\) is always comparable with \(a^*\) with respect to the order \(\leq\). If the relation \(\approx\) is trivial then \(\vartriangleleft\) coincides with \(\leq\). Given a field \(k\) one can define the category of \(k\)-representations of \(D\) in such a way that in case of the trivial relation \(\approx\) this category coincides with the usual category of \(k\)-representations of the poset \((D,\leq)\) (see also \textit{D. Simson} [Commun. Algebra 20, No. 12, 3541-3591 (1992; Zbl 0791.16011)]). The dyadic set \(D\) is said to be of finite type if it has only finitely many indecomposable representations up to isomorphism. The paper gives a criterion for finite type of dyadic sets expressed in combinatorial terms. It reduces to the well-known criterion of \textit{M. M. Kleiner} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklov 28, 32-41 (1972; Zbl 0345.06001)] in case when the relation \(\approx\) is trivial.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1505734
zbMATH Keywords: dyadic sets
zbMATH Keywords: representations of posets
zbMATH Keywords: finite representation type
zbMATH Keywords: vector space categories
zbMATH Keywords: categories of representations
reviewed by: Stanisław Kasjan
MaRDI profile type: Publication
cites work: Algebra VIII: Representations of finite-dimensional algebras. Transl. from the Russian
cites work: Tame and wild subspace problems
cites work: Elementary and multielementary representations of vectroids
cites work: Representation of partially ordered sets
cites work: Q4217048
full work available at URL: https://doi.org/10.1007/bf02482433
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1505734