Representations of bipartite completed posets
type: Item
instance of: scholarly article
title: Representations of bipartite completed posets
zbMATH Open document ID: 0671.18002
DOI: 10.1007/BF02566776
author: L. A. Nazarova
author: A. V. Rojter
published in: Commentarii Mathematici Helvetici
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140131
review text: A completed poset \({\tilde S}\) consists of a finite set S, a partial order relation \(S^{\leq}=\{(s,t)\in S^ 2:\) \(s\leq t\}\) on S and an equivalence relation \(\sim\) on S such that \(r\leq s\leq t\) and \((r,s)\sim (r',t')\) imply the existence of a unique \(s'\) satisfying \(r'\leq s'\leq t'\), \((r,s)\sim (r',s')\) and \((s,t)\sim (s',t')\). By a \(\mu\)-chain \({\mathring P}\) in a poset \(P\) we mean a subset \({\mathring P}\) consisting of m linearly ordered elements \(s_ 1<...<s_ m\) such that for each \(i\leq m\) the interval \([a_ i,a_{\mu (i)}]\) (in P) \(=\{s_ i,s_{i+1},...,s_{\mu (i)}\}\), where \(\mu:\{ 1,2,...,m\}\mapsto \{1,2,...,m\}\) is a non-decreasing function such that \(\mu(i)\geq i+1\) for each \(i\leq m.\) If \({\tilde S}\) is a completed poset we say that \({\tilde S}\) is a bipartite completed poset if there exist two finite posets P, Q equipped with \(\mu\)-chains \({\mathring P}=\{s_ 1<...<s_ m\}\), \({\mathring Q}=\{s'_ 1<...<s'_ m\}\), respectively, such that: i) \(S=P\coprod Q\) (=disjoint union), ii) \(S^{\leq}=P^{\leq}\cup Q^{\leq}\cup P\times Q\) (in particular if \(p\in P\) and \(q\in Q\) imply \(p<q)\), iii) \((s_ i,s_ j)\sim (s'_ i,s'_ j)\) if \(i\leq m\) and \(j\leq \mu (i)\); any other \((s,t)\in S^{\leq}\) is equivalent only to itself (we denote this fact by \({\tilde S} = P\triangleleft Q)\). The points of \({\mathring P}\) and \({\mathring Q}\) are called thick, those of \({\dot P}=P\setminus \mathring P\) and \({\dot Q} = Q\setminus \mathring Q\) thin. For each thick point \(s\in S\), we denote by \(s'\) the point of \(S\) such that \(s'\sim s\neq s'\). The quasidual of \({\tilde S} = P\triangleleft Q\) is by definition \({\tilde S}^* = Q\triangleleft P.\) If \({\tilde S} = P\triangleleft Q\), for each \(s\in S\) we set \(S(s)=\{t\in S: s\nleq t\nleq s\}\), endow \(S(s)\) with the order relation induced by \(S\) and formally add to \(S(s)\) a smallest element 0 and a largest element 1; we define \({\bar S}(s) = S(s)\cup…
Mathematics Subject Classification ID: 18B35
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4097608
zbMATH Keywords: bipartite completed poset
zbMATH Keywords: representation-finite
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
OpenAlex ID: W2083157323
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description: scientific article; zbMATH DE number 4097608