Representations of bipartite completed posets
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 \({\bar S}(s) = S(s)\cup \{0,1\}\). With this notation, we attach two posets \({\hat P}\) and \({\hat Q}\) to the components \(P\) and \(Q\): the poset \({\hat P}\) consists of the thin points \(s\in \dot P\) and of the pairs \((p,t)\) where \(p\in \mathring P\) and \(t\in \bar S(p')\). The subset \({\dot P}\) of \({\hat P}\) is equipped with the order induced by \({\tilde S}\) and set \(s\leq (p,t)\) iff \(s\leq p\), \((p,t)\leq s\) iff \(p\leq s\). Also \((p_ 1,t_ 1)\leq (p_ 2,t_ 2)\) in the following two cases: i) \(p_ 1<p_ 2\) and \((p_ 1,p_ 2)\nsim (p'_ 1,p'_ 2)\), ii) \(p_ 1\leq p_ 2\), \((p_ 1,p_ 2)\sim (p'_ 1,p'_ 2)\) and one of the conditions \(p'_ 1\leq t_ 2\neq 1\), \(0\neq t_ 1\leq p'_ 2\) or \(t_ 1\leq t_ 2\) holds. The description of \({\hat Q}\) is dual (and quasidual) to that of \({\hat P}\). In particular the elements of \({\hat Q}\) have the form \(t\in \dot Q\) or (q,s) where \(q\in \mathring Q\) and \(s\in \bar S(q').\) The main scope of this paper is to prove the following results: Theorem 1. The bipartite completed poset \({\tilde S} = P\triangleleft Q\) is representation-finite (i.e. admits only finitely many isomorphism classes of indecomposables) iff so are the posets \({\hat P}\) and \({\hat Q}\). Theorem 2. Let \({\tilde S} = P\triangleleft Q\) be a faithful bipartite completed poset (i.e. \({\tilde S}\) admits a faithful indecomposable representation). i) If the poset \(S(s)\) is linearly ordered for each \(s\in \mathring P\) (resp. \(s\in \mathring Q)\), then there is a natural bijection from \(\text{ind}{\tilde S}\setminus \text{ind}\dot P\) onto \(\text{ind}{\hat Q}\setminus \{\Phi_ 0\}\) (resp. from \(\text{Ind}{\tilde S}\setminus \text{ind}\dot Q\) onto \(\text{ind}{\hat P}\setminus \{\Phi_ 0\}\) \((\Phi_ 0 =\) the trivial representation). ii) If \({\tilde S}\) is representation-finite and if there exist thick points \(p\in \mathring P\) and \(q\in \mathring Q\) such that neither \(S(p)\) nor \(S(q)\) is linearly ordered, then \({\tilde S}\) is isomorphic to \({\tilde E}\) or \({\tilde E}^*\) (where \({\tilde E}\) is the bipartite completed poset in the case \(m=2\) and \(\mu(1)=\mu(2)=2)\).
scientific article; zbMATH DE number 4097608