Functors and dyadic sets of finite type
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.
scientific article; zbMATH DE number 1505734