Transitive systems and conclusive partial orders

Abstract

This article introduces the concept of a transitive system, which is, in essence, a special kind of a directed, weighted graph. The weights of the graph are elements of an abelian group and have a transitive property. We address the topic of completion of such a system, that is, a possibility of extension of the existing weights to a complete graph. We formulate our terms and theorems on sets equipped with reflexive and transitive relations on which are defined transitive functions into abelian groups. We show that it is possible to restrict our study to the partial orders and to finite cyclic groups. All considered classes of examples show independence of the completion possibility from the group. This observation leads us to the concept of conclusive partial orders. Corresponding posets predetermine the possibility of completion of transitive systems independently on a group. In particular, the completion issue may be reduced to consideration of the two-element group. The class of conclusive partial orders includes the posets with two maximal or two minimal elements. We conjecture that all finite posets are conclusive and settle the conjecture for the posets with ten or fewer elements. Our work is motivated by appearance of the transitive systems in our study of full algebras of matrices (Cigler et al., 2019).