Decompositions of Directed Graphs with Loops and Related Algebras

A correspondence between decompositions of complete directed graphs with loops into collections of closed trails which partition the edge set of the graph and the variety of all column latin groupoids is established. Subvarieties which consist of column latin groupoids arising from decompositions where only certain trail lengths occur are examined. For all positive integers \(m\), the set of values of \(n\) for which the complete directed graph with loops on a vertex set of cardinality \(n\) can be decomposed in this manner such that all the closed trails have length \(m\) is shown to be the set of all \(n\) for which \(m\) divides \(n^2\).