Commuting Regular Graphs for Non-commutative Semigroups

To study the commuting regularity of a semigroup, we use a graph. Indeed, we define a multi-graph for a semigroup and identify this graph for the semidirect product of two monogenic semigroups. For a non-group semigroup S, the ordered pair \((x, y)\) of the elements of \(S\) is called a commuting regular pair if for some \(z \in S\), \(xy = yxzyx\) holds, and \(S\) is called a commuting regular semigroup if every ordered pair of S is commuting regular. As a result of Abueida in 2013 concerning the heterogenous decomposition of uniform complete multi-graphs into the spanning edge-disjoint trees, we show that for a semigroup of order \(n\), the commuting regular graph of \(S\), \(\Gamma(S)\) has at most n spanning edge-disjoint trees.