Retractions of split graphs and End-orthodox split graphs (Q1850000)

A finite graph \(\Gamma\) is said to be split if its vertex set can be partitioned into two subsets \(K\) and \(I\), so that every pair of vertices of \(K\) is joined by an edge of \(\Gamma\), and no two vertices of \(I\) are. A monoid \(M\) is called regular if, given any \(a\in M\), there is an element \(b \in M\) so that \(aba=a\), and a regular monoid is called orthodox if the product of any two idempotents is idempotent. A graph is called End-orthodox if its endomorphism monoid is orthodox. In this paper, the author finds a necessary and sufficient condition for a split graph to be End-orthodox.