Completely (0-)simple semigroups and maximal quotient semigroups (Q1318912)

A semigroup \(S\) is called right (left) reductive if for \(s_ 1,s_ 2 \in S\) we have that \(s_ 1 t = s_ 2 t\) (\(ts_ 1 = ts_ 2\)) for all \(t \in S\) implies \(s_ 1 = s_ 2\). This is equivalent to saying that \(S\) has a maximal right (left) quotient semigroup. The author gives conditions for a completely 0-simple and a completely simple semigroup \(S\) to be right reductive in terms of the sandwich matrix \(P\) used in the matrix representation \(M^ 0[G;I,\Lambda;P]\) or \(M[G;I,\Lambda;P]\) of \(S\). In this case the maximal right quotient semigroup is described in either case as a certain column monomial matrix semigroup over \(G\).