Simplicity of wreath product of semigroups with fixed passive semigroup (Q1337475)

The wreath product \(T = (S \text{ wr } R\mid A_ R)\) of semigroups \(S\) and \(R\) by a right \(R\)-act \(A\), i.e., \(R\)-operand or \(R\)-system in other terminology, is defined as the set \(F(A,S) \times R\), where \(F(A,S)\) is the set of all mappings from \(A\) into \(S\), and the multiplication on \(T\) is given by the rule \((f,p)(g,q) = (f^ p g,pq)\) with \((f^ pg)(x) = f(x)g(xp)\) for all \(x \in A\). Main Theorem: The wreath product \((S \text{ wr }R \mid A_ R)\) is a simple semigroup for each simple semigroup \(R\) and each of its right acts \(A_ R\) iff the passive semigroup \(S\) is left simple.