Semidirect products of regular semigroups
Peter R. Jones, Peter Trotter · Transactions of the American Mathematical Society · 1997
Within the usual semidirect product S ∗ T S*T of regular semigroups S S and T T lies the set Reg ( S ∗ T ) \text {Reg}\, (S*T) of its regular elements. Whenever S S or T T is completely simple, Reg ( S ∗ T ) \text {Reg}\, (S*T) is a (regular) subsemigroup. It is this ‘product’ that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, U {\mathbf U} and V {\mathbf V} , the e-variety U ∗ V {\mathbf U}*{\mathbf V} generated by { Reg ( S ∗ T ) : S ∈ U , T ∈ V } \{\text {Reg}\, (S*T) : S \in {\mathbf U} ,