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Semidirect Products of Regular Semigroups
 Trans. Amer. Math. Soc
, 1999
"... Within the usual semidirect product S T of regular semigroups S and T lies the set Reg (S T ) of its regular elements. Whenever S or T is completely simple, 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 ..."
Abstract

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Within the usual semidirect product S T of regular semigroups S and T lies the set Reg (S T ) of its regular elements. Whenever S or T is completely simple, 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 and V , the evariety U V generated by fReg (S T ) : S 2 U; T 2 V g, is well defined if and only if either U or V is contained within the evariety CS of completely simple semigroups. General properties of this product, together with decompositions of many important evarieties, are obtained. For instance, as special cases of general results the evariety LI of locally inverse semigroups is decomposed as I RZ, where I is the variety of inverse semigroups and RZ is that of right zero semigroups; and the evariety ES of Esolid semigroups is decomposed as CR G, where CR is the variety of completely regular ...