Semidirect products of regular semigroups

Within the usual semidirect product $S*T$ of regular semigroups $S$ and $T$ lies the set $\text {Reg}\,% (S*T)$ of its regular elements. Whenever $S$ or $T$ is completely simple, $\text {Reg}\,% (S*T)$ 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, ${\mathbf U}%$ and ${\mathbf V}%$, the e-variety ${\mathbf U}*{\mathbf V}%$ generated by $\{\text {Reg}\,% (S*T) : S \in {\mathbf U}% , T \in {\mathbf V}% \}$ is well defined if and only if either ${\mathbf U} %$ or ${\mathbf V}%$ is contained within the e-variety ${\mathbf {CS}}%$ of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety $L{\mathbf I}%$ of locally inverse semigroups is decomposed as ${\mathbf I}% * {\mathbf {RZ}}%$, where ${\mathbf I}%$ is the variety of inverse semigroups and ${\mathbf {RZ}}%$ is that of right zero semigroups; and the e-variety ${\mathbf {ES}}%$ of $E$-solid semigroups is decomposed as ${\mathbf {CR}}*{\mathbf G}%$, where ${\mathbf {CR}}%$ is the variety of completely regular semigroups and ${\mathbf G}%$ is the variety of groups. In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products ${\mathbf U}% * {\mathbf V}%$ of the above type, as a semidirect product of e-free semigroups from ${\mathbf U}%$ and ${\mathbf V}%$, ``cut down to regular generators''. Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups, $E$-solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups, $E$-solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties. Similar techniques are also applied to describe the e-free semigroups in a different `semidirect' product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.