The Rhodes expansion and free objects in varieties of completely regular semigroups (Q755897)

Various operators on the lattice of varieties of completely regular semigroups have been intensively studied in the last few years. Those defined by certain Mal'cev products have played a particularly important role. In this paper, the object of study is the operator \({\mathcal V}\to \vec {\mathcal V}={\mathcal L}{\mathcal Z}\circ {\mathcal V}\), where \({\mathcal L}{\mathcal Z}\) is the variety of left zero semigroups. A dual operator is defined in the natural way. An alternative and useful view of \(\vec {\mathcal V}\) is as \({\mathcal V}^ K\cap {\mathcal V}^{T_ l}\) (see \textit{F. Pastijn} [J. Aust. Math. Soc., Ser. A 49, 24-42 (1990; Zbl 0706.20042)]); the main technical tool, however, is its interpretation in terms of fully invariant congruences, in the manner of \textit{L. Polák} [Semigroup Forum 32, 97- 123 (1985; Zbl 0564.20034)]. Among many interesting properties determined, it is shown that iteration of this operator and its dual yields a simply described sublattice of the filter generated by \({\mathcal V}\). The least upper bound of this lattice is just the variety \({\mathcal V}^ K={\mathcal B}\circ {\mathcal V}\) (where \({\mathcal B}\) is the variety of bands). The free object in \(\vec {\mathcal V}\) on a set X is obtained from that in \({\mathcal V}\) by means of the well-known Rhodes expansion, ``cut down to generators'' (in the completely regular sense). The free object on \({\mathcal V}^ K\) is thus obtained as a projective limit of such expansions and their duals.