\(E\)-free objects and \(E\)-locality for completely regular semigroups (Q1902169)

Let \(H\) be a variety of groups and let \(CR(H)\) denote the class of all completely regular semigroups all of whose subgroups belong to \(H\). This class is a variety of unary semigroups and forms an existence variety (e- variety), as well. The main result of the paper states that the e-variety \(CR(H)\) is e-local: each regular semigroupoid \(C\) (i.e. for each edge \(x\) there is an edge \(y\) such that \(x=xyx\)) all of whose local semigroups \(C_c\) belong to \(CR(H)\) divides a member of \(CR(H)\) in a ``regular'' way. Locality of varieties and pseudovarieties of monoids (and semigroups) has been introduced and discussed systematically by \textit{B. Tilson} [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)] and has been shown to be an indispensable tool for the study of semidirect product decompositions of varieties and pseudovarieties. The present paper deals with the rightful analogue of this concept in the context of regular semigroups and e-varieties, and the result will have important applications in the theory of semidirect product decompositions of e-varieties [see, e.g., \textit{P. R. Jones} and \textit{P. G. Trotter}, Semidirect products of regular semigroups, Trans. Am. Math. Soc. (to appear), and forthcoming papers]. The proof of the main result relies heavily on some knowledge of the bifree semigroup in \(CR(H)\) (termed ``e-free'' object in this paper), and more generally, of the bifree semigroupoid on a graph \(X\) in the e- variety \(lCR(H)\) of all regular semigroupoids whose local semigroups belong to \(CR(H)\). This knowledge, a word problem solution, is obtained as a preliminary result. Finally, it is pointed out that in all stages finiteness may be preserved so that the result is equally important in the context of finite (regular) semigroups, that is, in the context of (e-) pseudovarieties.