The bifree regular \(E\)-solid semigroups
Bifree objects have been introduced by \textit{J. Kaďourek} and \textit{M. Szendrei} for e-varieties (equivalently: bivarieties) of orthodox semigroups [Semigroup Forum 40, No. 3, 257-296 (1990; Zbl 0705.20052)], and this concept has turned out to be the adequate notion of free object in the context of e-varieties of regular semigroups (that is, classes of regular semigroups closed under the formation of direct products, regular subsemigroups and homomorphic images). Bifree objects do not exist in arbitrary e-varieties, however. \textit{Y. Yeh} [Int. J. Algebra Comput. 2, No. 4, 471-484 (1992; Zbl 0765.20030)] has shown that an e-variety \(\mathbf V\) contains bifree objects on any non-empty set \(X\) if and only if \(\mathbf V\) is contained in \(\mathbf L I\) (the e-variety of all locally inverse semigroups) or in \(\mathbf E S\) (the e-variety of all \(E\)-solid semigroups). To give concrete descriptions of the bifree objects in the critical e-varieties \(\mathbf L I\) and \(\mathbf E S\) is therefore of particular interest. The case \(\mathbf L I\) has been dealt with by the reviewer [J. Algebra 166, No. 3, 630-650 (1994; Zbl 0806.20052)], the present paper is devoted to the case \(\mathbf E S\). The construction of the bifree \(E\)-solid semigroup on a set \(X\) roughly is as follows: Let \(X\) be any non-empty set, let \(G = F \mathbf G(X)\) be the free group on \(X\) and let \(\mathcal G\) be the Cayley graph of \(G\). Let \(U^{\text{loc}}({\mathcal G})\) be the free locally unary semigroupoid on the graph \(\mathcal G\) (i.e. the free semigroupoid on \(\mathcal G\) such that a unary operation is defined on all loops). It is clear that the set of all edges of \(U^{\text{loc}}({\mathcal G})\) is contained in the free unary semigroup on the set \(A = \text{Arr}({\mathcal G}) = G \times X\) of all edges of \(\mathcal G\). Let \(\xi\) be the congruence on \(U^{\text{loc}}({\mathcal G})\) which arises if one restricts the fully invariant congruence on the free unary semigroup on \(A\) corresponding to the variety of all completely regular semigroups. Further, \(G\) acts on the quotient semigroupoid \(C = U^{\text{loc}}({\mathcal G})/\xi\) in a natural way by automorphisms on the left. This allows to apply a construction reminiscent of a semidirect product: put \[ S(X) = \{(a,g) \in C \times G \mid a \in C(1,g)\} \] and define a multiplication on \(S(X)\) by \((a,g)(b,h) = (a \cdot gb, gh)\). In section 4 the main result of the paper is proved: the semigroup \(S(X)\) together with the mapping \(\nu : X \cup X' \to S(X)\), \(x\mapsto ((1,x) \xi, x)\), \(x' \mapsto ((1, x^{-1}) \xi, x^{-1})\) is a model of the bifree \(E\)-solid semigroup on \(X\). As an immediate consequence one gets the important result that each regular \(E\)-solid semigroup divides a semidirect product of a completely regular semigroup by a group.
scientific article; zbMATH DE number 836809
scientific article