A unified approach to biidentities for e-varieties
An existence variety (or e-variety) of regular semigroups is a class of such semigroups closed under direct products, regular subsemigroups and morphic images. Two e-varieties are critical in this context: \(L{\mathcal I}\), the e-variety of all locally inverse semigroups and \(\mathcal{ES}\), the e-variety of all \(E\)-solid semigroups. Precisely these e-varieties (and their sub-e-varieties) contain bifree objects on any set \(X\). These are objects behaving like free objects do in usual varieties of universal algebras: for example, each member of the e-variety is a morphic image of some bifree object. The existence of these has been discovered by \textit{Y.~T.~Yeh} [Int. J. Algebra Comput. 2, No. 4, 471-484 (1992; Zbl 0765.20030)]. Yeh's results have been generalized by \textit{J.~Kadourek} and \textit{M.~B.~Szendrei} [Semigroup Forum 59, No. 3, 470-521 (1999; Zbl 0931.20049)] (for the \(E\)-solid case) and the reviewer [J. Algebra 166, No. 3, 630-650 (1994; Zbl 0806.20052)] (for the locally inverse case) in that suitable concepts of equational logic (including a Birkhoff type theorem) have been developed for either of these cases (and this also included a new proof of the existence of bifree objects). The language for these respective equational logics is different for the respective cases: in the \(E\)-solid case we have a doubled alphabet \(X\cup X'\) (representing elements and their inverses), a binary symbol \(\cdot\) (representing multiplication) and a unary symbol \(^{-1}\) (being interpreted as the partial unary operation of forming the group inverse inside the selfconjugate core of an \(E\)-solid semigroup). The set of terms has to be restricted appropriately according to the partiality of the unary operation \(^{-1}\). In the locally inverse case, the language consists of \(X\cup X'\) and \(\cdot\) as above and another binary symbol \(\wedge\) (being interpreted as the sandwich operation on a locally inverse semigroup). The paper under review consists of two main parts. In the first one, the authors present a unified approach to an equational logic for both of the above mentioned cases. Again we have \(X\cup X'\) and \(\cdot\) as above. As another ingredient the authors introduce a binary symbol \(s\) which is interpreted, in a locally inverse semigroup, as the partial binary operation of forming the sandwich element of two idempotents \(e,f\), and, in an \(E\)-solid semigroup, as the partial operation of forming the specific sandwich element \(e(fe)^{-1}f\) of \(e,f\). Again, the set of terms has to be restricted according to the partiality of the operation \(s\), however, there is a common such term algebra usable for both distinct cases. The usual theorems (including a Birkhoff theorem) then hold for this logic. It should be noticed that the price for this unification, namely the loss of the totality of the second binary operation in the locally inverse case may be high: \textit{M.~B.~Szendrei} and the reviewer [in Isr. J. Math. 108, 327-343 (1998; Zbl 0920.20051)] have shown that an equational theory of this form for a class \(\mathcal V\) may not be finitely axiomatizable even if \(\mathcal V\) is finitely axiomatizable as a class of first order structures. This never can happen if all operations considered are total. In the second part of the paper the authors continue work which has been initiated by \textit{J. Kadourek} [Int. J. Algebra Comput. 7, No. 1, 93-131 (1997; Zbl 0872.20050)], namely to study trifree objects, a generalization of bifree objects. Kadourek has shown that these always exist in any e-variety of locally orthodox semigroups (however, the theory is no longer so nice as in the bifree case: an analogue of the Birkhoff theorem does not hold). Here this has been completed by showing that all trifree objects exist in each e-variety of locally \(E\)-solid semigroups, and outside locally \(E\)-solid semigroups there do not exist trifree objects on more than two generators. For the latter, a finitely presented semigroup is used whose structure and behaviour is investigated by the use of a computer.
scientific article; zbMATH DE number 1423169