\(E\)-solid e-varieties are not finitely based
The authors prove a stronger theorem than is suggested by the title: no non-orthodox e-variety of \(E\)-solid semigroups is finitely based. We should explain the framework of this theorem. Y. T. Yeh showed that within the e-varieties of locally inverse semigroups and of \(E\)-solid semigroups (these terms implicitly including regularity), some of the standard concepts of universal algebra have natural analogues. For instance, sub-e-varieties of each (and only of each) possess ``bifree'' semigroups, which play a role akin to free objects in varieties. J. Ka\u{d}ourek and M. B. Szendrei introduced the syntactic notion of ``bi-identity'' for the e-varieties of orthodox semigroups. \textit{K. Auinger} [J. Algebra 166, No. 3, 630-650 (1994; Zbl 0806.20052)] founded an ``equational logic'' for locally inverse semigroups, involving the addition of a binary operation to the signature. \textit{J. Ka\u{d}ourek} and \textit{M. B. Szendrei} [Semigroup Forum 59, No. 3, 470-521 (1999)] founded such a logic for \(E\)-solid semigroups, by adding a partial unary operation to the signature, as follows. One characterization of \(E\)-solid semigroups is that their self-conjugate core -- the smallest subsemigroup that contains the idempotents and is closed under conjugation -- is completely regular: thus on this core is defined the unary operation of inversion. In a bifree \(E\)-solid semigroup the self-conjugate core is the kernel of the natural morphism upon the corresponding free group. Thus, starting with a countably infinite set \(X\), choosing a set \(X'\) in bijection with, and disjoint from, \(X\) and setting \(\overline X=X\cup X'\), form the free unary semigroup \(U(\overline X)\) on \(\overline X\), let \(K(X)\) denote the kernel of the natural morphism upon the free group on \(X\) and let \(T(X)\) be the smallest subsemigroup \(T\) of \(U(\overline X)\) that contains \(\overline X\) and contains \((u)^{-1}\) whenever \(u\in T\cap K(X)\). Then \(T(X)\) is a semigroup on which is defined a partial unary operation, with domain the kernel of the natural morphism upon the free group on \(X\). It is with respect to terms from this object that the results of this paper are found.
e-varieties of locally inverse semigroups
scientific article; zbMATH DE number 1270360
scientific article