On finite hyperidentity bases for varieties of semigroups
An identity \(E\) is hypersatisfied by a variety \(V\) if whenever the operation symbols in \(E\) are replaced by term functions of \(V\) of the appropriate arity then the resulting identity is an identity in \(V\); \(E\) is then said to be a hyperidentity of \(V\). The set of all hyperidentities of \(V\) is denoted by \(H(V)\). Just as with ordinary identities it is natural to ask whether \(H(V)\) is finitely based. This paper considers this in the special case where \(V\) is a variety of semigroups. This question has already been answered in the negative for several varieties of semigroups (including the variety of all semigroups [\textit{G. M. Bergman}, Aequationes Math. 23, 50-65 (1981; Zbl 0485.08005)]). This paper shows how Bergman's proof can be extended to cover all semigroup varieties except those satisfying an identity of the form \(x^ 2 = x^{2+m}\) for some positive \(m\). It is also shown that the variety of bands and all its subvarieties do not have finite bases of hyperidentities. In all these cases, the proofs depend on finding hyperidentities of arbitrarily high arity. The paper concludes by showing that the medial variety of semigroups (which is defined by the identity \(xyzw = xzyw\)) has the property that even its set of hyperidentities involving only one operation symbol, and that binary, is not finitely based.
scientific article; zbMATH DE number 724100