On finitely based groups and nonfinitely based quasivarieties
A set \(W\) of identities of a variety \(\mathcal V\) of groups is said to be a strict basis for \(\mathcal V\) if, modulo the group axioms, every identity of \(\mathcal V\) can be deduced from the members of \(W\) using only substitutions in which variables are replaced by variables. A variety is strictly finitely based if it has a finite strict basis, and a group is strictly finitely based if the variety which it generates is. The main result of the second section of the paper is that a finite group is strictly finitely based if and only if it is nilpotent (and hence, if and only if the variety it generates does not contain a variety of the form \({\mathcal A}_p{\mathcal A}_q\), where \(p\) and \(q\) are distinct primes). In the remaining sections a two-sorted algebra \(\mathbb{A}=\langle S,G,b\rangle\), where \(G\) is a finite group and \(b\colon G\times S\to S\) is a faithful left action of \(G\) on \(S\), is introduced. From this a 1-sorted algebra \(\mathbb{A}^*\), called an \(\mathbb{L}\)-algebra is constructed. Using the results of Section 2, it is shown that \(\mathbb{A}^*\) is finitely based if and only if \(G\) is nilpotent, and if \(G\) is not nilpotent, then no set of \(\mathbb{L}\)-algebras generating the same variety as \(\mathbb{A}^*\) can be contained in any finitely based, locally finite quasivariety. The smallest such example is of order only 18, arising from the symmetric group of order 3 and its action on a 3-element set. This example answers affirmatively questions of \textit{W. Rautenberg} [Stud. Logica 40, 315-353 (1981; Zbl 0493.03006)] and Wroński, and \textit{D. Pigozzi} [Trans. Am. Math. Soc. 310, No. 2, 499-533 (1988; Zbl 0706.08009)].
scientific article; zbMATH DE number 1203902
scientific article