Log-linear varieties of semigroups
Let the cardinality of the free algebra on \(n\) generators of the variety \(\mathcal V\) of algebras be denoted by \(f_ n({\mathcal V})\). Then \(\mathcal V\) is said to be log-linear if it is locally finite and there exists a constant \(c\) such that \(\log f_ n ({\mathcal V}) \leq cn\) for all \(n \geq 1\). The problem of characterizing log-linear varieties of semigroups was posed by \textit{G. Grätzer} and \textit{A. Kisielewicz} [Res. Expo. Math. 19, 57-88 (1992; Zbl 0772.08001)]. The main result of this paper solves this problem: Theorem. For any semigroup variety \(\mathcal V\) the following are equivalent: (i) \(\mathcal V\) is log-linear; (ii) \(\mathcal V\) satisfies the identities \(x^{\alpha + 1} \approx x^ \beta\), \(x_ 1 x_ 2 \dots x_ m \approx x_{\sigma(1)} x_{\sigma(2)} \dots x_{\sigma(m)}\), for some \(\alpha \geq \beta >0\) and some nontrivial permutation \(\sigma\) of the set \(\{1, \dots, m\}\); (iii) \(\mathcal V\) satisfies the identities \(x^{\alpha + 1} \approx x^ \beta\), \(x_ 1 x_ 2 \dots x_{i - 1} x_ i x_{i + 1} \dots x_ m \approx x_ 1 x_ 2 \dots x_{i - 1} x_{i + 1} x_ i x_{i + 2} \dots x_ m\), for some \(\alpha \geq \beta > 0\), \(m > i \geq 1\). An interesting consequence of this result is that every log-linear variety of semigroups is hereditarily finitely based. An example is given to show that the converse is false.
scientific article; zbMATH DE number 759387
scientific article