The ascending varietal chain of a variety of semigroups
Let V be a variety of algebras. For every cardinal number \(r<\omega\) let \(V_ r\) be the variety generated by the V-free algebra on r generators. The sequence \(V_ 1\subseteq V_ 2\subseteq...\subseteq V_ r\subseteq...\subseteq V_{\omega}=V\) is called the ascending varietal chain of V. \textit{B. Jónsson}, \textit{G. McNulty} and \textit{R. Quackenbush} [Can. J. Math. 27, 25-31 (1975; Zbl 0305.08003)] showed that for any set S of positive integers there is a variety V such that \(V_ n=V_{n+1}\) if and only if \(n\in S.\) In this paper it is shown that for any set S of integers greater than 1 there is a variety V of semigroups such that \(V_ n=V_{n+1}\) if and only if \(n\in S\). This is best possible in the sense that equality of \(V_ 1\) and \(V_ 2\) implies commutativity of V (and there are only countably many such varieties).
scientific article; zbMATH DE number 3993870