Minimal noncommutative varieties and power varieties
A variety of finite monoids is a class of finite monoids closed under taking submonoids, quotients and \textit{finite} direct products. A language L is a subset of a finitely generated free monoid. The variety theorem of Eilenberg sets up a one to one correspondence between varieties of finite monoids and classes of languages called, appropriately, varieties of languages. If V is a variety of monoids, let PV denote the variety generated by the power set monoids PM, \(M\in V\). The authors show that for any variety, one has \(P^ 3V=P^ 4V\). In fact, if V contains a noncommutative monoid, then \(P^ 3V\) is the variety of all finite monoids; and if V is commutative, then \(P^ 2V=P^ 3V\), a result obtained already by Perrot and Straubing. The proof of the above theorem uses the following result, which the authors also prove: if V is a minimal noncommutative variety, then V is generated either by a noncommutative metabelian group or by the syntactic monoid of one of the languages \(\{a,b\}^*a\), \(a\{a,b\}^*\) or \(\{\) a\(b\}\).
scientific article; zbMATH DE number 3865588