Power monoids and finite J-trivial monoids
A variety of finite monoids is a class of finite monoids closed under taking submonoids, quotients and \textit{finite} direct products. If M is a monoid, let \(P_ 1(M)\) denote the monoid of all subsets of M containing 1. If V is a variety, \(P_ 1V\) denotes the variety generated by the monoids \(P_ 1(M)\), \(M\in V\). Let J denote the variety of all J-trivial monoids. If V is a nontrivial commutative variety, then \(P_ 1V\) is the variety of all commutative J-trivial monoids (if V is trivial, so is \(P_ 1V)\). If V is a noncommutative variety, then either V contains a noncommutative aperiodic monoid or V contains a noncommutative group: in the first case, \(P_ 1V=J\); in the second case, \(P_ 1P_ 1V=J\), and if furthermore V contains all p-groups for some p, then \(P_ 1V=J\). The only unsettled question is whether there exists a variety V, containing a noncommutative group and such that \(P_ 1V\neq J\).
scientific article; zbMATH DE number 3865589