A classification of aperiodic power monoids (Q1340430)

For a monoid \(M\), its power monoid is the monoid with underlying set the power set of \(M\) and multiplication given by \(AB = \{ab: a \in A,\;b\in B\}\). For a pseudovariety \(\mathbf V\) of monoids, \(\mathbf P\mathbf V\) denotes the pseudovariety generated by the power monoids of monoids in \(\mathbf V\). Let \(\mathbf A\) be the pseudovariety of all aperiodic monoids, i.e. monoids without non-trivial subgroups. The main result of the paper classifies all pseudovarieties of the form \(\mathbf P \mathbf V\), where \(\mathbf V \subseteq\mathbf A\). It turns out that, while the restriction of the operator \(\mathbf P\) to pseudovarieties of band monoids is injective, there are only 5 different pseudovarieties of the form \(\mathbf P \mathbf V\) in case \(\mathbf V\) contains a monoid which is not a band. In particular, the set of all pseudovarieties of the form \(\mathbf P \mathbf V\), where \(\mathbf V \subseteq\mathbf A\), is countable infinite. Another important consequence of the classification is that the solution set of an equation of the form \(\mathbf P\mathbf X = \mathbf W\), where \(\mathbf W \subseteq\mathbf A\), is always an interval. Several interesting conjectures are formulated. Reviewer's remark: The definition of the pseudovariety \(\mathbf L\) on p. 358 should read \(\mathbf L = [[x(yx)^ \omega = (yx)^ \omega]]\).