On radical congruence systems (Q1300536)

A radical congruence system \(\rho\) for the class of all finite semigroups \(\mathbf S\) associates with each finite semigroup \(S\) a congruence \(\rho(S)\) such that four natural compatibility conditions are satisfied. A typical example of such a congruence system is \(\mu\) which associates with each finite semigroup \(S\) the greatest idempotent separating congruence \(\mu(S)\) on \(S\). Radical congruence systems play an important role in the study of the lattice of pseudovarieties of finite semigroups. For a pseudovariety \(\mathbf V\), a congruence system \(\rho\) is over \(\mathbf V\) if each idempotent \(\rho(S)\)-class is a member of \(\mathbf V\). Let \(\mathbf{IE}={\mathbf N}\vee{\mathbf G}\) be the pseudovariety of all nilextensions of groups (here \(\mathbf N\) is the pseudovariety of all finite nilpotent semigroups and \(\mathbf G\) is the pseudovariety of all finite groups); then \(\mu\) is over \(\mathbf{IE}\). Moreover, for each \(S\), \(\mu(S)\) is the greatest congruence on \(S\) which is over \(\mathbf{IE}\). The main result of the paper characterizes all radical congruence systems \(\rho\) for which \(\rho(S)\) is the greatest congruence over some pseudovariety as follows: the pseudovarieties \(\mathbf V\) such that each finite semigroup \(S\) admits a greatest congruence over \(\mathbf V\) and such that the resulting congruence system is a radical congruence system are precisely the pseudovariety \(\mathbf S\) of all finite semigroups and all pseudovarieties of the form \({\mathbf N}\vee{\mathbf H}\), \({\mathbf D}\vee{\mathbf H}\), \({\mathbf K}\vee{\mathbf H}\) and \(L{\mathbf H}\) where \(\mathbf H\) is any extension closed pseudovariety of groups, \(\mathbf D\) is the pseudovariety of all nilextensions of right zero semigroups, \(\mathbf K\) is the dual of \(\mathbf D\) and \(L{\mathbf H}\) is the pseudovariety of all nilextensions of completely simple semigroups with all subgroups being in \(\mathbf H\).