The intersection of pseudovarieties of central simple semigroups. (Q862379)

Given two pseudovarieties \(P_S(A)\) and \(P_S(B)\) that are generated by finite semigroups \(A\) and \(B\), respectively, it is in general very difficult to determine whether or not the intersection \(P_S(A)\cap P_S(B)\) is a pseudovariety generated by a finite semigroup. The authors consider the following Kublanovsky problem: find an operation \(\circ\) which given two finite simple semigroups \(A\) and \(B\) returns a finite simple semigroup \(A\circ B\) with \(P_S(A)\cap P_S(B)=P_S(A\circ B)\). They present a solution to this problem for finite central simple semigroups \(A\) and \(B\). (A completely simple semigroup \(S\) is called central if the product of any two idempotents of \(S\) lies in the center of the maximal subgroup containing it.) Moreover, the authors present several examples of critical semigroups, including an infinite class of critical monoids.