Type-II conjecture is true for finite \(\mathcal J\)-trivial monoids
A relation \(\varphi: M\to G\) from a finite monoid \(M\) to a finite group \(G\) is called a relational morphism if its graph is a submonoid of \(M\times G\) and it projects surjectively on the first component. The submonoid \(M_{II}\) of the ``type-II elements'' is defined as the set of all elements of \(M\) which are related with \(1\in G\) for every relational morphism \(\varphi: M\to G\). The ``weak type-II conjecture'' states that the function \(M\to M_{II}\) is computable. An approach to this conjecture proposed by \textit{J. Karnofsky} and \textit{J. Rhodes} [Semigroup Forum 24, 55-66 (1982; Zbl 0503.20028)] consists in taking the submonoid \(M_{II'}\) of constructible type-II elements which is defined as the least submonoid of \(M\) containing the idempotents and closed under the operation \(m\mapsto amb\) for any \(a,b\in M\) such that \(aba=a\) or \(bab=b\). While it is clear that \(M_{II'}\subseteq M_{II}\), the ``strong type- II conjecture'' asserts that \(M_{II'}=M_{II}\) for any finite monoid \(M\). The main theorem in the paper states that if every regular element of \(M\) lies in \(M_{II'}\), then \(M_{II'}=M_{II}\). As an application, it is shown that the strong type-II conjecture holds for the power monoid \({\mathcal P}(G)\) of any finite group \(G\). (The strong type-II conjecture has since been established by \textit{C. J. Ash} [Int. J. Algebra Comput. 1, No. 1, 127-146 (1991; Zbl 0722.20039)]).
scientific article; zbMATH DE number 130658
scientific article