A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups.
A semigroup is `nilpotent in the sense of Malcev, of step \(n\)' (more usually, in English, `of nilpotency class \(n\)') if it satisfies the identity \(U_n=V_n\), where the words \(U_n,V_n\) are defined recursively as follows: \(U_0=x\), \(V_0=y\), \dots, \(U_{k+1}=U_kz_{k+1}V_k\), \(V_{k+1}=V_kz_{k+1}U_k\). The Malcev Theorem in the title asserts that for groups this identity defines nilpotency of class \(n\) in the classical sense. The author first expands this theorem to describe the simple and 0-simple semigroups with this property (they are either nilpotent groups or Brandt semigroups over nilpotent groups, respectively). Most of the paper is devoted to the study of the preservation of nilpotency under the (extended standard) wreath product. Here is a sample theorem (Theorem 4.1). Suppose \(S\) is nilpotent in the usual semigroup sense. If \(R\) is nilpotent in the sense of Malcev, then their wreath product \(S\text{\,wr}_1R\) is nilpotent in the sense of Malcev (and, necessarily, the converse holds). If \(S\) is not nilpotent and \(S\text{\,wr}_1R\) is nilpotent in the sense of Malcev, then \(R\) is necessarily a finite nilpotent group and \(S\) is a uniformly periodic semigroup that is nilpotent in the sense of Malcev. The author goes on to find necessary and sufficient conditions that the wreath product be nilpotent in the sense of Malcev, under the hypothesis that each of \(R\) and \(S\) generates a variety with the property that all of its nilsemigroups are nilpotent.
scientific article; zbMATH DE number 6156634
scientific article