Finite semigroups that are minimal for not being Malcev nilpotent. (Q2878798)

Finite semigroups \(S\) that are not nilpotent in the sense of Malcev and such that every proper subsemigroup of \(S\) and every proper Rees factor of \(S\) is nilpotent are considered. Such an \(S\) is called a finite minimal non-nilpotent semigroup. Finite non-nilpotent groups all whose proper subgroups are nilpotent were characterized by O. Yu. Shmidt [see Theorem 6.5.7 in: \textit{W. R. Scott}, Group theory. 2nd ed., New York: Dover Publications (1987; Zbl 0641.20001)].NEWLINENEWLINE The main result of the paper shows that every finite semigroup \(S\) of the considered type that is not a group and is not a two-element right zero or left zero semigroup falls into one of the three classes of semigroups constructed in the paper. Every such \(S\) is a union (not necessarily 0-disjoint) of a completely 0-simple inverse ideal \(M\) whose nontrivial maximal subgroups are nilpotent and a 2-generated subsemigroup \(T\). The conditions describing these three classes of semigroups are in terms of the action of \(S\) on the set of \(\mathcal R\)-classes of \(M\). It is also shown that each class contains a semigroup that is not minimal non-nilpotent.