Finite semigroups that are minimal for not being Malcev nilpotent

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ϱ a congruence on S. Consider the sequence ϱ (0)⊆ϱ (1)⊆⋯⊆ϱ (n)⊆⋯ of congruences on S, where ϱ (0)=ϱ and, for an arbitrary non-negative integer n, ϱ (n+1) is defined by (a,b)∈ϱ (n+1) if and only if (xa,xb)∈ϱ (n) for all x∈S. We show that \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )\) for an arbitrary congruence ϱ on a semigroup S, where lrc(ϱ) denotes the least left reductive congruence on S containing ϱ. We focuse our attention on congruences ϱ on semigroups S for which the congruence \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is left reductive. We prove that, for a congruence ϱ on a semigroup S, \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is a left reductive congruence of S if and only if \(\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}\) is a left reductive congruence on the factor semigroup S/ϱ (here ι (S/ϱ) denotes the identity relation on S/ϱ). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S n =S n+1 is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.

By an ordered semigroup we mean a system (S, o, <) where S[o) is a semigroup, 5(<) is a totally ordered set, and the monotone condition -a ^ b and x in S imply ax ^ bx and xa ^ xb -is satisfied.An (ideal) extension of a semigroup 5 by a semigroup Г with zero is a semigroup I containing S as an ideal such that IJS, the Rees factor semigroup, is isomorphic with T.Let S and Г be disjoint semigroups, Г having a zero element 0, and let Г* = T\0.Let Ф be a single-valued mapping of Г* into S satisfying tt' ф 0 impHes (tt') cp = = t(pt'(p, and hi Z == S Kj T*.Let о be defined on I as follows (t, f e T"^; s, s' e S):