Semigroup Forum

, Volume 87, Issue 1, pp 129–148

# Left Reductive Congruences on Semigroups

• Attila Nagy
RESEARCH ARTICLE

## Abstract

A semigroup S is called a left reductive semigroup if, for all elements a,bS, the assumption “xa=xb for all xS” 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 xS. 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.

## Keywords

Left reductive semigroup Left reductive congruence Left zero semigroup Nilpotent semigroup Noetherian semigroup

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