Finite Regular Bands Are Finitely Related

Abstract

An algebra A is said to be finitely related if the clone Clo(A) of its term operations is determined by a finite set of finitary relations. We prove that each finite idempotent semigroup satisfying the identity xyxzx ≈ xyzx is finitely related.2010 Mathematics subject classification: primary 20M07; secondary 08A40, 05A05.

On finitely related semigroups

“…In this section we show finite relatedness for the larger class of regular bands (see Theorem 6.2). After submitting the present article we received notice that Dolinka independently proved the same result using a different approach [12].…”

Finite Regular Bands Are Finitely Related – Retraction

“…The main result of [1] in Theorem 4.5 is correct as stated and has been proved by a different method in [2].…”

“…The argument in [1] is incorrect. The author, having noticed an error in the proof of Lemma 4.2, has retracted the article.…”

Finitely related algebras in congruence modular varieties have few subpowers

“…A recent result of Aichinger, Mayr, and McKenzie [1] is another source of examples: it implies that all finite quasigroups and their expansions (like finite groups, rings and modules) are finitely related. There are both finitely related and non-finitely related (expansions of) semigroups [24,25,40]. The simplest example of a non-finitely related algebra is the two-element implication algebra ((0, 1); {→}), where → is the implication regarded as a binary operation (see [24] for an elementary proof).…”