On finitely related semigroups

Abstract

An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid.

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].…”

Finite bands are finitely related

“…Proof. We use induction on m. If m = 2, the result already follows from [17,Theorem 6.2]. Hence fix m ≥ 3 and assume that the statement is true for values of indices up to m − 1.…”

“…The varieties LRB and RRB are the varieties of left regular and right regular bands defined by identities xyx ≈ xy and xyx ≈ yx, respectively. Their join is the variety of regular bands that is the subject of Theorem 6.2 of [17]. As we remarked earlier, our approach will be inductive with respect to the chains formed by varieties A m , B m and their duals, with the latter result from [17] serving as a basis of that induction.…”

Finite Regular Bands Are Finitely Related

“…Note added in revision. In mid-February 2012, around two months after this note was submitted, I have learned from a personal communication with Peter Mayr (CAUL, Lisbon) that he independently proved the main result of this paper; this proof is included in the wider study [13] of finite semigroups with respect to the property of being finitely related.…”