Abstract
An algebraic structure is finitely related (has finite degree) if its term func- tions are determined by some finite set of finitary relations. We show that the fol- lowing finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semi- groups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid. This answers a question by Davey, Jackson, Pitkethly, and Szabo from Davey et al. (Semigroup Forum, 83(1):89-122, 2011).
Highlights
A semigroup term t in k variables is a word in the alphabet x1, . . . , xk
The set of all finitary term functions on S is called the clone of term operations of S, denoted by Clo(S)
[11] Davey, Jackson, Pitkethly, and Szabó started the investigation of clones of semigroups and the relations that determine them. They posed the question: Is the clone Clo(S) of all term functions of a finite semigroup S necessarily determined by a finite set of relations?
Summary
A semigroup term t in k variables is a word in the alphabet x1, . . . , xk. On a fixed semigroup S := S, · , such a term t induces a k-ary term operation tS : Sk → S by evaluation. They posed the question: Is the clone Clo(S) of all term functions of a finite semigroup S necessarily determined by a finite set of relations? They showed that the answer is yes for finite nilpotent semigroups and for finite commutative semigroups They gave an example of a semigroup expanded with an additional unary operation which is not finitely related. The 2-element implication algebra {0, 1}, → which generates a congruence distributive variety is not finitely related (see for example [11] for an elementary proof). In [4] Barto showed that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. In [1] some kind of converse to Valeriote’s conjecture is proved: Every finite algebra with few subpowers is finitely related
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