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Finite degree: algebras in general and semigroups in particular. (English) Zbl 1243.08003

An algebra \(A\) has finite degree if the clone \(\mathrm{Clo}(A)\) is determined by a finite set of finite relations on \(A\). A characterization of finite algebras having finite degree is given. It is proved that if two finite algebras generate the same variety then either both have finite degree or none of them has finite degree. All finite nilpotent semigroups, all finite commutative monoids and all finite commutative semigroups have finite degree. A finite algebra with bounded \(p_n\)-sequence has finite degree. The authors present an example of a five-element unary semigroup that has no finite degree. Also, examples showing that having finite degree is not a hereditary property for subalgebras, homomorphic images and direct products are given.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
20M14 Commutative semigroups
20M32 Algebraic monoids
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