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On finitely related semigroups. (English) Zbl 1278.20077

Let \(\mathbf A=\langle A,F\rangle\) be an algebra, where \(A\) is a set and \(F\) is a set of finitary operations on \(A\). The algebra \(\mathbf A\) is called finitely related if there exist subalgebras \(R_1,\dots,R_l\) of finitary powers of \(\mathbf A\) such that every operation on \(A\) that preserves every \(R_i\) (\(1\leq i\leq l\)) is a term operation on \(A\). Finitely related algebras were considered by many authors under varied names.
It is shown that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Some examples of not finitely related semigroups are presented, particularly, the 6-element Brandt monoid is not finitely related. Some open problems are formulated.

MSC:

20M07 Varieties and pseudovarieties of semigroups
08A40 Operations and polynomials in algebraic structures, primal algebras
20M05 Free semigroups, generators and relations, word problems
20M10 General structure theory for semigroups
20M14 Commutative semigroups
20M17 Regular semigroups
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