Higher commutators in semigroups with zero

A group is nilpotent if it has a finite central series. In universal algebra we have two generalizations of the notion of nilpotency for arbitrary algebras: nilpotency and supernilpotency. We obtain these definitions by using the term condition commutator, as introduced by Freese and McKenzie. The definition of supernilpotency uses a higher order commutator which was introduced by Bulatov.

We develop the notion of the higher commutator of ideals in semigroups with zero. Further on, we show that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero all three conditions of supernilpotency, nilpotency and nilpotency in the sense of semigroup theory, are equivalent. We also give a sufficient and necessary condition for solvability of semigroups with zero.

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