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September 10, 2018

The Relationship of Supernilpotence to Nilpotence (2)

Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent.