THE SUBPOWER MEMBERSHIP PROBLEM FOR MAL'CEV ALGEBRAS

Given tuples a1, …, ak and b in An for some algebraic structure A, the subpower membership problem asks whether b is in the subalgebra of An that is generated by a1, …, ak. For A a finite group, there is a folklore algorithm which decides this problem in time polynomial in n and k. We show that the subpower membership problem for any finite Mal'cev algebra is in NP and give a polynomial time algorithm for any finite Mal'cev algebra with finite signature and prime power size that has a nilpotent reduct. In particular, this yields a polynomial algorithm for finite rings, vector spaces, algebras over fields, Lie rings and for nilpotent loops of prime power order.

Characteristically Nilpotent Algebras

Our aim in this paper is to extend (Theorem 1.7) to general algebras a classical result of Lie algebras due to Léger and Togo [6]. This extension requires, in turn, extension to general algebras of the concept of characteristically nilpotent algebras introduced by Dixmier and Lister [3] for Lie algebras. Based on this extended concept, we introduce in § 2 a new concept of radical (and semisimplicity) for general algebras and Lie triple systems. We study in some detail the consequences of the newly introduced concepts, furnishing necessary examples. With a stronger notion of characteristically nilpotent Mal'cev algebra arising out of these concepts, we obtain (Proposition 3.6) for such an algebra the parallel to the Leger-Tôgô result mentioned at the outset. In § 4, we deal with a further generalization of the concept of characteristic nilpotency leading to extension of very recent results of Chao [1] and Tôgô [12].