Independence of algebras with edge term

In [A. L. Foster, The identities of — and unique subdirect factorization within — classes of universal algebras, Math. Z. 62 (1955) 171–188], two varieties [Formula: see text] of the same type are defined to be independent if there is a binary term [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give necessary and sufficient conditions for two finite algebras with a Mal’cev term (or, more generally, with an edge term) to generate independent varieties. In particular we show that the independence of finitely generated varieties with edge term can be decided by a polynomial time algorithm.