Mal’tsev products of varieties, I

We investigate the Mal’tsev product $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W of two varieties $$\mathcal {V}$$ V and $$\mathcal {W}$$ W of the same similarity type. While such a product is usually a quasivariety, it is not necessarily a variety. We give an equational base for the variety generated by $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W in terms of identities satisfied in $$\mathcal {V}$$ V and $$\mathcal {W}$$ W . Then the main result provides a new sufficient condition for $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W to be a variety: If $$\mathcal {W}$$ W is an idempotent variety and there are terms f(x, y) and g(x, y) such that $$\mathcal {W}$$ W satisfies the identity $$f(x,y) = g(x,y)$$ f ( x , y ) = g ( x , y ) and $$\mathcal {V}$$ V satisfies the identities $$f(x,y) = x$$ f ( x , y ) = x and $$g(x,y) = y$$ g ( x , y ) = y , then $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W is a variety. We provide a number of examples and applications of this result.