  The notion of a weakly Mal'tsev category, as it was introduced in 2008 by the
third author, is a generalization of the classical notion of a Mal'tsev
category. It is well-known that a variety of universal algebras is a Mal'tsev
category if and only if its theory admits a Mal'tsev term. In the main theorem
of this paper, we prove a syntactic characterization of the varieties that are
weakly Mal'tsev categories. We apply our result to the variety of distributive
lattices which was known to be a weakly Mal'tsev category before. By a result
of Z. Janelidze and the third author, a finitely complete category is weakly
Mal'tsev if and only if any internal strong reflexive relation is an
equivalence relation. In the last part of this paper, we give a syntactic
characterization of those varieties in which any regular reflexive relation is
an equivalence relation.
