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.
Keywords: weakly Mal'tsev category, weakly Mal'tsev variety, Mal'tsev condition, syntactic characterization, strong relation, pullback injection
2020 MSC: 08B05, 18E13 (primary); 18C10, 06B20, 06D99, 18D40 (secondary)
Theory and Applications of Categories, Vol. 42, 2024, No. 12, pp 314-353.
Published 2024-08-26.
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