In universal algebra, it is well known that varieties admitting a majority term admit several Mal'tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for regular categories. We prove a categorical version of Bergman's Double-projection Theorem: a regular category is a majority category if and only if every subobject S of a finite product A 1 × A 2 × · · · × A n is uniquely determined by its twofold projections. We also establish a categorical counterpart of the Pairwise Chinese Remainder Theorem for algebras, and characterize regular majority categories by the classical congruence equation α ∩ (β • γ) = (α ∩ β) • (α ∩ γ) due to A.F. Pixley.

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For a given variety V of algebras, we define a class relation to be a binary relation R ⊆ S 2 which is of the form R = S 2 ∩K for some congruence class K on A 2 , where A is an algebra in V such that S ⊆ A. In this paper we study the following property of V: every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal'tsev condition on the variety and in a suitable sense, it is a join of Chajda's egg-box property as well as Duda's direct decomposability of congruence classes.

For a variety V, it has been recently shown that binary products commute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points π : Pt(C) → C, if and only if Gumm's shifting lemma holds on pullbacks in V. In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain categorical anticommutativity condition. In particular, we show that this anticommutativity and its local version are Mal'tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corollary, every locally anticommutative variety V has directly decomposable congruence classes in the sense of Duda, and the converse holds if V is idempotent.

This paper is concerned with the problem of classifying left exact categories according to their 'matrix properties' -a particular category-theoretic property represented by integer matrices. We obtain an algorithm for deciding whether a conjunction of these matrix properties follows from another. Computer implementation of this algorithm allows one to peer into the complex structure of the poset of all 'matrix classes', i.e., the poset of all collections of left exact categories determined by these matrix properties. Among elements of this poset are the collections of Mal'tsev categories, majority categories, (left exact) arithmetical categories, as well as left exact extensions of various classes of varieties of universal algebras, obtained through a process of 'syntactical refinement' of familiar Mal'tsev conditions studied in universal algebra.