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In this work we generalise Jónsson's theorem for congruence distributive varieties of universal algebras. The linear Mal'tsev condition extracted from the ternary Jónsson terms give rise to matrix conditions Jn, n ≥ 1. We characterise regular categories C which satisfy (the matrix condition) Jn, for some n ≥ 1, through properties involving equivalence and reflexive relations on a same object in C. These properties on relations then allow us to show that, when C is an n-permutable category, C satisfies Jm, for some m ≥ 1, if and only if C is equivalence distributive. It turns out that regular categories C that satisfy Jn are such that the Trapezoid Lemma holds in C; consequently, every such C is factor permutable. This is a joint work with Michael Hoefnagel.
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