The matrix taxonomy of left exact categories
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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.
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