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.

中文翻译：

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左精确类别的矩阵分类法

本文涉及根据左精确类别的“矩阵属性”对它们进行分类的问题，左精确类别是由整数矩阵表示的特定类别理论属性。我们获得了一种算法，用于确定这些矩阵属性的合取是否遵循另一个。该算法的计算机实现允许人们窥视所有“矩阵类”的波塞特的复杂结构，即由这些矩阵属性确定的所有左精确类别集合的波塞特。该姿势的元素包括Mal'tsev类别，多数类别，（左精确）算术类别的集合，以及通过熟悉的“语法精炼”过程获得的通用代数各种类别的左精确扩展。玛尔