A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R→A 2 in it has this closedness property.
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Partially supported by South African National Research Foundation, and Georgian National Science Foundation (GNSF/ST06/3-004).
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Janelidze, Z. Closedness Properties of Internal Relations III: Pointed Protomodular Categories. Appl Categor Struct 15, 325–338 (2007). https://doi.org/10.1007/s10485-007-9072-9