Embedding theorems for Janelidze's matrix conditions

Abstract

As a first objective, we characterise those essentially algebraic categories which satisfy properties like being unital, strongly unital, n-permutable, subtractive or protomodular. For each such property, we obtain a Mal'tsev condition as an equivalent condition. Using the language of Janelidze matrix conditions, we treat many of these properties together.As a second objective, using these characterisations, we prove some embedding theorems for those properties in a regular context in the same style as we did in the companion paper [23]. Concrete examples of how to use these embedding theorems are given. Finally, to extend those embedding theorems to the exact context, we show that these properties are stable under the exact completion of a regular category.