We investigate $3$-permutability, in the sense of universal algebra, in an abstract categorical setting which unifies the pointed and the non-pointed contexts in categorical algebra. This leads to a unified treatment of regular subtractive categories and of regular Goursat categories, as well as of $E$-subtractive varieties (where $E$ is the set of constants in a variety) recently introduced by the fourth author. As an application, we show that ``ideals'' coincide with ``clots'' in any regular subtractive category, which can be considered as a pointed analogue of a known result for regular Goursat categories.
Keywords: Ideal of null morphisms, Goursat category, $3$-permutable variety, subtractive category, subtractive variety, ideal, clot, ideal determined category, good theory of ideals
2010 MSC: 18D99, 18C99, 18C05, 08B05
Theory and Applications of Categories, Vol. 27, 2012, No. 6, pp 80-96.