Symmetry of regular diamonds, the Goursat property, and subtractivity

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. © Marino Gran, Zurab Janelidze, Diana Rodelo and Aldo Ursini, 2012.