Browsing by Author "Janelidze, Zurab"
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- 3 x 3 lemma for star-exact sequencesPublication . Gran, Marino; Janelidze, Zurab; Rodelo, DianaA regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding 3 x 3 lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with "exact forks" and then, the corresponding 3 x 3 lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such 3 x 3 lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors.
- On difunctionality of class relationsPublication . Hoefnagel, Michael; Janelidze, Zurab; Rodelo, DianaFor a given variety V of algebras, we define a class relation to be a binary relation R subset of S(2)which is of the form R = S-2 boolean AND K for some congruence class K on A(2), where A is an algebra in V such that S subset of A. In this paper we study the following property of V : every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal'tsev condition on the variety and in a suitable sense, it is a join of Chajda's egg-box property as well as Duda's direct decomposability of congruence classes.