Approches catégoriques aux variétés soustractives

Details

Supervisors

Gran, Marino

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques, à finalité didactique

Abstract

enThe main objective of this thesis is to familiarize ourselves with subtractive varieties and categories in various mathematical contexts. To establish the framework in which we are working, we first introduce concepts from universal algebra, and then demonstrate the exactness of varieties of algebras. Within chapter three, we define the notion of a subtractive variety as proposed by A. Ursini, as well as its categorical counterpart from an article by Z. Janelidze. We subsequently establish certain characterizations of these objects, notably involving the permutation of congruences, a projective covering, and free algebras. Lastly, the final chapter explores the connection between subtractive categories and some homological lemmas. We demonstrate that subtractivity is equivalent to the lower nine lemma as well as the short five lemma for ideals.