Categorical Methods in Non Abelian Algebra

Publications

Further remarks on the "Smith is Huq" condition

Publication . Martins-Ferreira, Nelson; Van der Linden, Tim

We compare the Smith is Huq condition (SH) with three commutator conditions in semi-abelian categories: first an apparently weaker condition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be stronger than (SH). This leads to the even stronger condition that weighted commutators in the sense of Gran, Janelidze and Ursini are independent of the chosen weight, which is known to be false for groups but turns out to be true in any two-nilpotent semi-abelian category.

2015-08Journal article

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An observation on n-permutability

Publication . Martins-Ferreira, Nelson; Rodelo, Diana; Van der Linden, Tim

We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these conditions hold when the category is n-permutable for some n.

2014-05Journal article

Open access

Monoids and pointed S-protomodular categories

Publication . Bourn, Dominique; Martins-Ferreira, Nelson; Montoli, Andrea; Sobral, Manuela

We investigate the notion of pointed S-protomodular category, with respect to a suitable class S of points, and we prove that these categories satisfy, relatively to the class S, many partial aspects of the properties of Mal’tsev and protomodular categories, like the split short five lemma for S-split exact sequences, or the fact that a reflexive S-relation is transitive. The main examples of S-protomodular categories are the category of monoids and, more generally, any category of monoids with operations, where the class S is the class of Schreier points.

2016-04-11Journal article

Open access

On Algebraic and More General Categories Whose Split Epimorphisms Have Underlying Product Projections

Publication . Gray, J. R. A.; Martins-Ferreira, Nelson

We characterize those varieties of universal algebras where every split epimorphism considered as a map of sets is a product projection. In addition we obtain new characterizations of semi-abelian, protomodular, unital and subtractive varieties as well as varieties of right Ω-loops and biternary systems.

2015-06Journal article

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What is an ideal a zero-class of?

Publication . Martins-Ferreira, Nelson; Montoli, A.; Ursini, A.; Linden, T. Van der

We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the Smith is Huq condition: two surjective left split relations commute as soon as their zero-classes commute.

2017-03Journal article

Open access