Anticommutativity and n-schemes

2022

EJMS

Over the past years various authors have investigated the famous elementary result in group theory called Goursat's lemma for characterizing the subgroups of the direct product $A\times B$ of two groups $A,B$. Given a homomorphic relation $\rho = (R,A,B)$ where $A$ and $B$ are groups and $R$ is a subgroup of $A\times B.$ What can one say about the structure of $\rho$. In 1950 Riguet proved a theorem that allows us to obtain a characterization of $\rho$ induces by examining the sections of the direct factors. The purpose of this paper is two-fold. A first and more concrete aim is to provide a containment relation property between homomorphic relation. Indeed if $\rho,\sigma$ are homomorphic relations, we provide necessary and sufficient conditions for $\sigma\leq\rho$. A second and more abstract aim is to introduce a generalization of some notions in homological algebra. We define the concepts of $\theta$-exact. We also obtain some interesting results. We use these results to find a generalization of Lambek Lemma.

Homomorphic Relations and Goursat Lemma

2022

EJMS

Subgroups Inclusions in 3-Factors Direct Product

2022

EJMS

The aim of this paper is to use a correspondent theorem to characterize containment of a degenerate $2$-factor injective subdirect products. Namely, let $\Omega,\Lambda$ be degenerate 2-factor injective subdirect products of $ M_{1}\times M_{2}\times M_{3}$, we provide necessary and sufficient conditions for $\Omega\leq \Lambda.$ Based on a decomposition of the inclusion order on the subgroup lattice of a subdirect product as a relation product of three smaller partial orders, we induce a matrix product of three incidence matrices.

Geometrical Methods in Goursat Categories

2021

EJMS

Self Cite

The main aim of the paper is to show that the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, hold in any regular Mal'tsev categories. We prove that Mal'tsev categories may be characterized through variations of the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, that is classically expressed in terms of four congruences R, S1, S2 and T, and characterizes congruence modular varieties. The proof of this result in a varietal context may be obtained exclusively through the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3. This was shown by H.P. Gumm in Geometric Methods in Congruence Modular Algebras. We prove that for any 2n+1-permutable category $\mathcal{E}$, the category Equiv$(\mathcal{E})$ of equivalence relations in $\mathcal{E}$ is also a 2n+1-permutable category.