Homomorphic Relations and Goursat Lemma

  • Brice Réné Amougou Mbarga Laboratory of Algebra, Geometry and Application, University of Yaoundé I, P.O. Box : 812, Yaoundé, Cameroon
Keywords: Goursat, variety, relation, homomorphic, Mal'cev


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.


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How to Cite
Mbarga, B. R. A. (2022). Homomorphic Relations and Goursat Lemma. Earthline Journal of Mathematical Sciences, 10(1), 169-181. https://doi.org/10.34198/ejms.10122.169181