TY - JOUR
AU - Brice Réné Amougou Mbarga
PY - 2022/06/20
Y2 - 2022/08/07
TI - Homomorphic Relations and Goursat Lemma
JF - Earthline Journal of Mathematical Sciences
JA - EJMS
VL - 10
IS - 1
SE - Articles
DO - 10.34198/ejms.10122.169181
UR - https://earthlinepublishers.com/index.php/ejms/article/view/537
AB - 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.
ER -