Hypersubtraction and semi-direct product
Dominique Bourn
TL;DR
This work presents an extrinsic perspective on semi-direct products by leveraging the forgetful functor $V$ to characterize split epimorphisms through a semi-direct index, then strengthening the analysis with two algebraic frameworks: hyper-Słomínski settings and hypersubtractions. It systematically builds from ILO settings and their classifications (Class A/B) to structural recalls on protomodular and Mal'tsev categories, and finally to the notions of semi-direct index and hyperindex for left-exact functors, proving that semi-direct indices force protomodularity under conservativity and showing how these indices factor through specialized categories. The results unify several internal algebraic structures (prequandles, quandles, Alexander constructions) within protomodular/Mal'tsev contexts and clarify how various categorical properties control the existence and nature of semi-direct/hyperindexes. The framework has implications for understanding intrinsic versus extrinsic characterizations of extensions and actions in categorical algebra, with concrete illustrations in groups, rings, and quandles via forgetful and Alexander-type constructions.
Abstract
In this article, we introduce an extrinsic approach to the notion of semi-direct product, an intrinsic one (namely inside the category Gp of group itself) having been already done elsewhere. This will led us to focus our attention on two algebraic structures (hypersubtraction and hyper-Slominski settings) which will allow us to characterize this extrinsic explicitation.