In this article, we introduce an extrinsic approach to the notion of semi-direct product, an intrinsic one having been already done in [Bourn-Janelidze, 2003] and [Borceux-Janelidze-Kelly, 2005-1]. This led us to focus our attention on two algebraic structures, hyper-Slomínski setting and hypersubtraction which produce pointed protomodular categories, and will allow us to characterize this extrinsic approach.
Keywords: Group; Semi-direct product; Subtraction, Split epimorphism; Mal'tsev, protomodular and additive category; Hypersubtraction; S\l om\'inski setting; Quandle and prequandle; Latin square; Digroup; Skew brace; Commutation of structures.
2020 MSC: 08A05, 18E13, 20E34, 20J15.
Theory and Applications of Categories, Vol. 45, 2026, No. 35, pp 1407-1425.
Published 2026-06-03.
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