Semibiproducts are introduced as a generalization of biproducts. Classically, biproducts are defined in the context of additive categories ([1], p. 250). The appropriate context in which the notion of semibiproduct makes sense is the one of a concrete category over sets with maps and morphisms such that maps (and morphisms) can be added. The most natural example of this is the concrete category of magmas with underlying functions as maps. Its objects are magma structures (binary operations with no conditions), its morphisms are the structure preserving functions (magma homomorphisms), whereas its maps are all functions between the underlying sets of magma structures. Maps can be added component-wise using the underlying magma structure. The new notion of semibiproduct generalizes the classical notion of biproduct in the sense that some arrows in the biproduct diagram may be maps rather than morphisms but the conditions they satisfy are the same (it has been considered in [1], p. 263, as diagram (5.2) in the context of relative Abelian categories). Its purpose is to serve as a common setting in which group extensions and group split extensions can be treated in the same say. As it is well known, group split extensions are the same as semi-direct products whereas group extensions are equivalence classes of actions with a representative factor set. However, both concepts are instances of a semibiproduct which in the case of groups is characterized as a pseudo-action with a compatible factor system. If the extension splits then the factor system is trivial. It is shown that this approach can be generalized into the context of monoids where a new ingredient, which is invisible in groups, enters into the play [2]. The case of semigroups will be analysed as well.

References:

[1] Saunders MacLane, Homology, Berlin, 1963

[2] N. Martins-Ferreira, Semi-biproducts of monoids, arxiv.org/abs/2109.06278