Escola Superior de Tecnologia e Gestão, Centro para o Desenvolvimento Rápido e Sustentado do Produto, Instituto Politécnico de Leiria.
he classical notion of biproduct introduced by Mac Lane  for additive categories is generalized into a frame-work which permits to consider it in the category of semigroups and semigroup homomorphisms. The resulting concept is called semi-biproduct because two of the four arrows in the classical diagram of a biproduct may fail to be homomorphisms. In the particular case of groups it is a generalization of semi-direct product and corresponds precisely to group extensions (not necessarily split). Long ago, Schreier extensions were introduced in monoids to mimic the behaviour of group extensions. As shown recently , semi-biproducts of monoids generalize Schreier extensions and correspond to a certain kind of pseudo-actions with a factor system (as observed in groups) together with a new ingredient which is invisible in groups. This new ingredient is called a correction system and it is responsible for the strange phenomenon that can only be observed for monoids: an extension of X by B may not be in bijection with the cartesian product X times B. In this talk we will see how these results can be generalized to the case of semigroups. Although null objects are no longer present there is still some similarities that can be compared with the results obtained for groups and monoids.
 S. Mac Lane, Categories for the Working Mathematician, 2ed, Graduate Texts in Mathematics 5, Springer, 1998.
 N. Martins-Ferreira, Semi-biproducts of monoids, Preprint 2021, arxiv.org/abs/2109.06278.