A new notion of a (pointed) semibiproduct is introduced, which, in the case of groups amounts to an extension equipped with a set-theoretical section. When the section is a group homomorphism then a pointed semibiproduct is the same as a group split extension. The main result of the paper is a characterization of pointed semibiproducts of monoids using a structure that is a generalization of the action that is used in the definition of a semidirect product of groups.
Keywords: Semibiproduct, biproduct, semidirect product of groups and monoids, pointed semibiproduct, semibiproduct extension, pointed monoid action system, Schreier extension
2020 MSC: 18G50, 20M10, 20M32
Theory and Applications of Categories, Vol. 39, 2023, No. 6, pp 172-185.