On semibiproducts of magmas and semigroups

TL;DR

This paper extends the concept of semibiproducts from monoids and semigroups to magmas, revealing new insights into their classification and the role of magma-actions in this broader context.

Contribution

It introduces the study of semibiproducts in magmas and analyzes the relationship between representable and associative magma-actions.

Findings

Semibiproducts in magmas are more general than in semigroups.

Representable magma-actions classify a subclass of semibiproducts.

Further analysis is needed to fully understand semibiproducts in magmas.

Abstract

A generalization to the categorical notion of biproduct, called semibiproduct, which in the case of groups covers classical semidirect products, has recently been analysed in the category of monoids with surprising results in the classification of weakly Schreier extensions. The purpose of this paper is to extend the study of semibiproducts to the category of semigroups. However, it is observed that a further analysis into the category of magmas is required in attaining a full comprehension on the subject. Indeed, although there is a subclass of magma-actions, that we call representable, which classify all semibiproducts of magmas whose behaviour is similar to semigroups, it is nevertheless more general than the subclass of associative magma-actions.