Semi-biproducts in monoids

TL;DR

This paper introduces semi-biproducts in monoids, showing their equivalence to pseudo-actions, and explores their relation to Schreier extensions, generalizing concepts from groups and commutative monoids.

Contribution

It establishes a categorical equivalence between semi-biproducts and pseudo-actions in monoids, introducing correction systems and generalizing semi-direct products.

Findings

Semi-biproducts generalize semi-direct products and biproducts.

Schreier extensions are instances of semi-biproducts with trivial correction systems.

A new structure of map-transformations is developed from a 2-cell category structure.

Abstract

It is shown that the category of semi-biproducts in monoids is equivalent to a category of pseudo-actions. A semi-biproduct in monoids is at the same time a generalization of a semi-direct product in groups and a biproduct in commutative monoids. Every Schreier extension of monoids can be seen as an instance of a semi-biproduct; namely a semi-biproduct whose associated pseudo-action has a trivial correction system. A correction system is a new ingredient that must be inserted in order to obtain a pseudo-action out of a pre-action and a factor system. In groups, every correction system is trivial. Hence, semi-biproducts there are the same as semi-direct products with a factor system, which are nothing but group extensions. An attempt to establish a general context in which to define semi-biproducts is made. As a result, a new structure of map-transformations is obtained from a category with a 2-cell structure. Examples and first basic properties are briefly explored.