Semi-biproducts of monoids

TL;DR

This paper introduces semi-biproducts of monoids, a new concept generalizing biproducts by relaxing commutativity and homomorphism requirements, and shows their equivalence to pseudo-actions, which include pre-actions, factor systems, and correction systems.

Contribution

It defines semi-biproducts of monoids and establishes their equivalence to pseudo-actions, expanding the understanding of monoid structures beyond groups.

Findings

Semi-biproducts generalize biproducts by relaxing commutativity.

Pseudo-actions include pre-actions, factor systems, and correction systems.

In groups, correction systems are trivial, explaining the novelty of pseudo-actions.

Abstract

It is shown that the category of \emph{semi-biproducts} of monoids is equivalent to the category of \emph{pseudo-actions}. A semi-biproduct of monoids is a new notion, obtained through generalizing a biproduct of commutative monoids. By dropping commutativity and requiring some of the homomorphisms in the biproduct diagram to be merely identity-preserving maps, we obtain a semi-biproduct. A pseudo-action is a new notion as well. It consists of three ingredients: a pre-action, a factor system and a correction system. In the category of groups all correction systems are trivial. This is perhaps the reason why this notion, to the author's best knowledge, has never been considered before.