Pointed semibiproducts of monoids

TL;DR

This paper introduces semibiproducts of monoids as a unifying concept for various monoid extensions, establishing categorical equivalences and exploring their properties and classifications.

Contribution

It defines semibiproducts of monoids, generalizes existing extension theories, and establishes a new categorical equivalence with monoid action systems.

Findings

All group extensions are semibiproduct extensions.

Categorical equivalence between pointed semibiproducts and monoid action systems.

Complete list of 14 semibiproduct extensions of 2-element monoids.

Abstract

Semibiproducts of monoids are introduced here as a common generalization to biproducts (of abelian groups) and to semidirect products (of groups) for exploring a wide class of monoid extensions. More generally, abstract semibiproducts exist in any concrete category over sets in which map addition is meaningful thus reinterpreting Mac Lane's relative biproducts. In the pointed case they give rise to a special class of extensions called semibiproduct extensions. Not all monoid extensions are semibiproduct extensions but all group extensions are. A categorical equivalence is established between the category of pointed semibiproducts of monoids and the category of pointed monoid action systems, a new category of actions that emerges from the equivalence. The main difference to classical extension theory is that semibiproduct extensions are treated in the same way as split extensions, even though the section map may fail to be a homomorphism. A list with all the 14 semibiproduct extensions of 2-element monoids is provided.