Unitary magma actions

TL;DR

This paper introduces a new concept of action for unitary magmas, enabling the classification of split extensions and expanding understanding of their algebraic structure beyond traditional associative contexts.

Contribution

It presents a novel action concept for unitary magmas and explores split extensions where the middle object isn't necessarily a direct product, extending prior work in algebraic structures.

Findings

Classification of split extensions in unitary magmas

Identification of non-bijective middle objects in extensions

Connections to non-associative algebraic structures

Abstract

We introduce a novel concept of action for unitary magmas, facilitating the classification of various split extensions within this algebraic structure. Our method expands upon the recent study of split extensions and semidirect products of unitary magmas conducted by Gran, Janelidze, and Sobral. Building on their research, we explore split extensions in which the middle object does not necessarily maintain a bijective correspondence with the Cartesian product of its end objects. Although this phenomenon is not observed in groups or any associative semiabelian variety of universal algebra, it shares similarities with instances found in monoids through weakly Schreier extensions and certain exotic non-associative algebras, such as semi-left-loops. Our work seeks to contribute to the comprehension of split extensions in unitary magmas and may offer valuable insights for potential abstractions of categorical properties in more general contexts.