Unitary magma actions

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.