Overview
This study introduces a mathematical construct termed a "restriction semigroupoid" ($S$), designed to integrate the concepts of restriction semigroups and restriction categories within a unified algebraic framework. The research establishes a representation theorem for these semigroupoids, indicating that any restriction semigroupoid ($S$) is embeddable into a determined category of partial maps. Furthermore, the study details the construction of the Szendrei expansion, denoted as $Sz(S)$, for any given restriction semigroupoid $S$. A principal result concerns the factorization of premorphisms: it is shown that every premorphism relating two restriction semigroupoids, $S$ and $T$, possesses a unique factorization through a morphism between $Sz(S)$ and $T$.
Research Context
The concept of a restriction semigroupoid extends and unifies existing mathematical structures. Prior to this research, restriction semigroups and restriction categories were treated as distinct, albeit related, notions. The proposed restriction semigroupoid concept provides a common algebraic structure that encompasses properties and behaviors observed in both of these established mathematical entities. This unification aims to provide a more generalized framework for studying partial functions and related algebraic structures.
Approach
The research methodology involves defining the axiomatic properties of a restriction semigroupoid. Following this foundational definition, the study proceeds to construct a representation for these structures. This representation takes the form of an embedding into a category of partial maps, specifically a "determined category of partial maps." The construction of the Szendrei expansion $Sz(S)$ for a restriction semigroupoid $S$ is a central step in the theoretical development. This expansion is then utilized to prove a factorization theorem concerning premorphisms. The proof demonstrates that any premorphism from $S$ to $T$ can be uniquely decomposed into a specific path involving $Sz(S)$.
Findings
- The introduction of the concept of a restriction semigroupoid $S$ unifies the notions of restriction semigroups and restriction categories into a single, comprehensive structure.
- A representation theorem demonstrates that every restriction semigroupoid can be embedded into a determined category of partial maps. This embedding establishes a concrete realization for abstract restriction semigroupoids within a more familiar categorical setting.
- The Szendrei expansion $Sz(S)$ of a restriction semigroupoid $S$ has been successfully constructed.
- It has been established that each premorphism existing between two restriction semigroupoids, $S$ and $T$, can be uniquely factorized by a specific morphism positioned between the Szendrei expansion $Sz(S)$ and $T$. This factorization property provides a fundamental structural insight into the relationships between these algebraic objects.