Groupoid Factorizations in the Semigroup of Binary Systems

TL;DR

This paper introduces new factorization methods for binary systems (groupoids) within the semigroup of all groupoids on a set, revealing unique decompositions based on properties like non-idempotency and orientation.

Contribution

It presents novel factorization techniques for groupoids in the semigroup of binary systems, including applications to various algebraic structures.

Findings

Strong non-idempotent groupoids can be factorized into similar- and signature-derived factors.

Groupoids with the orientation property can be expressed as products of orient- and skew- factors.

The methods have applications to multiple algebraic systems such as B/BCH/BCI/BCK/BH/BI/d-algebras.

Abstract

Let $(X,\bullet )$ be a groupoid (binary algebra) and $Bin(X\dot{)}$ denote the collection of all groupoids defined on $X$. We introduce two methods of factorization for this binary system under the binary groupoid product \textquotedblleft $\diamond $\textquotedblright\ in the semigroup $\left( Bin\left( X\right) ,\diamond \right) $. We conclude that a strong non-idempotent groupoid can be represented as a product of its \textit{% similar-} and \textit{signature-} derived factors. Moreover, we show that a groupoid with the orientation property is a product of its \textit{orient-} and \textit{skew-} factors. These unique factorizations can be useful for various applications in other areas of study. Application to algebras such as $B/BCH/BCI/BCK/BH/BI/d$-algebra are widely given throughout this paper.