The $x$-divisor pseudographs of a commutative groupoid

Authors : John D LAGRANGE

Pages : 62-77

Doi:10.24330/ieja.325926

View : 11 | Download : 9

Publication Date : 2017-07-11

Article Type : Research Paper

Abstract :The notion of a zero-divisor graph is considered for commutative groupoids with zero. Moufang groupoids and certain medial groupoids with zero are shown to have connected zero-divisor graphs of diameters at most four and three, respectively. As $x$ ranges over the elements of a commutative groupoid $\mB$ insert ignore into journalissuearticles values(not necessarily with zero);, a system of pseudographs is obtained such that the vertices of a pseudograph are the elements of $\mB$ and vertices $a$ and $b$ are adjacent if and only if $ab=x$. These systems are completely characterized as being partitions of complete pseudographs $\overline{K}_{n}$ whose parts are indexed by the vertices of $\overline{K}_{n}$. Furthermore, morphisms are defined in the class of all such systems of pseudographs making it insert ignore into journalissuearticles values(categorically); isomorphic to the category of commutative groupoids, thereby combinatorializing the theory of commutative groupoids. Also, concepts of ``congruence` and ``direct product` that are compatible with those in the category of commutative groupoids are established for these systems of pseudographs.  
Keywords : Groupoid, zero divisor graph