x-divisor pseudographs of a commutative groupoid

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 A (not necessarily with zero), a system of pseudographs is obtained such that the vertices of a pseudograph are the elements of A and vertices a and b are adjacent if and only if ab = x. These systems are completely characterized as being partitions of complete pseudographs Kn whose parts are indexed by the vertices of Kn. Furthermore, morphisms are defined in the class of all such systems of pseudographs making it (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.