In [7], it has been recently defined a new graph  on monogenic semigroups  (with zero) having elements  The vertices are the nonzero elements  and, for  any two distinct vertices  and  are adjacent if  in  As a continuing study, in [3], it has been investigated some well known parameters (diameter, girth, radius, etc.) over  to the lexicographic product.

In the light of above references, our main aim in this paper is to extend these studies over  to the disjunctive product. In detail, we will investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the disjunctive product of any two graphs  and

monogenic semigroup, clique number, chromatic number, domination number, disjunctive product, co-normal product.