On graded Ω-groups

In this paper we study the notion of a graded ?-group (X;+ ?), but graded in the sense of M. Krasner, i.e., we impose nothing on the grading set except that it is nonempty, since operations of and the grading of (X,+) induce operations (generally partial) on the grading set. We prove that graded ?-groups in Krasner?s sense are determined up to isomorphism by their homogeneous parts, which, with respect to induced operations, represent partial structures called ?-homogroupoids, thus narrowing down the theory of graded -groups to the theory of ?-homogroupoids. This approach already proved to be useful in questions regarding A. V. Kelarev?s S-graded rings inducing S; where S is a partial cancellative groupoid. Particularly, in this paper we prove that the homogeneous subring of a Jacobson S-graded ring inducing S is Jacobson under certain assumptions. We also discuss the theory of prime radicals for ?-homogroupoids thus extending results of A. V. Mikhalev, I. N. Balaba and S. A. Pikhtilkov in a natural way. We study some classes of ?-homogroupoids for which the lower and upper weakly solvable radicals coincide and also, study the question of the homogeneity of the prime radical of a graded ring.