On graded $\Omega$-groups

Emil Ilić-Georgijević

In this paper we study the notion of a graded $\Omega$-group $(X,+,\Omega)$, 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 $\Omega$ and the grading of $(X,+)$ induce operations (generally partial) on the grading set. We prove that graded $\Omega$-groups in Krasner's sense are determined up to isomorphism by their homogeneous parts, which, with respect to induced operations, represent partial structures called $\Omega$-\emph{homogroupoids}, thus narrowing down the theory of graded $\Omega$-groups to the theory of $\Omega$-homogroupoids. This approach already proved to be useful in questions regarding A. V. Kelarev's $S$-graded rings inducing $0$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 $\Omega$-homogroupoids thus extending results of A. V. Mikhalev, I. N. Balaba and S. A. Pikhtilkov in a natural way. We study some classes of $\Omega$-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.