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We introduce the notion of a graded $\Omega$-group, but graded in the sense of M.~Krasner, that is, we do not assume neither the commutativity, nor the associativity, nor the existence of the neutral element in the set of grades. 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 general structures of their own called $\Omega$-\emph{homogroupoids}, thus narrowing down the theory of graded $\Omega$-groups to the theory of $\Omega$-homogroupoids. As an application, we 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.

Graded $\Omega$-group; $\Omega$-homogroupoid