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The natural generalization of this notion for a reductive algebraic group $G/k$ is an \"$F$-zip with $G$-structure\", a so-called $G$-zip introduced by R. Pink, T. Wedhorn, P. Ziegler. A $G$-zip $I$ over $S$ yields the stratification of the base scheme in loci, where $I$ has locally a constant isomorphism class for the fppf topology. 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