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Algebraic structure of well-known (standard) relativistic supersymmetries (SUSY) is associated with Z_2-graded Lie superalgebras. In this talk I am about to discuss alternative variants of relativistic SUSY based on Z_2xZ_2- graded Lie superalgebras. In particular, it will be demonstrated that anti-de-Sitter symmetry admits of only Z_2-graded superextension whereas in the case of de-Sitter symmetry there exists only Z_2xZ_2-graded superextension.