We start with definitions of the general notions of the theory of $\Bbb Z_{ 2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{ 2}$-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers \cite{ VO,OV} to representation theory of symmetric groups. The main example is the classical - theory of the projective representations of symmetric groups.
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