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Author’s summary: We define the notion of “graded algebra with symmetries”. This notion is a generalization of the extended differential forms. We prove that for a graded algebra with symmetries $$T$$, we associate a subalgebra $$\Omega^*$$ which generalizes the noncommutative differential forms. Using this algebra $$\Omega^*$$, we can define the Hochschild and cyclic homologies, cup $$i$$-products and the Steenrod squares.