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A chain bicomplex for A -algebras, which generalizes the Tsygan chain bicomplex in the theory of cyclic homology of associative algebras, is constructed by using the techniques of differential modules with ∞-simplicial faces and D -differential modules. For homotopy unital A -algebras, an exact sequence generalizing the Connes-Tsygan exact sequence for unital associative algebras is obtained. [ABSTRACT FROM AUTHOR]

The homotopy theory of ∞-simplicial coalgebras is developed; in terms of this theory, an additional structure on the tensor bigraded coalgebra of a graded module is described such that endowing the coalgebra with this structure is equivalent to endowing the given graded module with the structure of a homotopy unital A-algebra. [ABSTRACT FROM AUTHOR]

On the basis of the colored version of Koszul duality, the notion of a differential module with ∞-simplicial faces is introduced. By using the homotopy technique of differential Lie modules over colored coalgebras, the homotopy invariance of the structure of a differential module with ∞-simplicial faces is proved. A relationship between differential modules with ∞-simplicial faces and A-algebras is described. The notions of the chain realization of a differential module with ∞-simplicial faces and the tensor product of differential modules with ∞-simplicial faces are introduced. It is shown that the chain realization of a tensor differential module with ∞-simplicial faces constructed from an A-algebra and the B-construction over this A-algebra are isomorphic differential coalgebras. [ABSTRACT FROM AUTHOR]

It is proved in the paper that the multiplication operation on an arbitrary E-algebra can be extended to an E-algebra A-morphism. As a corollary, it is proved that every May algebra defined by an E-algebra is a Cartan object in the category of May algebras. [ABSTRACT FROM AUTHOR]

For a given ϕ-function ϕ(u), a condition on a ϕ-function ψ(u) is found such that it is necessary and sufficient for the following to hold: if f(x) → f(x) and ∥ f( x)∥⩽ M ( n=1, 2, ...) where M>0 is an absolute constant, then ∥ f( x)− f( x)∥→0(n→∞). An analogous condition for convergence in Orlicz spaces is obtained as a corollary. [ABSTRACT FROM AUTHOR]