  This work is devoted to study new bialgebra structures related to
2-associative algebras. A 2-associative algebra is a vector space equipped with
two associative multiplications. We discuss the notions of 2-associative
bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed
by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem,
the second was suggested by Loday and the third is a variation of the second
one. The main results of this paper are the construction of 2-associative
bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative
algebra and the classification of these structures in low dimensions.
