On Some Remarkable Operads Constructed from Baxter Operators 1

Language theory, symbolic dynamics, modelisation of viral insertion into the genetic code of a host cell motivate the introduction of new types of bialgebras, whose coalgebra parts are not necessarily coassociative. Similarly, certain algebraic descriptions of combinatorial objects such as weighted directed graphs through a coalgebraic formulation often require at least two coalgebras whose coproducts verify entanglement conditions. All these structures generate special types of associative algebras (or binary quadratic and non-symmetric operads) obtained from commuting Baxter operators. One of the aim of this article is to study what type of associative algebras appear when such or such coalgebraic structures are used to describe for instance combinatorial objects. Examples from symbolic dynamics, weighted directed graphs and matrix theory are given.