On some remarkable operads constructed from Baxter operators

TL;DR

This paper explores new operads derived from Baxter operators, motivated by applications in language theory, genetics, and combinatorics, focusing on non-coassociative bialgebras and their algebraic structures.

Contribution

It introduces novel types of operads constructed from Baxter operators, expanding the understanding of algebraic structures associated with non-coassociative coalgebras.

Findings

Identification of algebraic structures from non-coassociative coalgebras

Construction of operads related to combinatorial objects

Analysis of associative algebras arising from these operads

Abstract

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. One of the aim of this article is to study what type of (associative) algebras (and thus binary quadratic and non-symmetric operads) appear when such or such coalgebraic structures are used to decribe for instance combinatorial objects such as weighted directed graphs, trees, substitutions and so forth.