Combinatorial Semigroup Bialgebras

TL;DR

This paper introduces a general framework for combinatorial bialgebras based on commutative semigroups, unifying shuffle and stuffle algebras, and explores dual bases and Schützenberger's factorizations.

Contribution

It presents a new unified framework for combinatorial bialgebras using commutative semigroups, encompassing shuffle and stuffle algebras, and applies it to dual bases and factorizations.

Findings

Unified framework for combinatorial bialgebras

Construction of dual bases in this setting

Application to Schützenberger's factorizations

Abstract

This paper is devoted to the presentation of combinatorial bialgebras whose coproduct is defined with the help of a commutative semigroup. We consider this setting in order to give a general framework which admits as special cases the shuffle and stuffle algebras. We also consider the problem of constructing pairs of bases in duality and present Sch\"utzenberger's factorisations as an application.