Compatibility axioms for left-regular bands to construct Hopf algebras

TL;DR

This paper introduces compatibility axioms for left-regular bands to extend existing algebraic structures into Hopf algebras, enriching the theoretical framework for algebraic combinatorics.

Contribution

It adds compatibility axioms to existing algebraic axioms, enabling the construction of Hopf algebras from left-regular bands.

Findings

Established new compatibility axioms for left-regular bands.

Extended algebraic structures to Hopf algebras.

Provided a framework for constructing Hopf algebras from combinatorial objects.

Abstract

Aguiar and Mahajan provided several coalgebra axioms and algebra axioms for a family of left-regular bands to construct a commutative diagram of algebras and coalgebras. In this paper, we will add compatibility axioms to make it a diagram of Hopf algebras.