Hopf Semialgebras

TL;DR

This paper extends the theory of Hopf algebras to semialgebras over commutative semirings, introducing bisemialgebras and quantum monoids, and generalizing fundamental theorems to this broader context.

Contribution

It introduces Hopf semialgebras and bisemialgebras over semirings, generalizing key Hopf algebra results and defining quantum monoids in this new setting.

Findings

Generalized the Fundamental Theorem of Hopf Algebras to semialgebras.

Defined quantum monoids as non-commutative, non-cocommutative Hopf semialgebras.

Extended Hopf algebra concepts to a non-additive, semialgebraic context.

Abstract

In this paper, we introduce and investigate \emph{bisemialgebras}and\emph{\ Hopf semialgebras} over commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over rings including the main reconstruction theorems and the \emph{Fundamental Theorem of Hopf Algebras}. We also provide a notion of \emph{quantum monoids} as Hopf semialgebras which are neither commutative nor cocommutative; this extends the Hopf algebraic notion of a quantum group. The generalization to the semialgebraic context is neither trivial nor straightforward due to the non-additive nature of the base category of Abelian monoids which is also neither Puppe-exact nor homological and does not necessarily have enough injectives.