Hopf Semialgebras
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.
Cite
@article{arxiv.1304.6042,
title = {Hopf Semialgebras},
author = {Jawad Abuhlail and Nabeela Alsulaiman},
journal= {arXiv preprint arXiv:1304.6042},
year = {2013}
}