Hopf Semialgebras

In this paper, we introduce and investigate bisemialgebras and Hopf semialgebras over commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over commutative rings including the main reconstruction theorems and the Fundamental Theorem of Hopf Algebras. We also provide a notion of 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 nonadditive nature of the base category of commutative monoids which is also neither Puppe-exact nor homological and does not necessarily have enough injectives.