Hopf Semialgebras.

In this paper, we introduce and investigatebisemialgebrasandHopf semialgebrasover commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over commutative rings including the main reconstruction theorems and theFundamental Theorem of Hopf Algebras. We also provide a notion ofquantum monoidsas 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. [ABSTRACT FROM AUTHOR]