In this paper, we introduce and investigate emph{bisemialgebras}andemph{ Hopf semialgebras} over commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over rings including the main reconstru
ction 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.
