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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]

In this paper, we introduce and investigate semicorings over associative semirings and their categories of semicomodules. Our results generalize old and recent ones on corings over rings and their categories of comodules. The generalization is not straightforward and even subtle at several places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no nonzero injective objects. To overcome these and other difficulties, a combination of methods and techniques from categorical, homological and universal algebra is used including a new notion of exact sequences of semimodules over semirings. [ABSTRACT FROM AUTHOR]