Semiunital Semimonoidal Categories (Applications to Semirings and Semicorings)

The category $_{A}\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$ with the so called Takahashi's tensor product $-\boxtimes_{A}-,$ is semimonoidal but not monoidal. Although not a unit in $_{A}\mathbb{S}%_{A},$ the base semialgebra $A$ has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories $(\mathcal{V}%, \bullet, I)$ as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call $\mathbb{J}$-monads ($\mathbb{J}$-% comonads) with respect to the endo-functor $\mathbb{J}:=\mathbf{I}\bullet -\simeq -\bullet \mathbf{I}:\mathcal{V}\longrightarrow \mathcal{V}.$ This motivated also introducing a more generalized notion of monads (comonads) in arbitrary categories with respect to arbitrary endo-functors. Applications to the semiunital semimonoidal variety $(_{A}\mathbb{S}%_{A},\boxtimes_{A},A)$ provide us with examples of semiunital $A$-semirings (semicounital $A$-semicorings) and semiunitary semimodules (semicounitary semicomodules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules).