Semiunital semimonoidal categories (Applications to semirings and semicorings)

The category _AS_A of bisemimodules over a semialgebra A; with the so called Takahashi's tensor-like product - {squared times}_A -; is semimonoidal but not monoidal. Al- though not a unit in ASA; the base semialgebra A has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we inves- tigate semiunital semimonoidal categories (V, ·, I) as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call J-monads (J-comonads) with respect to the endo-functor J:= I·-~-·I: V → 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 (_AS_A, {squared times}_A;A) provide us with examples of semiunital A-semirings (semicounital A-semicorings) and semiunitary semimodules (semicounitary semicomod-ules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules).