Monadicity of the Category of Ordered Semigroups over Posets

This paper investigates the categorical structure of ordered semigroups by analysing the forgetful functor U: OrdSgp → Pos to partially ordered sets. We prove that U is monadic in the sense of Beck: ordered semigroups are precisely the Eilenberg–Moore algebras for a naturally defined monad on Pos. After recalling the free ordered semigroup construction and the resulting adjunction, we verify Beck’s conditions constructively. In particular, we show that U creates and preserves U-split coequalizers by endowing the underlying split coequalizer in Pos with a canonical ordered semigroup structure. The proof is elementary, avoids any choice principles, and makes explicit use of the order-theoretic features of Pos. The resulting monadic description clarifies how order and multiplication interact categorically, and provides a convenient framework for transporting properties and constructions between ordered semigroups and posets.