Partially ordered monads and powerset Kleene algebras

Monads are used for various applications in computer science, and well-known is e.g. the interpretation of morphisms in the Kleisli category of the term monad as variable substitutions assigning variables to terms. An application building upon this observation is the equivalence between most general unifiers and co-equalizers in this category [13]. In this paper we will use monads with additional structure, namely partially ordered monads. We show how partially ordered monads can be used in order to obtain a generalised notion of Kleene powerset algebras building upon more general powerset functor settings beyond strings [11] and relations [14].