Partially Ordered Monads for Monadic Topologies, Rough Sets and Kleene Algebras

In this paper we will show that partially ordered monads contain sufficient structure for modelling monadic topologies, rough sets and Kleene algebras. Convergence represented by extension structures over partially ordered monads includes notions of regularity and compactness. A compactification theory can be developed. Rough sets [Z. Pawlak, Rough sets, Int. J. Computer and Information Sciences 5 (1982) 341356] are modelled in a generalized setting with set functors. Further, we show how partially ordered monads can be used in order to obtain monad based examples of Kleene algebras building upon a wide range of set functors far beyond just strings [S. C. Kleene, Representation of events in nerve nets and finite automata, In: Automata Studies (Eds. C. E. Shannon, J. McCarthy), Princeton University Press, 1956, 3-41] and relations [A. Tarski, On the calculus of relations, J. Symbolic Logic 6 (1941), 65-106].