nLab
monadic algebra

Idea

Monadic algebras are type one modal algebras, in which the single operator behaves like a closure operation in a topological space, (so these are closure algebras) and, in addition, elements are closed if and only if they are open.

Definitions

Definition

A monadic algebra is a closure algebra, $(𝔹,m)$, which satisfies: for all $x$, $x\le lmx$, where, as usual, $l$ is a shorthand for $\neg m\neg$.