The monad of saturated prefilters.

This paper addresses the problem whether the functor of (quantale-valued) saturated prefilters is a monad on the category of sets. It is shown that when the quantale is the unit interval equipped with a continuous t-norm &, the following conditions are equivalent: (1) The functor of saturated prefilters is a monad. (2) The functor of $\top$-filters is a monad. (3) The functor of bounded saturated prefilters is a monad. (4) The implication operator of $\&$ is continuous at each point off the diagonal.