The Relationship Between Various Filter Notions on a GL-Monoid

The notion of a generalised filter is extended to the setting of a GL-monoid. It is shown that there exists a one-to-one correspondence between the collection of generalised filters on a set X and the collection of strongly stratified L-filters on X. Specialising to the case where L is the closed unit intervalw0, cxviewed as a Heyting algebra, we show that any strongly stratified w0, cx-filter on X can be uniquely identified with a saturated filter on I X with characteristic value c. In this way, the notion of a generalised filter unifies various filter notions. In particular, necessity measures and finitely additive probability measures are specific examples of generalised filters.