A closure algebra C is a boolean algebra B together with a unary closure operator, and additional axioms, the Kuratowski axioms, that the closure operator must satisfy. (The Wikipedia article prefers the term "interior algebra".)
Since C is a boolean algebra, we may define a boolean filter F on C in the usual way, as a non-empty set of elements of C which contains:
the meet of x and y whenever x and y are both in F;
the join of x and y, for any x in F and y in C.
The definition of a boolean filter makes no mention of the closure operator. This raises the question of whether there is another kind of filter, to be called a closure filter, which adds another condition, relating to the closure operator, to the definition of a boolean filter. For example, it might require that:
- the interior of x is in F, for any x in F.
where interior is defined in terms of closure as usual. Or perhaps this is not the right choice for an additional condition.
If we then define a closure ultrafilter as a maximal closure filter, we might find that these have different properties from maximal boolean filters.
I have not been able to locate any research along these lines, and would appreciate hearing of any references.