CCLCP basis-space status of the blurry-filter construction

Establish whether, for every sorted complemented algebra, the basis space of blurry K-filters is a countable complemented locally compact Polish basis space, thereby determining whether the construction A → (A) yields a full converse to the blurry duality theorem.

Background

The blurry duality theorem proves that a countable complemented locally compact Polish basis space can be recovered, up to equivalence, from the sorted complemented algebra constructed from it. The converse would require showing that the blurry-filter basis space associated with an arbitrary sorted complemented algebra satisfies the stronger CCLCP axioms. The paper identifies this as the main obstacle to obtaining a complete analogue of Stone’s representation theorem for all sorted complemented algebras and to characterizing algebra isomorphism directly through basis-space equivalence.