Pre-BZ and degenerate BZ posets: Applications to fuzzy sets and unsharp quantum theories

Two different generalizations of Brouwer-Zadeh posets (BZ posets) introduce d. The former ( called pre-BZ poset) arises from topological spaces whose s tandard power set orthocomplimented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure.This complementation satisfies only some properties of the algerbraic version of an intuitionastic negation, an d can be considered as a generalized version form of Brouwer negation. The latter ( called degenerate BZ poset ) arises from the so-called special eff ects on a Hilbert space. It is shown that the standard Brouwer negation for effect operators produces a degenerate BZ poset with respect to the order induced from the partial sum operation.