Brouwer-Zadeh posets and three-valued Łukasiewicz posets

This paper is a study of the structure of Brouwer-Zadeh (or BZ-) poset, i.e. a poset endowed with two non-usual orthocomplementations. These two orthocomplementations allow definition of two unary operators which can be considered as algebraic counterparts of the necessity and possibility operators of modal logic. A construction of how to induce a three-valued BZ-poset from a BZ-poset is given. The examples of the BZ-lattice of all generalized characteristic functionals on a reference space and of the BZ-poset of all generalized orthogonal projections on a Hilbert space are dealt with