Classical and quantum degrees of truth: a new look at the effects of a Hilbert space

Speaker: Roberto Giuntini (University of Cagliari).

We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ^∗-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ^∗-lattices and their reducts; in particular, we prove some embedding results for PBZ^∗-lattices and provide an initial description of the lattice of PBZ^∗-varieties.