We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of e¤ects 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 sharpnessthat are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*- lattices and their reducts; in particular, we prove some embedding re- sults for PBZ*-lattices and provide an initial description of the lattice of PBZ*-varieties.
A new view of effects in a Hilbert space
GIUNTINI, ROBERTO;LEDDA, ANTONIO;PAOLI, FRANCESCO
2016-01-01
Abstract
We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of e¤ects 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 sharpnessthat are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*- lattices and their reducts; in particular, we prove some embedding re- sults for PBZ*-lattices and provide an initial description of the lattice of PBZ*-varieties.
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