PBZ*-lattices: ordinal and horizontal sums

Roberto Giuntini;Claudia Muresan;Francesco Paoli

2021-01-01

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Abstract

PBZ∗–lattices are lattices with additional operations that arise in the context of the unsharp approach to quantum logic. They include orthomodular lattices and Kleene algebras with an extra unary operation. We study in the framework of PBZ∗ –lattices two constructions — the ordinal sum construction and the horizontal sum construction — that have been widely used in the investigation of both quantum structures and residuated structures. We provide axiomatisations of the varieties generated by certain sums of PBZ∗ –lattices, in particular of the variety generated by all horizontal sums of an orthomodular lattice and an antiortholattice.