nLab orthomodular lattice

Definition

An orthomodular lattice is an orthocomplemented lattice in which in addition a weak form of modularity holds in that

$$a \leq c$$

implies that

$$a \vee (a^\perp \wedge c) = c \,.$$

Examples

The canonical example are the Hilbert lattices that interpret Birkhoff-vonNeumann quantum logic.

References

• P. Pták and S. Pulmannová, Orthomodular structures as quantum logics, ser. Fundamental theories of physics. Kluwer Academic Publishers, 1991.