On the set representation of an orthomodular poset

John Harding, Pavel Pták
2001 Colloquium Mathematicum  
Let P be an orthomodular poset and let B be a Boolean subalgebra of P . A mapping s : P → 0, 1 is said to be a centrally additive B-state if it is order preserving, satisfies s(a ′ ) = 1 − s(a), is additive on couples that contain a central element, and restricts to a state on B. It is shown that, for any Boolean subalgebra B of P , P has an abundance of two-valued centrally additive B-states. This answers positively a question raised in [13, Open question, p. 13]. As a consequence one obtains
more » ... somewhat better set representation of orthomodular posets and a better extension theorem than in [2, 12, 13] . Further improvement in the Boolean vein is hardly possible as the concluding example shows. 2000 Mathematics Subject Classification: 06C15, 81P10.
doi:10.4064/cm89-2-8 fatcat:fauu4xvoprd3zbnu2qhle4m5fe