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 a 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-2-8,
author = {John Harding and Pavel Pt\'ak},
title = {On the set representation of an orthomodular poset},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {233-240},
zbl = {0984.06005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-2-8}
}
John Harding; Pavel Pták. On the set representation of an orthomodular poset. Colloquium Mathematicae, Tome 89 (2001) pp. 233-240. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-2-8/