On the set representation of an orthomodular poset

Tom 89 / 2001

John Harding, Pavel Pták Colloquium Mathematicum 89 (2001), 233-240 MSC: 06C15, 81P10. DOI: 10.4064/cm89-2-8


Let $P$ be an orthomodular poset and let $B$ be a Boolean subalgebra of $P$. A mapping $s:P \to \langle 0, 1 \rangle $ 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.


  • John HardingDepartment of Mathematical Sciences
    New Mexico State University
    Las Cruces, NM 88003, U.S.A.
  • Pavel PtákDepartment of Mathematics
    Faculty of Electrical Engineering
    Czech Technical University
    Technická 2
    16627 Praha 6, Czech Republic

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