Partially additive states on orthomodular posets

Volume 62 / 1991

Josef Tkadlec Colloquium Mathematicum 62 (1991), 7-14 DOI: 10.4064/cm-62-1-7-14


We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also be viewed as a generalization of [6]. Then we prove an extension theorem for B-states giving, as a by-product, a topological proof of a classical Boolean result.


  • Josef Tkadlec

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