A note on a function representation of orthomodular posets
Josef Tkadlec · Czech digital mathematics library · 1989
In the papers [1], [2] the authors give axioms for a set of functions to characterize an orthomodular poset with “enough” states. In the attempt to improve the characterization, D. Strojewski [3] offers (seemingly) more lucid conditions and derives several consequences. However, his crucial auxiliary result does not seem to be correct. In this note we construct the appropriate counterexample and give the correct version of the representation theorem. Let us first review the basic notions. By an orthomodular poset we mean a triple (L,≤,′ ) such that (a) (L,≤) is a partially ordered set with a greatest element 1, (b) the operation ′ : L → L is an orthocomplementation, for every a, b ∈ L we have a′′ = a and a ≤ b implies b′ ≤ a′. (c) the least upper bound exists for every pair of orthogonal elements in L, (d) b = a ∨ (b ∧ a′) for every a, b ∈ L with a ≤ b. By a state we mean a function s : L → [0, 1] such that s(1) = 1 and s(a∨b) = s(a) + s(b) for each pair of the orthogonal elements a, b ∈ L. Recall finally that a subset of states is called full if for every a, b ∈ L, a 6≤ b, the subset contains such a state s that s(a) 6≤ s(b).