A note on a function representation of orthomodular posets. (English) Zbl 0672.03048

Tkadlec, Josef

Math. Slovaca 39, No. 1, 27-29 (1989).

Let S be a set. The author gives four conditions which are necessary and sufficient for a partially ordered set \(P\subseteq [0,1]^ S\) to be orthomodular and admit a full set of states. This corrects an error in an earlier paper by D. Strojewski [Bull. Pol. Acad. Sci., Math. 33, 341-348 (1985; Zbl 0581.06007)] which makes the same claim for just the first three conditions. An example is given which illustrates the necessity of the fourth condition. If \(f,g\in P\) are orthogonal then for P to be orthomodular their join must be \(f+g\), hence \(f+g\in P\) is required; this is condition three. The example shows that this is not enough to guarantee that the join of f and g exists.

Reviewer: M.Roddy

MSC:

03G12	Quantum logic
06A06	Partial orders, general
06C15	Complemented lattices, orthocomplemented lattices and posets

Keywords:

orthomodular poset; full set of states

Citations:

Zbl 0581.06007

References:

[1]	MACZYŃSKI M. J.: On some numerical characterization of Boolean algebras. Colloquium Mathematicum 27, 1973, 207-210. · Zbl 0264.06010
[2]	MACZYŃSKI M. J., TRACZYK T.: A characterization of orthomodular partially ordered sets admitting a full set of states. Bull. Acad. Polon. Sci., Ser. Sci. Math., Astronom., Phys. 21, 1973, 3-8. · Zbl 0265.06003
[3]	STROJEWSKI D.: Numerical representations of orthomodular lattices and Boolean algebras with infinite operations. Bull. Acad. Polon. Sci., Ser. Sci. Math., Astronom., Phys. 33, 1985, 341-348. · Zbl 0581.06007