Abstract
We study lexicographic effect algebras which are intervals in lexicographic products \(H\,\overrightarrow{\times }\,G\), where (H, u) is a unital po-group and G is a monotone \(\sigma \)-complete po-group with interpolation. We prove that there is a one-to-one correspondence between observables, which are a special kind of \(\sigma \)-homomorphisms and analogues of measurable functions, and spectral resolutions which are systems \(\{x_t : t \in {\mathbb {R}}\}\) of elements of a lexicographic effect algebra that are monotone, “left continuous”, and going to 0 if \(t\rightarrow -\infty \) and to 1 if \(t\rightarrow +\infty \). We show that this correspondence in lexicographic effect algebras holds only for spectral resolutions with the finiteness property. Otherwise, they do not determine any observable. Whence, the information involved in a spectral resolution with the finiteness property completely describes information about an observable.
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The authors are very indebted to anonymous referees for their suggestions and remarks that improve the readability of the paper.
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Presented by S. Pulmannova.
The paper has been supported by the grant of the Slovak Research and Development Agency under contract APVV-16-0073 and the Grant VEGA No. 2/0069/16 SAV, A.D., and by Grant CZ.02.2.69/0.0/0.0/16-027/0008482 SPP 8197200115, D.L.
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Dvurečenskij, A., Lachman, D. Observables on lexicographic effect algebras. Algebra Univers. 80, 49 (2019). https://doi.org/10.1007/s00012-019-0628-y
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DOI: https://doi.org/10.1007/s00012-019-0628-y
Keywords
- Effect algebra
- Lexicographic effect algebra
- Monotone \(\sigma \)-complete po-group
- Observable
- Spectral resolution
- Finiteness property