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Algebra universalis

, 80:49 | Cite as

Observables on lexicographic effect algebras

  • Anatolij DvurečenskijEmail author
  • Dominik Lachman
Article
  • 27 Downloads

Abstract

We study lexicographic effect algebras which are intervals in lexicographic products \(H\,\overrightarrow{\times }\,G\), where (Hu) 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.

Keywords

Effect algebra Lexicographic effect algebra Monotone \(\sigma \)-complete po-group Observable Spectral resolution Finiteness property 

Mathematics Subject Classification

03G12 03B50 06C15 81P15 

Notes

Acknowledgements

The authors are very indebted to anonymous referees for their suggestions and remarks that improve the readability of the paper.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia
  2. 2.Faculty of SciencesPalacký University OlomoucOlomoucCzech Republic

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