| Title:
|
A stateless orthomodular poset with the augmentation property (English) |
| Author:
|
Burešová, Dominika |
| Author:
|
Navara, Mirko |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2026 |
| Pages:
|
1-6 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the matrix representation of an effect algebra $L$ (in particular, an orthomodular poset), if $L$ admits a state, then its matrix $M(L)$ has the augmentation property ($\text{rank}(M(L)) = \text{rank}(M(L)|1)$). We show that the reverse implication is not true, thus answering an open question published on effect algebras. (English) |
| Keyword:
|
effect algebra |
| Keyword:
|
orthomodular poset |
| Keyword:
|
state |
| Keyword:
|
rank of a matrix |
| MSC:
|
06C15 |
| MSC:
|
81P10 |
| MSC:
|
81P15 |
| DOI:
|
10.14736/kyb-2026-1-0001 |
| . |
| Date available:
|
2026-03-03T14:11:36Z |
| Last updated:
|
2026-03-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153529 |
| . |
| Reference:
|
[1] Bińczak, G., Kaleta, J., Zembrzuski, A.: Matrix representation of finite effect algebras..Kybernetika 59 (2023), 737-751. |
| Reference:
|
[2] Busch, P., Lahti, P. J., Mittelstaedt, P.: The Quantum Theory of Measurement..Springer Science and Business Media, 1996. |
| Reference:
|
[3] Simone, A. De, Pták, P.: Group-valued measures on coarse-grained quantum logics..Czechoslovak Math. J. 57 (2007), 2, 737-746. 10.1007/s10587-007-0110-4 |
| Reference:
|
[4] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures..Kluwer Acad. Publ./Ister Sci., Dordrecht/Bratislava 2000. Zbl 0987.81005 |
| Reference:
|
[5] Greechie, R. J.: Orthomodular lattices admitting no states..J. Comb. Theor. Ser. A 10 (1971), 119-132. 10.1016/0097-3165(71)90015-X |
| Reference:
|
[6] Gudder, S.: Effect test spaces and effect algebras..Found. Phys. 27 (1997), 287-304. |
| Reference:
|
[7] Kalmbach, G.: Orthomodular Lattices..Academic Press, London 1983. |
| Reference:
|
[8] Mayet, R.: Personal communication..1993. |
| Reference:
|
[9] Navara, M.: An orthomodular lattice admitting no group-valued measure..Proc. Amer. Math. Soc. 122 (1994), 1, 7-12. 10.1090/S0002-9939-1994-1191871-X |
| Reference:
|
[10] Navara, M.: Existence of states on quantum structures..Inform. Sci. 179 (2009), 508-514. |
| Reference:
|
[11] Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics..Kluwer Academic Publishers, Dordrecht/Boston/London 1991. Zbl 0743.03039 |
| Reference:
|
[12] Svozil, K.: Quantum Logic..Springer-Verlag, Singapore 1998. |
| Reference:
|
[13] Weber, H.: There are orthomodular lattices without non-trivial group-valued states: a computer-based construction..J. Math. Anal. Appl. 183 (1994), 1, 89-93. |
| . |
