| Title:
|
A Cantor-Bernstein theorem for $\sigma$-complete MV-algebras (English) |
| Author:
|
de Simone, A. |
| Author:
|
Mundici, D. |
| Author:
|
Navara, M. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
53 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
437-447 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The Cantor-Bernstein theorem was extended to $\sigma $-complete boolean algebras by Sikorski and Tarski. Chang’s MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Łukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to $\sigma $-complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras. (English) |
| Keyword:
|
Cantor-Bernstein theorem |
| Keyword:
|
MV-algebra |
| Keyword:
|
boolean element of an MV-algebra |
| Keyword:
|
partition of unity |
| Keyword:
|
direct product decomposition |
| Keyword:
|
$\sigma $-complete MV-algebra |
| MSC:
|
03G20 |
| MSC:
|
06C15 |
| MSC:
|
06D30 |
| MSC:
|
06D35 |
| idZBL:
|
Zbl 1024.06003 |
| idMR:
|
MR1983464 |
| . |
| Date available:
|
2009-09-24T11:03:10Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127812 |
| . |
| Reference:
|
[1] R. Cignoli and D. Mundici: An invitation to Chang’s MV-algebras.In: Advances in Algebra and Model Theory, M. Droste, R. Göbel (eds.), Gordon and Breach Publishing Group, Reading, UK, 1997, pp. 171–197. MR 1683528 |
| Reference:
|
[2] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-valued Reasoning. Trends in Logic. Vol. 7.Kluwer Academic Publishers, Dordrecht, 1999. MR 1786097 |
| Reference:
|
[3] W. Hanf: On some fundamental problems concerning isomorphism of boolean algebras.Math. Scand. 5 (1957), 205–217. Zbl 0081.26101, MR 0108451, 10.7146/math.scand.a-10496 |
| Reference:
|
[4] J. Jakubík: Cantor-Bernstein theorem for $MV$-algebras.Czechoslovak Math. J. 49(124) (1999), 517–526. MR 1708370, 10.1023/A:1022467218309 |
| Reference:
|
[5] S. Kinoshita: A solution to a problem of Sikorski.Fund. Math. 40 (1953), 39–41. MR 0060809, 10.4064/fm-40-1-39-41 |
| Reference:
|
[6] A. Levy: Basic Set Theory. Perspectives in Mathematical Logic.Springer-Verlag, Berlin, 1979. MR 0533962 |
| Reference:
|
[7] D. Mundici: Interpretation of AF $C^{*}$-algebras in Łukasiewicz sentential calculus.J. Funct. Anal. 65 (1986), 15–63. Zbl 0597.46059, MR 0819173, 10.1016/0022-1236(86)90015-7 |
| Reference:
|
[8] R. Sikorski: Boolean Algebras.Springer-Verlag. Ergebnisse Math. Grenzgeb., Berlin, 1960. Zbl 0087.02503, MR 0126393 |
| Reference:
|
[9] R. Sikorski: A generalization of a theorem of Banach and Cantor-Bernstein.Colloq. Math. 1 (1948), 140–144 and 242. MR 0027264, 10.4064/cm-1-2-140-144 |
| Reference:
|
[10] A. Tarski: Cardinal Algebras.Oxford University Press, New York, 1949. Zbl 0041.34502, MR 0029954 |
| . |
