| Title:
|
On idempotent modifications of $MV$-algebras (English) |
| Author:
|
Jakubík, Ján |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
243-252 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The notion of idempotent modification of an algebra was introduced by Ježek. He proved that the idempotent modification of a group is subdirectly irreducible. For an $MV$-algebra $\mathcal A$ we denote by $\mathcal A^{\prime }, A$ and $\ell (\mathcal A)$ the idempotent modification, the underlying set or the underlying lattice of $\mathcal A$, respectively. In the present paper we prove that if $\mathcal A$ is semisimple and $\ell (\mathcal A)$ is a chain, then $\mathcal A^{\prime }$ is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras. (English) |
| Keyword:
|
$MV$-algebra |
| Keyword:
|
idempotent modification |
| Keyword:
|
subdirect reducibility |
| MSC:
|
03G25 |
| MSC:
|
06D35 |
| idZBL:
|
Zbl 1174.06317 |
| idMR:
|
MR2309963 |
| . |
| Date available:
|
2009-09-24T11:45:26Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128169 |
| . |
| Reference:
|
[1] G. Cattaneo and F. Lombardo: Independent axiomatization of $MV$-algebras.Tatra Mt. Math. Publ. 15 (1998), 227–232. MR 1655091 |
| Reference:
|
[2] C. C. Chang: Algebraic analysis of many valued logics.Trans. Amer. Math. Soc. 88 (1958), 467–490. Zbl 0084.00704, MR 0094302, 10.1090/S0002-9947-1958-0094302-9 |
| Reference:
|
[3] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundation of Many Valued Reasoning.Kluwer Academic Publ., Dordrecht, 2000. MR 1786097 |
| Reference:
|
[4] A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structure.Kluwer Academic Publ., Dordrecht and Ister, Bratislava, 2000. MR 1861369 |
| Reference:
|
[5] L. Fuchs: Partially Ordered Algebraic Systems.Pergamon Press, Oxford-New York-London-Paris, 1963. Zbl 0137.02001, MR 0171864 |
| Reference:
|
[6] D. Glushankof: Cyclic ordered groups and $MV$-algebras.Czechoslovak Math. J. 43 (1993), 249–263. MR 1211747 |
| Reference:
|
[8] J. Ježek: A note on idempotent modifications of groups.Czechoslovak Math. J. 54 (2004), 229–231. MR 2040234, 10.1023/B:CMAJ.0000027262.04069.10 |
| . |
