Geodesic
On idempotent modifications of $MV$-algebras
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 243-252 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 03G25, 06D35
Keywords: $MV$-algebra; idempotent modification; subdirect reducibility 
@article{CMJ_2007_57_1_a19,
     author = {Jakub{\'\i}k, J\'an},
     title = {On idempotent modifications of $MV$-algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {243--252},
     year = {2007},
     volume = {57},
     number = {1},
     mrnumber = {2309963},
     zbl = {1174.06317},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a19/}
}
TY  - JOUR
AU  - Jakubík, Ján
TI  - On idempotent modifications of $MV$-algebras
JO  - Czechoslovak Mathematical Journal
PY  - 2007
SP  - 243
EP  - 252
VL  - 57
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a19/
LA  - en
ID  - CMJ_2007_57_1_a19
ER  - 
%0 Journal Article
%A Jakubík, Ján
%T On idempotent modifications of $MV$-algebras
%J Czechoslovak Mathematical Journal
%D 2007
%P 243-252
%V 57
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a19/
%G en
%F CMJ_2007_57_1_a19
Jakubík, Ján. On idempotent modifications of $MV$-algebras. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 243-252. http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a19/

[1] G. Cattaneo and F. Lombardo: Independent axiomatization of $MV$-algebras. Tatra Mt. Math. Publ. 15 (1998), 227–232. | MR

[2] C. C. Chang: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88 (1958), 467–490. | DOI | MR | Zbl

[3] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundation of Many Valued Reasoning. Kluwer Academic Publ., Dordrecht, 2000. | MR

[4] A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structure. Kluwer Academic Publ., Dordrecht and Ister, Bratislava, 2000. | MR

[5] L. Fuchs: Partially Ordered Algebraic Systems. Pergamon Press, Oxford-New York-London-Paris, 1963. | MR | Zbl

[6] D. Glushankof: Cyclic ordered groups and $MV$-algebras. Czechoslovak Math. J. 43 (1993), 249–263. | MR

[8] J. Ježek: A note on idempotent modifications of groups. Czechoslovak Math. J. 54 (2004), 229–231. | DOI | MR