On idempotent modifications of M V -algebras

Ján Jakubík

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 243-252
  • ISSN: 0011-4642

Abstract

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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 M V -algebra 𝒜 we denote by 𝒜 ' , A and ( 𝒜 ) the idempotent modification, the underlying set or the underlying lattice of 𝒜 , respectively. In the present paper we prove that if 𝒜 is semisimple and ( 𝒜 ) is a chain, then 𝒜 ' is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.

How to cite

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Jakubík, Ján. "On idempotent modifications of $MV$-algebras." Czechoslovak Mathematical Journal 57.1 (2007): 243-252. <http://eudml.org/doc/31127>.

@article{Jakubík2007,
abstract = {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.},
author = {Jakubík, Ján},
journal = {Czechoslovak Mathematical Journal},
keywords = {$MV$-algebra; idempotent modification; subdirect reducibility; MV-algebra; idempotent modification; subdirect irreducibility},
language = {eng},
number = {1},
pages = {243-252},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On idempotent modifications of $MV$-algebras},
url = {http://eudml.org/doc/31127},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Jakubík, Ján
TI - On idempotent modifications of $MV$-algebras
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 243
EP - 252
AB - 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.
LA - eng
KW - $MV$-algebra; idempotent modification; subdirect reducibility; MV-algebra; idempotent modification; subdirect irreducibility
UR - http://eudml.org/doc/31127
ER -

References

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  1. Independent axiomatization of M V -algebras, Tatra Mt. Math. Publ. 15 (1998), 227–232. (1998) MR1655091
  2. 10.1090/S0002-9947-1958-0094302-9, Trans. Amer. Math. Soc. 88 (1958), 467–490. (1958) Zbl0084.00704MR0094302DOI10.1090/S0002-9947-1958-0094302-9
  3. Algebraic Foundation of Many Valued Reasoning, Kluwer Academic Publ., Dordrecht, 2000. (2000) MR1786097
  4. New Trends in Quantum Structure, Kluwer Academic Publ., Dordrecht and Ister, Bratislava, 2000. (2000) MR1861369
  5. Partially Ordered Algebraic Systems, Pergamon Press, Oxford-New York-London-Paris, 1963. (1963) Zbl0137.02001MR0171864
  6. Cyclic ordered groups and M V -algebras, Czechoslovak Math. J. 43 (1993), 249–263. (1993) MR1211747
  7. 10.1023/B:CMAJ.0000027262.04069.10, Czechoslovak Math. J. 54 (2004), 229–231. (2004) MR2040234DOI10.1023/B:CMAJ.0000027262.04069.10

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