| Title:
|
Annihilators in normal autometrized algebras (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Rachůnek, Jiří |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
111-120 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The concepts of an annihilator and a relative annihilator in an autometrized $l$-algebra are introduced. It is shown that every relative annihilator in a normal autometrized $l$-algebra $\mathcal {A}$ is an ideal of $\mathcal {A}$ and every principal ideal of $\mathcal {A}$ is an annihilator of $\mathcal {A}$. The set of all annihilators of $\mathcal {A}$ forms a complete lattice. The concept of an $I$-polar is introduced for every ideal $I$ of $\mathcal {A}$. The set of all $I$-polars is a complete lattice which becomes a two-element chain provided $I$ is prime. The $I$-polars are characterized as pseudocomplements in the lattice of all ideals of $\mathcal {A}$ containing $I$. (English) |
| Keyword:
|
autometrized algebra |
| Keyword:
|
annihilator |
| Keyword:
|
relative annihilator |
| Keyword:
|
ideal |
| Keyword:
|
polar |
| MSC:
|
06F05 |
| idZBL:
|
Zbl 1079.06502 |
| idMR:
|
MR1814636 |
| . |
| Date available:
|
2009-09-24T10:40:17Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127630 |
| . |
| Reference:
|
[1] R. Balbes and P. Dwinger: Distributive Lattices.Univ. of Missouri Press, 1974. MR 0373985 |
| Reference:
|
[2] I. Chajda: Indexed annihilators in lattices.Arch. Math. (Brno) 31 (1995), 259–262. Zbl 0860.06005, MR 1390584 |
| Reference:
|
[3] M. E. Hansen: Minimal prime ideals in autometrized algebras.Czechoslovak Math. J. 44 (119) (1994), 81–90. Zbl 0814.06011, MR 1257938 |
| Reference:
|
[4] T. Kovář: Normal autometrized $l$-algebras.Math. Slovaca (to appear). MR 1857295 |
| Reference:
|
[5] M. Mandelker: Relative annihilators in lattices.Duke Math. J. 49 (1979), 377–386. MR 0256951 |
| Reference:
|
[6] J. Rachůnek: Prime ideals in autometrized algebras.Czechoslovak Math. J. 37 (112) (1987), 65–69. MR 0875128 |
| Reference:
|
[7] J. Rachůnek: Polars in autometrized algebras.Czechoslovak Math. J. 39 (114) (1989), 681–685. MR 1018003 |
| Reference:
|
[8] K. L. N. Swamy: A general theory of autometrized algebras.Math. Ann. 157 (1964), 65–74. Zbl 0135.02602, MR 0170842, 10.1007/BF01362667 |
| Reference:
|
[9] K. L. N. Swamy and N. P. Rao: Ideals in autometrized algebras.J. Austral. Math. Soc. (Ser. A) 24 (1977), 362–374. MR 0469843, 10.1017/S1446788700020383 |
| . |
