Remarks on special ideals in lattices

Ladislav Beran

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 4, page 607-615
  • ISSN: 0010-2628

Abstract

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The author studies some characteristic properties of semiprime ideals. The semiprimeness is also used to characterize distributive and modular lattices. Prime ideals are described as the meet-irreducible semiprime ideals. In relatively complemented lattices they are characterized as the maximal semiprime ideals. D -radicals of ideals are introduced and investigated. In particular, the prime radicals are determined by means of C ^ -radicals. In addition, a necessary and sufficient condition for the equality of prime radicals is obtained.

How to cite

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Beran, Ladislav. "Remarks on special ideals in lattices." Commentationes Mathematicae Universitatis Carolinae 35.4 (1994): 607-615. <http://eudml.org/doc/247629>.

@article{Beran1994,
abstract = {The author studies some characteristic properties of semiprime ideals. The semiprimeness is also used to characterize distributive and modular lattices. Prime ideals are described as the meet-irreducible semiprime ideals. In relatively complemented lattices they are characterized as the maximal semiprime ideals. $D$-radicals of ideals are introduced and investigated. In particular, the prime radicals are determined by means of $\hat\{C\}$-radicals. In addition, a necessary and sufficient condition for the equality of prime radicals is obtained.},
author = {Beran, Ladislav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semiprime ideal; prime ideal; congruence of a lattice; allele; lattice polynomial; meet-irreducible element; kernel; forbidden exterior quotients; $D$-radical; prime radical; prime ideal; congruence; allele; lattice polynomial; kernel; forbidden exterior quotients; distributive lattices; semiprime ideals; modular lattices; -radicals; prime radicals},
language = {eng},
number = {4},
pages = {607-615},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Remarks on special ideals in lattices},
url = {http://eudml.org/doc/247629},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Beran, Ladislav
TI - Remarks on special ideals in lattices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 4
SP - 607
EP - 615
AB - The author studies some characteristic properties of semiprime ideals. The semiprimeness is also used to characterize distributive and modular lattices. Prime ideals are described as the meet-irreducible semiprime ideals. In relatively complemented lattices they are characterized as the maximal semiprime ideals. $D$-radicals of ideals are introduced and investigated. In particular, the prime radicals are determined by means of $\hat{C}$-radicals. In addition, a necessary and sufficient condition for the equality of prime radicals is obtained.
LA - eng
KW - semiprime ideal; prime ideal; congruence of a lattice; allele; lattice polynomial; meet-irreducible element; kernel; forbidden exterior quotients; $D$-radical; prime radical; prime ideal; congruence; allele; lattice polynomial; kernel; forbidden exterior quotients; distributive lattices; semiprime ideals; modular lattices; -radicals; prime radicals
UR - http://eudml.org/doc/247629
ER -

References

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  1. Beran L., Orthomodular Lattices (Algebraic Approach), Reidel Dordrecht (1985). (1985) Zbl0558.06008MR0784029
  2. Beran L., Distributivity in finitely generated orthomodular lattices, Comment. Math. Univ. Carolinae 28 (1987), 433-435. (1987) Zbl0624.06008MR0912572
  3. Beran L., On semiprime ideals in lattices, J. Pure Appl. Algebra 64 (1990), 223-227. (1990) Zbl0703.06003MR1061299
  4. Beran L., On the rhomboidal heredity in ideal lattices, Comment. Math. Univ. Carolinae 33 (1992), 723-726. (1992) Zbl0782.06007MR1240194
  5. Birkhoff G., Lattice Theory, 3rd ed., American Math. Soc. Colloq. Publ., vol. XXV, Providence, 1967. Zbl0537.06001MR0227053
  6. Chevalier G., Semiprime ideals in orthomodular lattices, Comment. Math. Univ. Carolinae 29 (1988), 379-386. (1988) Zbl0655.06008MR0957406
  7. Dubreil-Jacotin M.L., Lesieur L., Croisot R., Leçons sur la théorie des treillis, des structures algébriques ordonnées et des treillis géometriques, Gauthier-Villars Paris (1953). (1953) Zbl0051.26005MR0057838
  8. Rav Y., Semiprime ideals in general lattices, J. Pure Appl. Algebra 56 (1989), 105-118. (1989) Zbl0665.06006MR0979666

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