Radicals and complete distributivity in relatively normal lattices

Jiří Rachůnek

Mathematica Bohemica (2003)

  • Volume: 128, Issue: 4, page 401-410
  • ISSN: 0862-7959

Abstract

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Lattices in the class ℐℛ𝒩 of algebraic, distributive lattices whose compact elements form relatively normal lattices are investigated. We deal mainly with the lattices in ℐℛ𝒩 the greatest element of which is compact. The distributive radicals of algebraic lattices are introduced and for the lattices in ℐℛ𝒩 with the sublattice of compact elements satisfying the conditional join-infinite distributive law they are compared with two other kinds of radicals. Connections between complete distributivity of algebraic lattices and the distributive radicals are described. The general results can be applied e.g. to M V -algebras, G M V -algebras and unital ℓ -groups.

How to cite

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Rachůnek, Jiří. "Radicals and complete distributivity in relatively normal lattices." Mathematica Bohemica 128.4 (2003): 401-410. <http://eudml.org/doc/249215>.

@article{Rachůnek2003,
abstract = {Lattices in the class $\mathcal \{IRN\}$ of algebraic, distributive lattices whose compact elements form relatively normal lattices are investigated. We deal mainly with the lattices in $\mathcal \{IRN\}$ the greatest element of which is compact. The distributive radicals of algebraic lattices are introduced and for the lattices in $\mathcal \{IRN\}$ with the sublattice of compact elements satisfying the conditional join-infinite distributive law they are compared with two other kinds of radicals. Connections between complete distributivity of algebraic lattices and the distributive radicals are described. The general results can be applied e.g. to $MV$-algebras, $GMV$-algebras and unital $\ell $-groups.},
author = {Rachůnek, Jiří},
journal = {Mathematica Bohemica},
keywords = {relatively normal lattice; algebraic lattice; complete distributivity; closed element; radical; relatively normal lattice; algebraic lattice; complete distributivity; closed element; radical},
language = {eng},
number = {4},
pages = {401-410},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Radicals and complete distributivity in relatively normal lattices},
url = {http://eudml.org/doc/249215},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Rachůnek, Jiří
TI - Radicals and complete distributivity in relatively normal lattices
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 4
SP - 401
EP - 410
AB - Lattices in the class $\mathcal {IRN}$ of algebraic, distributive lattices whose compact elements form relatively normal lattices are investigated. We deal mainly with the lattices in $\mathcal {IRN}$ the greatest element of which is compact. The distributive radicals of algebraic lattices are introduced and for the lattices in $\mathcal {IRN}$ with the sublattice of compact elements satisfying the conditional join-infinite distributive law they are compared with two other kinds of radicals. Connections between complete distributivity of algebraic lattices and the distributive radicals are described. The general results can be applied e.g. to $MV$-algebras, $GMV$-algebras and unital $\ell $-groups.
LA - eng
KW - relatively normal lattice; algebraic lattice; complete distributivity; closed element; radical; relatively normal lattice; algebraic lattice; complete distributivity; closed element; radical
UR - http://eudml.org/doc/249215
ER -

References

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  1. Lattice-Ordered Groups, D. Reidel Publ., Dordrecht, 1988. (1988) MR0937703
  2. Distributive Lattices, Univ. of Missouri Press, Columbia, Missouri, 1974. (1974) MR0373985
  3. Groupes et Anneaux Réticulés, Springer, Berlin, 1977. (1977) MR0552653
  4. 10.1090/S0002-9947-1958-0094302-9, Trans. Amer. Math. Soc. 88 (1958), 467–490. (1958) MR0094302DOI10.1090/S0002-9947-1958-0094302-9
  5. Algebraic Foundations of Many-Valued Reasoning, Kluwer Acad. Publ., Dordrecht, 2000. (2000) MR1786097
  6. Closed ideals of M V -algebras, Advances in Contemporary Logic and Computer Science, Contemp. Math., vol. 235, AMS, Providence, 1999, pp. 99–112. (1999) MR1721208
  7. New Trends in Quantum Structures, Kluwer Acad. Publ., Dordrecht, 2000. (2000) MR1861369
  8. 10.1016/0012-365X(95)00221-H, Discrete Math. 161 (1996), 87–100. (1996) MR1420523DOI10.1016/0012-365X(95)00221-H
  9. Pseudo- M V algebras: A non-commutative extension of M V -algebras, Proc. Fourth Inter. Symp. Econ. Inform., May 6–9, 1999, INFOREC Printing House, Bucharest, 1999, pp. 961–968. (1999) 
  10. Pseudo- M V algebras, Multiple Valued Logic 6 (2001), 95–135. (2001) MR1817439
  11. Partially Ordered Groups, World Scientific, Singapore, 1999. (1999) Zbl0933.06010MR1791008
  12. 10.1090/S0002-9947-1994-1211409-2, Trans. Amer. Math. Soc. 341 (1994), 519–548. (1994) MR1211409DOI10.1090/S0002-9947-1994-1211409-2
  13. 10.1007/BF02945124, Algebra Universalis 3 (1973), 247–260. (1973) Zbl0317.06004MR0349503DOI10.1007/BF02945124
  14. L’arithmétique des filtres et les espaces topologiques, De Segundo Symp. Mathematicas-Villavicencio, Mendoza, Buenos Aires, 1954, pp. 129–162. (1954) Zbl0058.38503MR0074805
  15. L’arithmétique des filtres et les espaces topologiques I–II. Notas de Logica Mathematica, vol. 29–30, 1974, . 
  16. Linear finitely separated objects of subcategories of domains, Math. Slovaca 46 (1996), 457–490. (1996) Zbl0890.06007MR1451036
  17. 10.1023/A:1021766309509, Czechoslovak Math. J. 52 (2002), 255–273. (2002) Zbl1012.06012MR1905434DOI10.1023/A:1021766309509
  18. 10.1007/PL00012447, Algebra Universalis 48 (2002), 151–169. (2002) Zbl1058.06015MR1929902DOI10.1007/PL00012447
  19. Radicals in non-commutative generalizations of M V -algebras, Math. Slovaca 52 (2002), 135–144. (2002) Zbl1008.06011MR1935113
  20. 10.1007/BF01190765, Algebra Universalis 33 (1995), 40–67. (1995) MR1303631DOI10.1007/BF01190765

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