# Radicals and complete distributivity in relatively normal lattices

Mathematica Bohemica (2003)

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

## Access Full Article

top## Abstract

top## How to cite

topRachůnek, 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

top- Lattice-Ordered Groups, D. Reidel Publ., Dordrecht, 1988. (1988) MR0937703
- Distributive Lattices, Univ. of Missouri Press, Columbia, Missouri, 1974. (1974) MR0373985
- Groupes et Anneaux Réticulés, Springer, Berlin, 1977. (1977) MR0552653
- 10.1090/S0002-9947-1958-0094302-9, Trans. Amer. Math. Soc. 88 (1958), 467–490. (1958) MR0094302DOI10.1090/S0002-9947-1958-0094302-9
- Algebraic Foundations of Many-Valued Reasoning, Kluwer Acad. Publ., Dordrecht, 2000. (2000) MR1786097
- Closed ideals of $MV$-algebras, Advances in Contemporary Logic and Computer Science, Contemp. Math., vol. 235, AMS, Providence, 1999, pp. 99–112. (1999) MR1721208
- New Trends in Quantum Structures, Kluwer Acad. Publ., Dordrecht, 2000. (2000) MR1861369
- 10.1016/0012-365X(95)00221-H, Discrete Math. 161 (1996), 87–100. (1996) MR1420523DOI10.1016/0012-365X(95)00221-H
- Pseudo-$MV$ algebras: A non-commutative extension of $MV$-algebras, Proc. Fourth Inter. Symp. Econ. Inform., May 6–9, 1999, INFOREC Printing House, Bucharest, 1999, pp. 961–968. (1999)
- Pseudo-$MV$ algebras, Multiple Valued Logic 6 (2001), 95–135. (2001) MR1817439
- Partially Ordered Groups, World Scientific, Singapore, 1999. (1999) Zbl0933.06010MR1791008
- 10.1090/S0002-9947-1994-1211409-2, Trans. Amer. Math. Soc. 341 (1994), 519–548. (1994) MR1211409DOI10.1090/S0002-9947-1994-1211409-2
- 10.1007/BF02945124, Algebra Universalis 3 (1973), 247–260. (1973) Zbl0317.06004MR0349503DOI10.1007/BF02945124
- L’arithmétique des filtres et les espaces topologiques, De Segundo Symp. Mathematicas-Villavicencio, Mendoza, Buenos Aires, 1954, pp. 129–162. (1954) Zbl0058.38503MR0074805
- L’arithmétique des filtres et les espaces topologiques I–II. Notas de Logica Mathematica, vol. 29–30, 1974, .
- Linear finitely separated objects of subcategories of domains, Math. Slovaca 46 (1996), 457–490. (1996) Zbl0890.06007MR1451036
- 10.1023/A:1021766309509, Czechoslovak Math. J. 52 (2002), 255–273. (2002) Zbl1012.06012DOI10.1023/A:1021766309509
- 10.1007/PL00012447, Algebra Universalis 48 (2002), 151–169. (2002) Zbl1058.06015MR1929902DOI10.1007/PL00012447
- Radicals in non-commutative generalizations of $MV$-algebras, Math. Slovaca 52 (2002), 135–144. (2002) Zbl1008.06011MR1935113
- 10.1007/BF01190765, Algebra Universalis 33 (1995), 40–67. (1995) MR1303631DOI10.1007/BF01190765

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.