| Title:
|
Radicals and complete distributivity in relatively normal lattices (English) |
| Author:
|
Rachůnek, Jiří |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
128 |
| Issue:
|
4 |
| Year:
|
2003 |
| Pages:
|
401-410 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
relatively normal lattice |
| Keyword:
|
algebraic lattice |
| Keyword:
|
complete distributivity |
| Keyword:
|
closed element |
| Keyword:
|
radical |
| MSC:
|
06D15 |
| MSC:
|
06D20 |
| MSC:
|
06D35 |
| MSC:
|
06F05 |
| MSC:
|
06F15 |
| idZBL:
|
Zbl 1052.06009 |
| idMR:
|
MR2032477 |
| DOI:
|
10.21136/MB.2003.134005 |
| . |
| Date available:
|
2009-09-24T22:11:18Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134005 |
| . |
| Reference:
|
[1] Anderson, M., Feil, T.: Lattice-Ordered Groups.D. Reidel Publ., Dordrecht, 1988. MR 0937703 |
| Reference:
|
[2] Balbes, R., Dwinger, P.: Distributive Lattices.Univ. of Missouri Press, Columbia, Missouri, 1974. MR 0373985 |
| Reference:
|
[3] Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés.Springer, Berlin, 1977. MR 0552653 |
| Reference:
|
[4] Chang, C. C.: Algebraic analysis of many valued logic.Trans. Amer. Math. Soc. 88 (1958), 467–490. MR 0094302, 10.1090/S0002-9947-1958-0094302-9 |
| Reference:
|
[5] Cignoli, R. O. L., D’Ottaviano, I. M. L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning.Kluwer Acad. Publ., Dordrecht, 2000. MR 1786097 |
| Reference:
|
[6] Di Nola, A., Georgescu, G., Sessa, S.: Closed ideals of $MV$-algebras.Advances in Contemporary Logic and Computer Science, Contemp. Math., vol. 235, AMS, Providence, 1999, pp. 99–112. MR 1721208 |
| Reference:
|
[7] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures.Kluwer Acad. Publ., Dordrecht, 2000. MR 1861369 |
| Reference:
|
[8] Filipoiu, A., Georgescu, G.: On values in relatively normal lattices.Discrete Math. 161 (1996), 87–100. MR 1420523, 10.1016/0012-365X(95)00221-H |
| Reference:
|
[9] Georgescu, G., Iorgulescu, A.: 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. |
| Reference:
|
[10] Georgescu, G., Iorgulescu, A.: Pseudo-$MV$ algebras.Multiple Valued Logic 6 (2001), 95–135. MR 1817439 |
| Reference:
|
[11] Glass, A. M. W.: Partially Ordered Groups.World Scientific, Singapore, 1999. Zbl 0933.06010, MR 1791008 |
| Reference:
|
[12] Hart, J. B., Tsinakis, C.: Decompositions for relatively normal lattices.Trans. Amer. Math. Soc. 341 (1994), 519–548. MR 1211409, 10.1090/S0002-9947-1994-1211409-2 |
| Reference:
|
[13] Martinez, J.: Archimedean lattices.Algebra Universalis 3 (1973), 247–260. Zbl 0317.06004, MR 0349503, 10.1007/BF02945124 |
| Reference:
|
[14] Monteiro, A.: L’arithmétique des filtres et les espaces topologiques.De Segundo Symp. Mathematicas-Villavicencio, Mendoza, Buenos Aires, 1954, pp. 129–162. Zbl 0058.38503, MR 0074805 |
| Reference:
|
[15] Monteiro, A.: L’arithmétique des filtres et les espaces topologiques I–II. Notas de Logica Mathematica, vol. 29–30, 1974.. |
| Reference:
|
[16] Paseka, J.: Linear finitely separated objects of subcategories of domains.Math. Slovaca 46 (1996), 457–490. Zbl 0890.06007, MR 1451036 |
| Reference:
|
[17] Rachůnek, J.: A non-commutative generalization of $MV$-algebras.Czechoslovak Math. J. 52 (2002), 255–273. Zbl 1012.06012, MR 1905434, 10.1023/A:1021766309509 |
| Reference:
|
[18] Rachůnek, J.: Prime spectra of non-commutative generalizations of $MV$-algebras.Algebra Universalis 48 (2002), 151–169. Zbl 1058.06015, MR 1929902, 10.1007/PL00012447 |
| Reference:
|
[19] Rachůnek, J.: Radicals in non-commutative generalizations of $MV$-algebras.Math. Slovaca 52 (2002), 135–144. Zbl 1008.06011, MR 1935113 |
| Reference:
|
[20] Snodgrass, J. T., Tsinakis, C.: The finite basis theorem for relative normal lattices.Algebra Universalis 33 (1995), 40–67. MR 1303631, 10.1007/BF01190765 |
| . |
