| Title:
|
Higher degrees of distributivity in $MV$-algebras (English) |
| Author:
|
Jakubík, Ján |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
53 |
| Issue:
|
3 |
| Year:
|
2003 |
| Pages:
|
641-653 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we deal with the of an $MV$-algebra $\mathcal A$, where $\alpha $ and $\beta $ are nonzero cardinals. It is proved that if $\mathcal A$ is singular and $(\alpha,2)$-distributive, then it is . We show that if $\mathcal A$ is complete then it can be represented as a direct product of $MV$-algebras which are homogeneous with respect to higher degrees of distributivity. (English) |
| Keyword:
|
$MV$-algebra |
| Keyword:
|
archimedean $MV$-algebra |
| Keyword:
|
completeness |
| Keyword:
|
singular $MV$-algebra |
| Keyword:
|
higher degrees of distributivity |
| MSC:
|
06D10 |
| MSC:
|
06D35 |
| MSC:
|
06F20 |
| idZBL:
|
Zbl 1080.06014 |
| idMR:
|
MR2000060 |
| . |
| Date available:
|
2009-09-24T11:05:24Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127830 |
| . |
| Reference:
|
[1] D. Byrd and J. T. Lloyd: Closed subgroups and complete distributivity in lattice-ordered groups.Math. Z. 101 (1967), 123–130. MR 0218284, 10.1007/BF01136029 |
| Reference:
|
[2] R. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-valued Reasoning. Trends in Logic, Studia Logica Library, Vol. 7.Kluwer Academic Publishers, Dordrecht, 2000. MR 1786097 |
| Reference:
|
[3] P. Conrad: The relationship between the radical of a lattice ordered group and complete distributivity.Pacific J. Math. 14 (1964), 494–499. Zbl 0122.03701, MR 0166279, 10.2140/pjm.1964.14.493 |
| Reference:
|
[4] P. Conrad: Lattice Ordered Groups.Tulane University, 1970. Zbl 0258.06011 |
| Reference:
|
[5] L. Fuchs: Partially Ordered Algebraic Systems.Pergamon Press, Oxford, 1963. Zbl 0137.02001, MR 0171864 |
| Reference:
|
[6] J. Jakubík: Higher degrees of distributivity in lattices and lattice ordered groups.Czechoslovak Math. J. 18 (1968), 356–376. MR 0225690 |
| Reference:
|
[7] J. Jakubík: Distributivity in lattice ordered groups.Czechoslovak Math. J. 22 (1972), 108–125. MR 0325487 |
| Reference:
|
[8] J. Jakubík: Direct product decompositions of $MV$-algebras.Czechoslovak Math. J. 44 (1994), 725–739. |
| Reference:
|
[9] J. Jakubík: On complete $MV$-algebras.Czechoslovak Math. J. 45 (1995), 473–480. MR 1344513 |
| Reference:
|
[10] J. Jakubík: On archimedean $MV$-algebras.Czechoslovak Math. J. 48 (1998), 575–582. MR 1637871, 10.1023/A:1022436113418 |
| Reference:
|
[11] J. Jakubík: Complete generators and maximal completions of $MV$-algebras.Czechoslovak Math. J. 48 (1998), 597–608. MR 1637863, 10.1023/A:1022440214327 |
| Reference:
|
[12] J. T. Lloyd: Complete distributivity in certain infinite permutation groups.Michigan Math. J. 14 (1967), 393–400. Zbl 0167.30202, MR 0219462, 10.1307/mmj/1028999839 |
| Reference:
|
[13] R. S. Pierce: Distributivity in Boolean algebras.Pacific J. Math. 7 (1957), 983–992. Zbl 0086.02803, MR 0089180, 10.2140/pjm.1957.7.983 |
| Reference:
|
[14] R. Sikorski: Boolean Algebras, Second Edition.Springer Verlag, Berlin, 1964. MR 0126393 |
| Reference:
|
[15] F. Šik: Über subdirekte Summen geordneter Gruppen.Czechoslovak Math. J. 10 (1960), 400–424. MR 0123626 |
| Reference:
|
[16] E. C. Weinberg: Higher degrees of distributivity in lattices of continuous functions.Trans. Amer. Math. Soc. 104 (1962), 334–346. Zbl 0105.09401, MR 0138569, 10.1090/S0002-9947-1962-0138569-8 |
| Reference:
|
[17] E. C. Weinberg: Completely distributive lattice ordered groups.Pacific J. Math. 12 (1962), 1131–1148. Zbl 0111.24301, MR 0147549, 10.2140/pjm.1962.12.1131 |
| . |
