| Title:
|
On some types of radical classes (English) |
| Author:
|
Jakubík, Ján |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
3 |
| Year:
|
2008 |
| Pages:
|
833-848 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\frak m$ be an infinite cardinal. We denote by $C_\frak m$ the collection of all $\frak m$-representable Boolean algebras. Further, let $C_\frak m^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_\frak m$. In this paper we prove that $C_\frak m^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras. (English) |
| Keyword:
|
Boolean algebra |
| Keyword:
|
generalized Boolean algebra |
| Keyword:
|
$\frak m$-representability |
| Keyword:
|
lattice ordered group |
| Keyword:
|
generalized $MV$-algebra |
| Keyword:
|
radical class |
| MSC:
|
06D35 |
| MSC:
|
06E05 |
| MSC:
|
06F20 |
| idZBL:
|
Zbl 1174.06322 |
| idMR:
|
MR2455941 |
| . |
| Date available:
|
2010-07-20T14:11:52Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140424 |
| . |
| Reference:
|
[1] Chang, C. C.: On the representation of $\alpha$-complete Boolean algebras.Trans. Amer. Math. Soc. 85 (1957), 208-218. Zbl 0080.25502, MR 0086792 |
| Reference:
|
[2] Conrad, P.: $K$-radical classes of lattice ordered groups.In: Proc. Conf. Carbondale, Lecture Notes Math 848 Springer Verlag New York (1981), 186-207. Zbl 0455.06010, MR 0613186 |
| Reference:
|
[3] Conrad, P., Darnel, M. R.: Subgroups and hulls of Specker lattice ordered groups.Czech. Math. J. 51 (2001), 395-413. Zbl 0978.06011, MR 1844319, 10.1023/A:1013759300701 |
| Reference:
|
[4] Darnel, M.: Closure operators on radicals of lattice ordered groups.Czech. Math. J. 37 (1987), 51-64. MR 0875127 |
| Reference:
|
[5] Dvurečenskij, A.: Pseudo $MV$-algebras are intervals in $\ell$-groups.J. Austral. Math. Soc. 72 (2002), 427-445. MR 1902211, 10.1017/S1446788700036806 |
| Reference:
|
[6] Georgescu, G., Iorgulescu, A.: Pseudo $MV$-algebras: a noncommutative extension of $MV$-algebras.Proc. Fourth Int. Symp. Econ. Inf., Bucharest (1999), 961-968. Zbl 0985.06007, MR 1730100 |
| Reference:
|
[7] Georgescu, G., Iorgulescu, A.: Pseudo $MV$-algebras.Multiple Valued Logic 6 (2001), 95-135. Zbl 1014.06008, MR 1817439 |
| Reference:
|
[8] Jakubík, J.: Radical mappings and radical classes of lattice ordered groups.Symposia Math. 21 Academic Press New York-London (1977), 451-477. MR 0491397 |
| Reference:
|
[9] Jakubík, J.: Radical classes of generalized Boolean algebras.Czech. Math. J. 48 (1998), 253-268. MR 1624315, 10.1023/A:1022885303504 |
| Reference:
|
[10] Jakubík, J.: Radical classes of $MV$-algebras.Czech. Math. J. 49 (1999), 191-211. MR 1676805, 10.1023/A:1022428713092 |
| Reference:
|
[11] Jakubík, J.: Direct product decompositions of pseudo $MV$-algebras.Archivum Math. 37 (2001), 131-142. MR 1838410 |
| Reference:
|
[12] Jakubík, J.: Torsion classes of Specker lattice ordered groups.Czech. Math. J. 52 (2002), 469-482. MR 1923254, 10.1023/A:1021711326115 |
| Reference:
|
[13] Loomis, L. H.: On the representation of $\sigma$-complete Boolean algebras.Bull. Amer. Math. Soc. 53 (1947), 757-760. Zbl 0033.01103, MR 0021084, 10.1090/S0002-9904-1947-08866-2 |
| Reference:
|
[14] Pierce, R. S.: Representation theorems for certain Boolean algebras.Proc. Amer. Math. Soc. 10 (1959), 42-50. Zbl 0091.03102, MR 0106862, 10.1090/S0002-9939-1959-0106862-6 |
| Reference:
|
[15] Rachůnek, J.: A non-commutative generalization of $MV$-algebras.Czech. Math. J. 52 (2002), 255-273. MR 1905434, 10.1023/A:1021766309509 |
| Reference:
|
[16] Scott, D.: A new characterization of $\alpha$-representable Boolean algebras.Bull. Amer. Math. Soc. 61 (1955), 522-523. |
| Reference:
|
[17] E. C. Smith, Jr.: A distributivity condition for Boolean algebras.Ann. Math. 64 (1956), 551-561. Zbl 0074.02105, MR 0086047, 10.2307/1969602 |
| Reference:
|
[18] Sikorski, R.: On the representation of Boolean algebras as fields of set.Fund. Math. 35 (1958), 247-258. MR 0028374, 10.4064/fm-35-1-247-258 |
| Reference:
|
[19] Sikorski, R.: Distributivity and representability.Fund. Math. 48 (1959), 95-103. MR 0109799, 10.4064/fm-48-1-91-103 |
| Reference:
|
[20] Sikorski, R.: Boolean Algebras.Second Edition Springer Verlag Berlin-Göttingen-Heidelberg-New York (1964). Zbl 0123.01303 |
| Reference:
|
[21] Ton, Dao Rong: Product radical classes of $\ell$-groups.Czech. Math. J. 42 (1992), 129-142. MR 1152176 |
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