| Title:
|
Relatively coarse sequential convergence (English) |
| Author:
|
Frič, Roman |
| Author:
|
Zanolin, Fabio |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
1997 |
| Pages:
|
395-408 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion of the group of rational numbers (equipped with the usual metric convergence) is complete; (ii) there are exactly $\exp \exp \omega $ such completions; (iii) the real line is the only one of them the convergence of which is Fréchet. Analogous results hold for the relatively coarse dense field precompletions of the subfield of all complex numbers both coordinates of which are rational numbers. (English) |
| Keyword:
|
Sequential convergence: compatible- |
| Keyword:
|
coarse- |
| Keyword:
|
relatively coarse- |
| Keyword:
|
FLUSH-group |
| Keyword:
|
FLUSH-ring |
| Keyword:
|
completion |
| Keyword:
|
extension |
| MSC:
|
16W99 |
| MSC:
|
20K35 |
| MSC:
|
20K45 |
| MSC:
|
54A20 |
| MSC:
|
54H11 |
| MSC:
|
54H13 |
| idZBL:
|
Zbl 0897.54002 |
| idMR:
|
MR1461420 |
| . |
| Date available:
|
2009-09-24T10:06:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127365 |
| . |
| Reference:
|
[1] Banaschewski, B.: Minimal topological algebras.Math. Ann. 211 (1974), 107–114. Zbl 0275.46043, MR 0357520, 10.1007/BF01344165 |
| Reference:
|
[2] Contessa, M. and F. Zanolin: On some remarks about a not completable convergence ring. General Topology and its Relations to Modern Analysis and Algebra V.Proc. Fifth Prague Topological Sympos., Prague 1981), Heldermann Verlag, Berlin, 1982, pp. 107–114. MR 0698397 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[9] Frič, R. and F. Zanolin: Coarse convergence groups.Convergence Structures 1984 (Proc. Conf. on Convergence, Bechyně 1984), Akademie-Verlag Berlin, 1985, pp. 107–114. MR 0835476 |
| Reference:
|
[10] Frič, R. and F. Zanolin: Sequential convergence in free groups.Rend. Istit. Mat. Univ. Trieste 18 (1986), 200–218. MR 0928331 |
| Reference:
|
[11] Frič, R. and F. Zanolin: Coarse sequential convergence in groups, etc..Czechoslovak Math. J. 40 (1990), 459–467. MR 1065025 |
| Reference:
|
[12] Frič, R. and F. Zanolin: Strict completions of $L_0^*$-groups.Czechoslovak Math. J. 42 (1992), 589–598. MR 1182190 |
| Reference:
|
[13] Frič, R. and F. Zanolin: Minimal pseudotopological groups. (Presented at Categorical Topology and its Relation to Analysis, Algebra and Combinatorics, Praha 1988, unpublished.).(19$\infty $). |
| Reference:
|
[14] Jakubík, J.: On convergence in linear spaces.Mat.-Fyz. Časopis Slovensk. Akad. Vied 6 (1956), 57–67. (Slovak. Russian summary) MR 0084718 |
| Reference:
|
[15] Novák, J.: On completion of convergence commutative groups.General Topology and its Relations to Modern Analysis and Algebra III (Proc. Third Prague Topological Sympos., 1971), Academia, Praha, 1972, pp. 335–340. MR 0365451 |
| Reference:
|
[16] Simon, P. and F. Zanolin: A coarse convergence group need not be precompact.Czechoslovak Math. J. 37 (1987), 480–486. MR 0904772 |
| . |
