| Title:
|
An observation on Krull and derived dimensions of some topological lattices (English) |
| Author:
|
Rostami, M. |
| Author:
|
Rodrigues, Ilda I. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
329-334 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(L, \le)$, be an algebraic lattice. It is well-known that $(L, \le)$ with its topological structure is topologically scattered if and only if $(L, \le)$ is ordered scattered with respect to its algebraic structure. In this note we prove that, if $L$ is a distributive algebraic lattice in which every element is the infimum of finitely many primes, then $L$ has Krull-dimension if and only if $L$ has derived dimension. We also prove the same result for $\operatorname{\it spec} L$, the set of all prime elements of $L$. Hence the dimensions on the lattice and on the spectrum coincide. (English) |
| Keyword:
|
Krull dimension |
| Keyword:
|
derived dimension |
| Keyword:
|
inductive dimension |
| Keyword:
|
scattered spaces and algebraic lattices |
| MSC:
|
06-xx |
| MSC:
|
06B30 |
| MSC:
|
16U20 |
| MSC:
|
54C25 |
| MSC:
|
54G12 |
| idZBL:
|
Zbl 1249.06010 |
| idMR:
|
MR2876953 |
| . |
| Date available:
|
2011-12-16T15:21:08Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141779 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| . |
