| Title:
|
Dimension in algebraic frames, II: Applications to frames of ideals in $C(X)$ (English) |
| Author:
|
Martínez, Jorge |
| Author:
|
Zenk, Eric R. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
607-636 |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper continues the investigation into Krull-style dimensions in algebraic frames. Let $L$ be an algebraic frame. $\operatorname{dim}(L)$ is the supremum of the lengths $k$ of sequences $p_0< p_1< \cdots <p_k$ of (proper) prime elements of $L$. Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of $L$ in terms of the dimensions of certain boundary quotients of $L$. This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily compact. This result applies to the frame $\Cal C_z(X)$ of all $z$-ideals of $C(X)$, provided the underlying Tychonoff space $X$ is Lindelöf. If the space $X$ is compact, then it is shown that the dimension of $\Cal C_z(X)$ is at most $n$ if and only if $X$ is scattered of Cantor-Bendixson index at most $n+1$. If $X$ is the topological union of spaces $X_i$, then the dimension of $\Cal C_z(X)$ is the supremum of the dimensions of the $\Cal C_z(X_i)$. This and other results apply to the frame of all $d$-ideals $\Cal C_d(X)$ of $C(X)$, however, not the characterization in terms of boundaries. An explanation of this is given within, thus marking some of the differences between these two frames and their dimensions. (English) |
| Keyword:
|
dimension of a frame |
| Keyword:
|
$z$-ideals |
| Keyword:
|
scattered space |
| Keyword:
|
natural typing of open sets |
| MSC:
|
03G10 |
| MSC:
|
06D22 |
| MSC:
|
16P60 |
| MSC:
|
54B35 |
| MSC:
|
54C30 |
| idZBL:
|
Zbl 1121.06009 |
| idMR:
|
MR2259494 |
| . |
| Date available:
|
2009-05-05T16:53:51Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119554 |
| . |
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