# Dimension in algebraic frames, II: Applications to frames of ideals in $C\left(X\right)$

Commentationes Mathematicae Universitatis Carolinae (2005)

- Volume: 46, Issue: 4, page 607-636
- ISSN: 0010-2628

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topMartinez, Jorge, and Zenk, Eric R.. "Dimension in algebraic frames, II: Applications to frames of ideals in $C(X)$." Commentationes Mathematicae Universitatis Carolinae 46.4 (2005): 607-636. <http://eudml.org/doc/249551>.

@article{Martinez2005,

abstract = {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 $\mathcal \{C\}_z(X)$ 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 $\mathcal \{C\}_z(X)$ 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 $\mathcal \{C\}_z(X)$ is the supremum of the dimensions of the $\mathcal \{C\}_z(X_i)$. This and other results apply to the frame of all $d$-ideals $\mathcal \{C\}_d(X)$ 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.},

author = {Martinez, Jorge, Zenk, Eric R.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {dimension of a frame; $z$-ideals; scattered space; natural typing of open sets; dimension of a frame; -ideals; scattered space},

language = {eng},

number = {4},

pages = {607-636},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Dimension in algebraic frames, II: Applications to frames of ideals in $C(X)$},

url = {http://eudml.org/doc/249551},

volume = {46},

year = {2005},

}

TY - JOUR

AU - Martinez, Jorge

AU - Zenk, Eric R.

TI - Dimension in algebraic frames, II: Applications to frames of ideals in $C(X)$

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2005

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 46

IS - 4

SP - 607

EP - 636

AB - 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 $\mathcal {C}_z(X)$ 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 $\mathcal {C}_z(X)$ 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 $\mathcal {C}_z(X)$ is the supremum of the dimensions of the $\mathcal {C}_z(X_i)$. This and other results apply to the frame of all $d$-ideals $\mathcal {C}_d(X)$ 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.

LA - eng

KW - dimension of a frame; $z$-ideals; scattered space; natural typing of open sets; dimension of a frame; -ideals; scattered space

UR - http://eudml.org/doc/249551

ER -

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