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

Jorge Martinez; Eric R. Zenk

Commentationes Mathematicae Universitatis Carolinae (2005)

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

Abstract

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This paper continues the investigation into Krull-style dimensions in algebraic frames. Let L be an algebraic frame. dim ( L ) is the supremum of the lengths k of sequences p 0 < p 1 < ⋯ < 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 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 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 𝒞 z ( X ) is the supremum of the dimensions of the 𝒞 z ( X i ) . This and other results apply to the frame of all d -ideals 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.

How to cite

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Martinez, 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 -

References

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