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Keywords:
dimension of a frame; $z$-ideals; scattered space; natural typing of open sets
dimension of a frame; $z$-ideals; scattered space; natural typing of open sets
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.
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