# Article

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Keywords:
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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