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Keywords:
ring of real-valued continuous functions on a frame; coz-disjoint; coz-dense and coz-spatial frames; zero sets in pointfree topology; $z$-ideal; strongly $z$-ideal
ring of real-valued continuous functions on a frame; coz-disjoint; coz-dense and coz-spatial frames; zero sets in pointfree topology; $z$-ideal; strongly $z$-ideal
Summary:
Let $\mathcal{R}L$ be the ring of real-valued continuous functions on a frame $L$. The aim of this paper is to study the relation between minimality of ideals $I$ of $\mathcal{R}L$ and the set of all zero sets in $L$ determined by elements of $I$. To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$, it is proved that the $f$-ring $\mathcal{R}L$ is isomorphic to the $f$-ring $ C(\Sigma L)$ of all real continuous functions on the topological space $\Sigma L$. Finally, a one-one correspondence is presented between the set of isolated points of $\Sigma L$ and the set of atoms of $L$.
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