• Korean Journal of Mathematics
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• v.15 no.2
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• pp.87-100
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• 2007

We introduce a concept of countably strong inclusions ${\triangleleft}$ and that of ${\triangleleft}-{\sigma}$-ideals and prove that the subframe $S({\triangleleft})$ of the frame ${\sigma}IdL$ of ${\sigma}$-ideals is a Lindel$\ddot{o}$fication of a frame L. We also deal with conditions for which the converse holds. We show that any countably approximating regular $D({\aleph}_1)$ frame has the smallest countably strong inclusion and any frame which has the smallest $D({\aleph}_1)$ Lindel$\ddot{o}$fication is countably approximating regular $D({\aleph}_1)$.

• Journal for History of Mathematics
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• v.16 no.1
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• pp.63-72
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• 2003

This paper is a sequel to [11]. We introduce $\sigma$-coherent frames, stably countably approximating frames and strongly Lindelof frames, and show that a stably countably approximating frame is a strongly Lindelof frame. We also show that a complete chain in a Lindelof frame if and only if it is a strongly Lindelof frame by using the concept of strong convergence of filters. Finally, using the concepts of super compact frames and filter compact frames, we introduce an example of a strongly Lindelof frame which is not a stably countably approximating frame.

• Communications of the Korean Mathematical Society
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• v.17 no.2
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• pp.295-308
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• 2002

Using the Countably way below relation, we show that the category $\sigma$-CFrm of $\sigma$-coherent frames and $\sigma$-coherent homomorphisms is coreflective n the category Frm of frames and frame homomorphisms. Introducting the concept of stably countably approximating frames which are exactly retracts of $\sigma$-coherent frames, it is shown that the category SCAFrm of stably countably approximating frames and $\sigma$-proper frame homomorphisms is coreflective in Frm. Finally we introduce strongly Lindelof frames and show that they are precisely lax retracts of $\sigma$-coherent frames.