• 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.

• 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.

• Journal for History of Mathematics
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• v.13 no.2
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• pp.95-104
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• 2000

This paper is a sequel to [24]. It is well known that the order structure plays the important role in the study of various mathematical structures. In 1972, Scott has introduced a concept of continuous lattices and has shown the equivalence between continuous lattices and injective $T_0-spaces$. There have been many efforts made to generalize continuous lattices and extend corresponding properties to them. We introduce another class of frames, namely countably approximating frames, generalizing continuous frames and study its basic properties.

• 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 $D({\aleph}_1)$.