On c-completely regular frames
Publish place: Journal of Frame and Matrix Theory، Vol: 1، Issue: 1
Publish Year: 1402
Type: Journal paper
Language: English
View: 78
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Document National Code:
JR_JFMT-1-1_004
Index date: 25 November 2024
On c-completely regular frames abstract
Motivated by definitions of countable completely regular spaces and completely below relations of frames, we define what we call a c-completely below relation, denoted by \prec\!\!\prec_c, in between two elements of a frame. We show that a\prec\!\!\prec_c b for two elements a, b of a frame L if and only if there is \alpha\in\mathcal{R}L such that \coz\alpha\wedge a=0 and \coz(\alpha-{\bf1})\leq b where the set \{r\in\mathbb{R} : \coz(\alpha-{\bf r})\ne 1\} is countable. We say a frame L is a c-completely regular frame if a=\bigvee \limits_{x\prec\!\!\prec_ca}x for any a\in L. It is shown that a frame L is a c-completely regular frame if and only if it is a zero-dimensional frame. An ideal I of a frame L is said to be c-completely regular if a\in I implies a\prec\!\!\prec_c b for some b\in I. The set of all c-completely regular ideals of a frame L, denoted by {\mathrm{c-CRegId}}(L), is a compact regular frame and it is a compactification for L whenever it is a c-completely regular frame. We denote this compactification by \beta_cL and it is isomorphic to the frame \beta_0L, that is, Stone-Banaschewski compactification of L. Finally, we show that open and closed quotients of a c-completely regular frame are c-completely regular.
On c-completely regular frames Keywords:
frame , c-completely regular frame and space , c-completely below relation , c-completely regular ideals , zero-dimensional frame , compactification of frame
On c-completely regular frames authors
Mostafa Abedi
Esfarayen University of Technology, Esfarayen, North Khorasan, Iran.
Ali Akbar Estaji
Ali Akbar Estaji, Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.