Ompactification of Completely Regular Frames based on their Cozero Part

Authors
Mostafa Abedi 1
Ali Akbar Estaji 2
1 Esfarayen University of Technology
2 Hakim Sabzevari University
10.29252/mmr.5.1.53
Abstract
Let L be a frame. We denoted the set of all regular ideals of cozL by rId(cozL) . The aim of this paper is to study these ideals. For a frame L , we show that rId(cozL) is a compact completely regular frame and the map jc : rId(cozL)→L given by jc (I)=⋁I is a compactification of L which is isomorphism to its Stone–Čech compactification and is proved that jc have a right adjoint rc : L → rId(cozL) , given by rc(a)={x∈cozL : x≺≺a}. Moreover we identify prime and compact elements of rId(cozL) and we investigate the relation between regular ideals of cozL and P-frames. In addition it is shown that a frame L is a P-frame if and only if any ideal of cozL is regular../files/site1/files/51/%D8%B9%D8%A7%D8%A8%D8%AF%DB%8C.pdf
Keywords
Compactification
Completely regular frame
Regular ideal
Cozero part of a frame
Stone–Čech compactification
1. Johnstone P. T., Stone spaces, vol 3, Cambridge University Press, (1986). 2. Banaschewski B., Compactification of frames, Mathematische Nachrichten, 149 (1) (1990) 105-115. 3. Banaschewski B. and Mulvey C., Stone-Čech compactification of locales I, Houston Journal of Mathematics, 6 (3) (1980) 301-312. 4. Banaschewski B. and Mulvey C., Stone-Čech compactification of locales II, Journal of Pure and Applied Algebra, 33 (2) (1984) 107-122. 5. Banaschewski B. and Mulvey C. J., The Stone–Čech compactification of locales III, Journal of Pure and Applied Algebra, 185 (1) (2003) 25-33. 6. Banaschewski B., The real numbers in pointfree topology, Textos Mat, Sér B 12, (1997). 7. Estaji A., Karimi Feizabadi A., and Abedi M, Zero sets in pointfree topology and strongly z-ideals, Bulletin of the Iranian Mathematical Society, 41 (5) (2015) 1071-1084. 8. Picado J. and Pultr A., Frames and Locales: topology without points, Springer Science & Business Media, (2011). 9 . Gillman L. and Jerison M., Rings of continuous functions. Springer, (1976). 10. Ebrahimi M. M. and Mahmoudi M., Frame, Technical Report, Shahid Beheshti University, (1995). 11. Banaschewski B. and Gilmour C., Pseudocompactness and the cozero part of a frame, Commentationes Mathematicae Universitatis Carolinae , 37(1996) 577-588. 12. Azarpanah F., Karamzadeh O., and Aliabad A. R., On z◦-ideals in C (X), Fundamenta Mathematicae, 160 (1) (1999) 15-25. 13. Azarpanah F., On almost P-space, Far East J Math Sci Special, (2000) 121-132. 14. Dube T., Concerning P-frames, essential P-frames, and strongly zero-dimensional frames, Algebra universalis, 61 (1) (2009) 115-138.