Strongly fixed ideals in C ( L ) and compact frames

A. A. Estaji; A. Karimi Feizabadi; M. Abedi

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 1, page 1-12
  • ISSN: 0044-8753

Abstract

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Let C ( L ) be the ring of real-valued continuous functions on a frame L . In this paper, strongly fixed ideals and characterization of maximal ideals of C ( L ) which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of C ( L ) , is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than spatiality and they are equivalent in the case of conjunctive frames. Assuming Axiom of Choice, compact frames are weakly spatial.

How to cite

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Estaji, A. A., Karimi Feizabadi, A., and Abedi, M.. "Strongly fixed ideals in $ C (L)$ and compact frames." Archivum Mathematicum 051.1 (2015): 1-12. <http://eudml.org/doc/270063>.

@article{Estaji2015,
abstract = {Let $C(L)$ be the ring of real-valued continuous functions on a frame $L$. In this paper, strongly fixed ideals and characterization of maximal ideals of $C(L)$ which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of $C(L)$, is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than spatiality and they are equivalent in the case of conjunctive frames. Assuming Axiom of Choice, compact frames are weakly spatial.},
author = {Estaji, A. A., Karimi Feizabadi, A., Abedi, M.},
journal = {Archivum Mathematicum},
keywords = {frame; ring of real-valued continuous functions; weakly spatial frame; fixed and strongly fixed ideal},
language = {eng},
number = {1},
pages = {1-12},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Strongly fixed ideals in $ C (L)$ and compact frames},
url = {http://eudml.org/doc/270063},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Estaji, A. A.
AU - Karimi Feizabadi, A.
AU - Abedi, M.
TI - Strongly fixed ideals in $ C (L)$ and compact frames
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 1
SP - 1
EP - 12
AB - Let $C(L)$ be the ring of real-valued continuous functions on a frame $L$. In this paper, strongly fixed ideals and characterization of maximal ideals of $C(L)$ which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of $C(L)$, is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than spatiality and they are equivalent in the case of conjunctive frames. Assuming Axiom of Choice, compact frames are weakly spatial.
LA - eng
KW - frame; ring of real-valued continuous functions; weakly spatial frame; fixed and strongly fixed ideal
UR - http://eudml.org/doc/270063
ER -

References

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  1. Banaschewski, B., 10.1007/BF00334858, Order 2 (2) (1985), 211–213. (1985) Zbl0576.06010MR0815866DOI10.1007/BF00334858
  2. Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered algebraic structures, (Curacoa, 1995), Kluwer Acad. Publ., Dordrecht, 1997, pp. 123–148. (1997) Zbl0870.06017MR1445110
  3. Banaschewski, B., The real numbers in pointfree topology, Textos de Mathemática (Série B), Vol. 12, University of Coimbra, Departmento de Mathemática, Coimbra, 1997. (1997) Zbl0891.54009MR1621835
  4. Banaschewski, B., Gilmour, C.R.A., Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 577–587. (1996) Zbl0881.54018MR1426922
  5. Dube, T., 10.1007/s00012-009-2093-5, Algebra Universalis 60 (2009), 145–162. (2009) Zbl1186.06006MR2491419DOI10.1007/s00012-009-2093-5
  6. Dube, T., 10.1007/s10474-010-0024-8, Acta Math. Hungar. 129 (2010), 205–226. (2010) Zbl1299.06021MR2737723DOI10.1007/s10474-010-0024-8
  7. Dube, T., 10.1007/s00012-011-0127-2, Algebra Universalis 65 (2011), 263–276. (2011) MR2793399DOI10.1007/s00012-011-0127-2
  8. Dube, T., Real ideal in pointfree rings of continuous functions, Bull. Asut. Math. Soc. 83 (2011), 338–352. (2011) MR2784791
  9. Dube, T., Extending and contracting maximal ideals in the function rings of pointfree topology, Bull. Math. Soc. Sci. Math. Roumanie 55 (103) (4) (2012), 365–374. (2012) Zbl1274.06038MR2963403
  10. Ebrahimi, M.M., Karimi, A., Pointfree prime representation of real Riesz maps, Algebra Universalis 2005 (54), 291–299. (1954) MR2219412
  11. Estaji, A.A., Feizabadi, A. Karimi, Abedi, M., Zero sets in pointfree topology and strongly z -ideals, accepted in Bulletin of the Iranian Mathematical Society. 
  12. Garcáa, J. Gutiérrez, Picado, J., How to deal with the ring of (continuous) real-valued functions in terms of scales, Proceedings of the Workshop in Applied Topology WiAT'10, 2010, pp. 19–30. (2010) 
  13. Gillman, L., Jerison, M., Rings of continuous functions, Springer Verlag, 1979. (1979) MR0407579
  14. Johnstone, P.T., Stone Spaces, Cambridge Univ. Press, 1982. (1982) Zbl0499.54001MR0698074
  15. Paseka, J., Conjunctivity in quantales, Arch. Math. (Brno) 24 (4) (1988), 173–179. (1988) Zbl0663.06011MR0983235
  16. Paseka, J., Šmarda, B., T 2 -frames and almost compact frames, Czechoslovak Math. J. 42 (3) (1992), 385–402. (1992) MR1179302
  17. Picado, J., Pultr, A., Frames and Locales: topology without points, Frontiers in Mathematics, Springer, Basel, 2012. (2012) Zbl1231.06018MR2868166
  18. Simmons, H., The lattice theoretical part of topological separation properties, Proc. Edinburgh Math. Soc. (2) 21 (1978), 41–48. (1978) MR0493959

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