Mathematica Bohemica, Vol. 142, No. 3, pp. 323-336, 2017

When spectra of lattices of $z$-ideals are Stone-Čech compactifications

Themba Dube

Received January 23, 2016.  First published January 23, 2017.

Abstract:  Let $X$ be a completely regular Hausdorff space and, as usual, let $C(X)$ denote the ring of real-valued continuous functions on $X$. The lattice of $z$-ideals of $C(X)$ has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $\beta X$ precisely when $X$ is a $P$-space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$-ideal if whenever two elements have the same annihilator and one of the elements belongs to the ideal, then so does the other. We characterize when the spectrum of the lattice of $d$-ideals of $C(X)$ is the Stone-Čech compactification of the largest dense sublocale of the locale determined by $X$. It is precisely when the closure of every open set of $X$ is the closure of some cozero-set of $X$.
Keywords:  completely regular frame; coherent frame; $z$-ideal; $d$-ideal; Stone-Čech compactification; booleanization
Classification MSC:  06D22, 54E17, 13A15, 18A40
DOI:  10.21136/MB.2017.0009-16

[1] R. N. Ball, J. Walters-Wayland: $C$- and $C^*$-quotients in pointfree topology. Diss. Math. (Rozprawy Matematyczne) 412 (2002), 1-62. DOI 10.4064/dm412-0-1 | MR 1952051 | Zbl 1012.54025
  [2] B. Banaschewski: Pointfree topology and the spectra of $f$-rings. Ordered algebraic structures. Proc. Curaçao Conf., 1995 (C. W. Holland et al., eds.). Kluwer Academic Publishers, Dordrecht (1997), 123-148. DOI 10.1007/978-94-011-5640-0_5 | MR 1445110 | Zbl 0870.06017
  [3] B. Banaschewski: The Real Numbers in Pointfree Topology. Texts in Mathematics. Series B, vol. 12. Departamento de Matemática, Universidade de Coimbra, Coimbra (1997). MR 1621835 | Zbl 0891.54009
  [4] T. Dube: Concerning $P$-frames, essential $P$-frames, and strongly zero-dimensional frames. Algebra Univers. 61 (2009), 115-138. DOI 10.1007/s00012-009-0006-2 | MR 2551788 | Zbl 1190.06007
  [5] T. Dube: Notes on pointfree disconnectivity with a ring-theoretic slant. Appl. Categ. Struct. 18 (2010), 55-72. DOI 10.1007/s10485-008-9162-3 | MR 2586718 | Zbl 1188.06005
  [6] T. Dube: On the ideal of functions with compact support in pointfree function rings. Acta Math. Hung. 129 (2010), 205-226. DOI 10.1007/s10474-010-0024-8 | MR 2737723 | Zbl 1299.06021
  [7] T. Dube, O. Ighedo: Comments regarding $d$-ideals of certain $f$-rings. J. Algebra Appl. 12 (2013), Article ID 1350008, 16 pages. DOI 10.1142/S0219498813500084 | MR 3063447 | Zbl 1284.06046
  [8] T. Dube, O. Ighedo: On $z$-ideals of pointfree function rings. Bull. Iran. Math. Soc. 40 (2014), 657-675. MR 3224080 | Zbl 1309.13004
  [9] T. Dube, O. Ighedo: Two functors induced by certain ideals of function rings. Appl. Categ. Struct. 22 (2014), 663-681. DOI 10.1007/s10485-013-9320-0 | MR 3227611 | Zbl 1306.06007
  [10] T. Dube, O. Ighedo: On lattices of $z$-ideals of function rings. Accepted in Math. Slovaca.
  [11] G. Gruenhage: Products of cozero complemented spaces. Houston J. Math. 32 (2006), 757-773. MR 2247908 | Zbl 1109.54007
  [12] A. W. Hager, J. Martínez: Fraction-dense algebras and spaces. Can. J. Math. 45 (1993), 977-996. DOI 10.4153/CJM-1993-054-6 | MR 1239910 | Zbl 0795.06017
  [13] A. W. Hager, J. Martínez: Patch-generated frames and projectable hulls. Appl. Categ. Struct. 15 (2007), 49-80. DOI 10.1007/s10485-007-9062-y | MR 2306538 | Zbl 1122.06007
  [14] P. T. Johnstone: Stone Spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press, Cambridge (1982). MR 0698074 | Zbl 0499.54001
  [15] J. Martínez, E. R. Zenk: When an algebraic frame is regular. Algebra Univers. 50 (2003), 231-257. DOI 10.1007/s00012-003-1841-1 | MR 2037528 | Zbl 1092.06011
  [16] J. Martínez, E. R. Zenk: Dimension in algebraic frames II: Applications to frames of ideals in $C(X)$. Commentat. Math. Univ. Carol. 46 (2005), 607-636. MR 2259494 | Zbl 1121.06009
  [17] J. Picado, A. Pultr: Frames and Locales. Topology without Points. Frontiers in Mathematics. Birkhäuser/Springer Basel AG, Basel (2012). DOI 10.1007/978-3-0348-0154-6 | MR 2868166 | Zbl 1231.06018
  [18] T. Plewe: Localic triquotient maps are effective descent maps. Math. Proc. Camb. Philos. Soc. 122 (1997), 17-43. DOI 10.1017/S0305004196001648 | MR 1443584 | Zbl 0878.54005

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Affiliations:   Themba Dube, Department of Mathematical Sciences, University of South Africa, Room C6-33, GJ Gerwel Building, Science Campus, 0003 Pretoria, South Africa, e-mail:

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