Abstract
A cozero sublocale C of a completely regular locale L will here be called coz-complemented if there is a cozero sublocale D of L such that \(C\cap D\) is the void sublocale and \(C\vee D\) is dense in L. Following terminology in spaces, we say L is a cloz-locale if every coz-complemented sublocale of L has an open closure. We characterize cloz-locales in terms of certain embeddings, and also in terms of ring-theoretic properties of the ring \({\mathcal {R}}L\) of real-valued continuous functions on L.
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References
Ball, R.N., Walters-Wayland, J.: \(C\)- and \(C^{*}\)-quotients in pointfree topology. Diss. Math., vol. 42 (2002)
Banaschewski, B.: The real numbers in pointfree topology. Textos de Matemática, Série B, No. 12, Departamento de Matemática da Universidade de Coimbra (1997)
Banaschewski, B., Gilmour, C.: Pseudocompactness and the cozero part of a frame. Comment. Math. Univ. Carolin. 37, 577–587 (1996)
Banaschewski, B., Gilmour, C.: Realcompactness and the cozero part of a frame. Appl. Categ. Struct. 9, 395–417 (2001)
Banaschewski, B., Pultr, A.: Booleanization. Cah. Topol. Géom. Différ. Catég. 37, 41–60 (1996)
Dashiell, F., Hager, A., Henriksen, M.: Order-Cauchy completeness of rings and vector lattices of continuous functions. Can. J. Math. 32, 657–685 (1980)
Dube, T.: Some ring-theoretic properties of almost P-frames. Algebra Univ. 60, 145–162 (2009)
Dube, T.: Notes on pointfree disconnectivity with a ring-theoretic slant. Appl. Categ. Struct. 18, 55–72 (2010)
Dube, T., Ighedo, O.: Concerning the summand intersection property in function rings. Houst. J. Math. 44, 1029–1049 (2018)
Dube, T., Matlabyana, M.: Cozero complemented frames. Topol. Appl. 160, 1345–1352 (2013)
Dube, T., Matlabyana, M.: Notes concerning characterizations of Quasi-\(F\) frames. Quaest. Math. 32(4), 551–567 (2009)
Gutiérrez García, J., Picado, J.: Rings of real functions in pointfree topology. J. Pure Appl. Algebra. 158, 2264–2278 (2011)
Henriksen, M., Vermeer, J., Woods, R.G.: Wallman covers of compact spaces. Diss. Math. (Rozprawy Mat.), vol. 280 (1989)
Henriksen, M., Vermeer, J., Woods, R.G.: Quasi \(F\)-covers of Tychonoff spaces. Trans. Am. Math. Soc. 303, 779–803 (1987)
Ighedo, O., Mugochi, M.M.: On some parallelism between complete regularity and zero-dimensionality. Quaest. Math. 41(3), 423–435 (2018)
Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)
Knox, M., Levy, W., McGovern, W. Wm., Shapiro, J.: Generalizations of complemented rings with applications to rings of continuous functions. J. Algebra Appl. 8 (2009)
Madden, J., Vermeer, J.: Lindelöf locales and realcompactness. Math. Proc. Camb. Philos. Soc. 99, 473–480 (1986)
Picado, J., Pultr, A.: Frames and Locales: Topology Without Points, Frontiers in Mathematics. Springer, Basel (2012)
Steinberg, S.A.: Lattice-ordered Rings and Modules. Springer, New York (2010)
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The authors do wish to thank the referee for the insightful comments which have led to an improved presentation of this paper.
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Presented by Vincenzo Marra.
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The authors would also like to acknowledge funding from the Unisa Topology Research Chair that enabled Mugochi to visit Ighedo in June 2018, and enabled Ighedo to visit Mugochi in October 2018, during which periods most of the results in this paper were obtained.
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Ighedo, O., Mugochi, M.M. Locales whose coz-complemented cozero sublocales have open closures. Algebra Univers. 81, 17 (2020). https://doi.org/10.1007/s00012-020-00655-9
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DOI: https://doi.org/10.1007/s00012-020-00655-9