The function ring functors of pointfree topology revisited

Document Type : Research Paper

Author

Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.

10.29252/cgasa.11.1.19

Abstract

This paper establishes two new connections between the familiar function ring functor ${\mathfrak R}$ on the category ${\bf CRFrm}$ of completely regular frames and the category {\bf CR}${\mathbf \sigma}${\bf Frm} of completely regular $\sigma$-frames as well as their counterparts for the analogous functor ${\mathfrak Z}$ on the category {\bf ODFrm} of 0-dimensional frames, given by the integer-valued functions, and for the related functors ${\mathfrak R}^*$ and ${\mathfrak Z}^*$ corresponding to the bounded functions.  Further it is shown that some familiar facts concerning these functors are simple consequences of the present results.

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References

[1] Ball, R.N. and Walters-Wayland, J.L., "C- and C-Quotients in Pointfree Topology", Diss. Math. 412, 2002.
[2] Banaschewski, B., On the function ring functor in pointfree topology, Appl. Categ. Structures 13 (2005), 305-328.
[3] Banaschewski, B., On the maps of pointfree topology which preserve the rings of integervalued continuous functions, Appl. Categ. Structures 26 (2018), 477-489.
[4] Picado, J. and Pultr, A., "Frames and Locales", Birkhäuser, Springer Basel AG, 2012.

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