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Algebra universalis

, 80:43 | Cite as

Further thoughts on the ring \({\mathcal {R}}_c (L)\) in frames

  • Ali Akbar EstajiEmail author
  • Maryam Robat Sarpoushi
  • Mahtab Elyasi
Article
  • 40 Downloads

Abstract

Let C(X) denote the ring of all real-valued continuous functions on a topological space X and \({\mathcal {R}} (L)\) be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring \({\mathcal {R}}_c (L)\) is introduced as a sub-f-ring of \({\mathcal {R}} ( L)\) as a pointfree analogue to the subring \(C_c(X)\) of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring \({\mathcal {R}}_c (L)\). In order to do so we introduce the set \(R_{\alpha } := \{ r \in {\mathbb {R}} : {{\,\mathrm{coz}\,}}(\alpha - \mathbf{r}) \not = \top \} \) for every \(\alpha \in {\mathcal {R}} (L)\). We prove that \(R_{\alpha } \) is a countable subset of \({\mathbb {R}}\) for every \(\alpha \in {\mathcal {R}}_c (L)\). Next, we show that if L is a compact frame, then \(R_{\alpha } \) is a finite subset of \({\mathbb {R}}\) for every \(\alpha \in {\mathcal {R}}_c (L)\). Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and \(C_c(X) \cong C_c(Y )\) in pointfree topology. Finally, we prove that, for some frame L, the ring \({\mathcal {R}}_c (L)\) may not be isomorphic to \({\mathcal {R}} (M)\), for any given frame M.

Keywords

Zero-dimensional frame Compact frame Connected frame Ring of real-valued continuous functions Countable image 

Mathematics Subject Classification

06D22 54C05 54C30 

Notes

Acknowledgements

We appreciate the referee for his thorough comments and for taking the time and effort to review our manuscript.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ali Akbar Estaji
    • 1
    Email author
  • Maryam Robat Sarpoushi
    • 1
  • Mahtab Elyasi
    • 1
  1. 1.Faculty of Mathematics and Computer SciencesHakim Sabzevari UniversitySabzevarIran

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