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


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.


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

Mathematics Subject Classification

06D22 54C05 54C30 



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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