| | | |

## Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity

Let $\mathcal F_{\mathcal{P}}( L)$ be the set of all frame maps from $\mathcal P(\mathbb R)$ to $L$, which is an $f$-ring. In this paper, we introduce the subrings $\mathcal F_{{\mathcal{P}}_{\infty}}( L)$ of all frame maps from $\mathcal P(\mathbb R)$ to $L$ which vanish at infinity and $\mathcal F_{{\mathcal{P}}_{K}}( L)$ of all frame maps from $\mathcal P(\mathbb R)$ to $L$ with compact support. We prove $\mathcal F_{{\mathcal{P}}_{\infty}}( L)$ is a subring of $\mathcal F_{\mathcal{P}}(L)$ that may not be an ideal of $\mathcal F_{\mathcal{P}}(L)$ in general and we obtain necessary and sufficient conditions for $\mathcal F_{{\mathcal{P}}_{\infty}}( L)$ to be an ideal of $\mathcal F_{\mathcal{P}}( L)$. Also, we show that $\mathcal F_{{\mathcal{P}}_{K}}( L)$ is an ideal of $\mathcal F_{\mathcal{P}}( L)$ and it is a regular ring. For $f\in\mathcal F_{\mathcal{P}}( L)$, we obtain a sufficient condition for $f$ to be an element of $F_{{\mathcal{P}}_{\infty}}( L)$ ($\mathcal F_{{\mathcal{P}}_{K}}( L)$). Next, we give necessary and sufficient conditions for a frame to be compact. We introduce $\mathcal F_{\mathcal{P}}$-pseudocompact and next we establish equivalent condition for an $\mathcal F_{\mathcal{P}}$-pseudocompact frame to be a compact frame. Finally, we study when for some frame $L$ with $\mathcal F_{{\mathcal{P}}_{\infty}} (L)\neq(0)$, there is a locally compact frame $M$ such that $\mathcal F_{{\mathcal{P}}_{\infty}}( L)\cong\mathcal F_{{\mathcal{P}}_{\infty}}(M)$ and $\mathcal F_{{\mathcal{P}}_{K}}( L)\cong\mathcal F_{{\mathcal{P}}_{K}}(M)$.
Frame, compact frame, locally compact frame, zero-dimensional frame, vanish at infinity
• [1] A.R. Aliabad, F. Azarpanah and M. Namdari, Rings of continuous functions vanishing at infinity, Comment. Mat. Univ. Carolinae 45 (3), 519–533, 2004.
• [2] R.N. Ball and J. Walters-Wayland, $C$- and $C^*$-quotients in pointfree topology, Dissertationes Math. (Rozprawy Mat.) 412, 1–61, 2002.
• [3] B. Banaschewski, Remarks Concerning Certain Function Rings in Pointfree Topology, Appl. Categor. Struct, 26 (5), 873–881, 2018.
• [4] B. Banaschewski, The real numbers in pointfree topology, Textos de Mathematica (Series B) 12, 1–96, 1997.
• [5] T. Dube, On the ideal of functions with compact support in pointfree function rings, Acta Math. Hungar 129 (3), 205–226, 2010.
• [6] T. Dube, Extending and contracting maximal ideals in the function rings of pointfree topology, Bull. Math. Soc. Sci. Math. Roumanie 55 (103) No.4, 365–374, 2012.
• [7] A.A. Estaji, M. Abedi and A. Mahmoudi Darghadam, On self-injectivity of the $f$-ring ${\mathbf Frm}(\mathcal{P}(\mathbb{R}), L)$, Math. Slovaca Accepted.
• [8] A.A. Estaji and A. Mahmoudi Darghadam, Rings of continuous functions vanishing at infinity on a frame, Quaest. Math., 2018, DOI:10.2989/16073606.2018.1509151.
• [9] A.A. Estaji and A. Mahmoudi Darghadam, Ring of real measurable functions vanishing at infinity on a measurable space, submitted.
• [10] A.As. Estaji, E. Hashemi and A.A. Estaji, On property (A) and the socle of the $f$-ring ${\mathbf Frm}(\mathcal{P}(\mathbb{R}), L)$, Categ. Gen. Algebr. Struct. Appl. 8 (1), 61–80, January 2018.
• [11] A. Karimi Feizabadi, A.A. Estaji and M. Zarghani, The ring of real-valued functions on a frame, Categ. Gen. Algebr. Struct. Appl. 5 (1), 85–102, July 2016.
• [12] J. Picado and A. Pultr, Frames and locales: Topology without points, Frontiers in Mathematics, Springer Basel, 2012.
Birincil Dil en Matematik Matematik Orcid: 0000-0002-0376-5477Yazar: Ali Akbar ESTAJİ Kurum: Hakim Sabzevari UniversityÜlke: Iran Orcid: 0000-0001-9416-6041Yazar: Ahmad Mahmoudi DARGHADAM Kurum: Hakim Sabzevari UniversityÜlke: Iran Yayımlanma Tarihi : 2 Nisan 2020
 Bibtex @araştırma makalesi { hujms624015, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2020}, volume = {49}, pages = {854 - 868}, doi = {10.15672/hujms.624015}, title = {Rings of frame maps from \$\\mathcal\{P\}(\\mathbb\{R\})\$ to frames which vanish at infinity}, key = {cite}, author = {ESTAJİ, Ali Akbar and DARGHADAM, Ahmad Mahmoudi} } APA ESTAJİ, A , DARGHADAM, A . (2020). Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity. Hacettepe Journal of Mathematics and Statistics , 49 (2) , 854-868 . DOI: 10.15672/hujms.624015 MLA ESTAJİ, A , DARGHADAM, A . "Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 854-868 Chicago ESTAJİ, A , DARGHADAM, A . "Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 854-868 RIS TY - JOUR T1 - Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity AU - Ali Akbar ESTAJİ , Ahmad Mahmoudi DARGHADAM Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.624015 DO - 10.15672/hujms.624015 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 854 EP - 868 VL - 49 IS - 2 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.624015 UR - https://doi.org/10.15672/hujms.624015 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity %A Ali Akbar ESTAJİ , Ahmad Mahmoudi DARGHADAM %T Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 2 %R doi: 10.15672/hujms.624015 %U 10.15672/hujms.624015 ISNAD ESTAJİ, Ali Akbar , DARGHADAM, Ahmad Mahmoudi . "Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity". Hacettepe Journal of Mathematics and Statistics 49 / 2 (Nisan 2020): 854-868 . https://doi.org/10.15672/hujms.624015 AMA ESTAJİ A , DARGHADAM A . Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 854-868. Vancouver ESTAJİ A , DARGHADAM A . Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 868-854.

Makalenin Yazarları
[1]
[2]