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

Year 2020, Volume: 49 Issue: 2, 854 - 868, 02.04.2020

Ali Akbar Estaji Ahmad Mahmoudi Darghadam

https://doi.org/10.15672/hujms.624015

Abstract

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 $\mathcal F_{{\mathcal{P}}_{K}}( L)\cong\mathcal F_{{\mathcal{P}}_{K}}(M)$.

Keywords

Frame, compact frame, locally compact frame, zero-dimensional frame, vanish at infinity

References

• [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.