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

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

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