The ring of real-valued functions on a frame

• Main Authors: Abolghasem Karimi Feizabadi, Ali Akbar Estaji, Mohammad Zarghani
• Format: Article
• Language: English
• Published: Shahid Beheshti University 2016-07-01
• Series: Categories and General Algebraic Structures with Applications
• Subjects: Frame; $f$-ring; Ring of real-valued functions
• Online Access: http://www.cgasa.ir/article_14685_3fb4a5800764dc102f3cbb565968e45d.pdf
• ISSN: 2345-5853; 2345-5861

In this paper, we define and study the notion of the real-valued functions on a frame $L$. We show that $F(L) $, consisting of all frame homomorphisms from the power set of $mathbb{R}$ to a frame $ L$, is an $f$-ring, as a generalization of all functions from a set $X$ into $mathbb R$. Also, we show that $F(L) $ is isomorphic to a sub-$f$-ring of $mathcal{R}(L)$, the ring of real-valued continuous functions on $L$. Furthermore, for every frame $L$, there exists a Boolean frame $B$ such that $F(L)$ is a sub-$f$-ring of $ F(B)$.