Estaji, A., Karimi Feizabadi, A., Zarghani, M. (2016). The ring of real-continuous functions on a topoframe. Categories and General Algebraic Structures with Applications, 4(1), 75-94.

Ali Akbar Estaji; Abolghasem Karimi Feizabadi; Mohammad Zarghani. "The ring of real-continuous functions on a topoframe". Categories and General Algebraic Structures with Applications, 4, 1, 2016, 75-94.

Estaji, A., Karimi Feizabadi, A., Zarghani, M. (2016). 'The ring of real-continuous functions on a topoframe', Categories and General Algebraic Structures with Applications, 4(1), pp. 75-94.

Estaji, A., Karimi Feizabadi, A., Zarghani, M. The ring of real-continuous functions on a topoframe. Categories and General Algebraic Structures with Applications, 2016; 4(1): 75-94.

The ring of real-continuous functions on a topoframe

^{1}Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

^{2}Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.

^{3}Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

Abstract

A topoframe, denoted by $L_{ \tau}$, is a pair $(L, \tau)$ consisting of a frame $L$ and a subframe $ \tau $ all of whose elements are complementary elements in $L$. In this paper, we define and study the notions of a $\tau $-real-continuous function on a frame $L$ and the set of real continuous functions $\mathcal{R}L_\tau $ as an $f$-ring. We show that $\mathcal{R}L_{ \tau}$ is actually a generalization of the ring $C(X)$ of all real-valued continuous functions on a completely regular Hausdorff space $X$. In addition, we show that $\mathcal{R}L_{ \tau}$ is isomorphic to a sub-$f$-ring of $\mathcal{R}\tau .$ Let ${\tau}$ be a topoframe on a frame $L$. The frame map $\alpha\in\mathcal{R}\tau $ is called $L$-{\it extendable} real continuous function if and only if for every $r\in \mathbb{R}$, $\bigvee^{L}_{r\in \mathbb R} (\alpha(-,r)\vee\alpha(r,-))'=\top.$ Finally, we prove that $\mathcal{R}^{L}{\tau}\cong \mathcal{R}L_{\tau}$ as $f$-rings, where $\mathcal{R}^{L}{\tau}$ is the set all of $L$-extendable real continuous functions of $ \mathcal{R}\tau $.

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