Representation of real Riesz maps on a strong f-ring by prime elements of a frame

Abstract

In classical topology, it is proved that for a topological space X, every bounded Riesz map $\varphi :C (X) \rightarrow {\mathbb {R}}$ is of the from ${\hat{x}}$ for a point $x\in X$. In this paper, our main purpose is to prove a version of this result by lattice-valued maps. A ring representation of the from $A\rightarrow {\mathbb {R}}$ is constructed. This representation is denoted by $\widetilde{p_c}$ that is an onto f-ring homomorphism for every $p\in \Sigma L$, where its index c, denotes a cozero lattice-valued map. Also, it is shown that for every Riesz map $\phi :A\rightarrow {\mathbb {R}}$ and $c\in F(A, L)$ with specific properties, there exists $p\in \Sigma L$ such that $\phi =\phi (1)\widetilde{p_c}$.