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

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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 \(\phi =\phi (1)\widetilde{p_c}\).

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The authors would like to thank the referee for careful reading and valuable comments and suggestions relating to this work. Also, we thank Prof. M. M. Ebrahimi for a thorough scrutiny of the first version of this paper, and for comments which have improved the exposition.

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Correspondence to Akbar Ali Estaji.

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Presented by W.Wm. McGovern.

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Ali Estaji, A., Karimi Feizabadi, A. & Emamverdi, B. Representation of real Riesz maps on a strong f-ring by prime elements of a frame. Algebra Univers. 79, 14 (2018).

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