@article {
author = {Karimi Feizabadi, Abolghasem and Estaji, Ali Akbar and Emamverdi, Batool},
title = {$\mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps},
journal = {Categories and General Algebraic Structures with Applications},
volume = {7},
number = {Special Issue on the Occasion of Banaschewski's 90th Birthday (II)},
pages = {141-163},
year = {2017},
publisher = {Shahid Beheshti University},
issn = {2345-5853},
eissn = {2345-5861},
doi = {},
abstract = {In this paper, for each {\it lattice-valued map} $A\rightarrow L$ with some properties, a ring representation $A\rightarrow \mathcal{R}L$ is constructed. This representation is denoted by $\tau_c$ which is an $f$-ring homomorphism and a $\mathbb Q$-linear map, where its index $c$, mentions to a lattice-valued map. We use the notation $\delta_{pq}^{a}=(a -p)^{+}\wedge (q-a)^{+}$, where $p, q\in \mathbb Q$ and $a\in A$, that is nominated as {\it interval projection}. To get a well-defined $f$-ring homomorphism $\tau_c$, we need such concepts as {\it bounded}, {\it continuous}, and $\mathbb Q$-{\it compatible} for $c$, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued map $c_{\phi}:A\rightarrow L $ for each $f$-ring homomorphism $\phi: A\rightarrow \mathcal{R}L$. It is proved that $c_{\tau_c}=c^r$ and $\tau_{c_{\phi}}=\phi$, which they make a kind of correspondence relation between ring representations $A\rightarrow \mathcal{R}L$ and the lattice-valued maps $A\rightarrow L$, Where the mapping $c^r:A\rightarrow L$ is called a {\it realization} of $c$. It is shown that $\tau_{c^r}=\tau_c$ and $c^{rr}=c^r$. Finally, we describe how $\tau_c$ can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski. },
keywords = {frame,cozero lattice-valued map,strong $f$-ring,interval projection,bounded,continuous,$mathbb{Q}$-compatible,coz-compatible},
url = {http://cgasa.sbu.ac.ir/article_38548.html},
eprint = {http://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf}
}