%0 Journal Article
%T $mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Karimi Feizabadi, Abolghasem
%A Estaji, Ali Akbar
%A Emamverdi, Batool
%D 2017
%\ 07/01/2017
%V 7
%N Special Issue on the Occasion of Banaschewski's 90th Birthday (II)
%P 141-163
%! $mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps
%K frame
%K cozero lattice-valued map
%K strong $f$-ring
%K interval projection
%K bounded
%K continuous
%K $mathbb{Q}$-compatible
%K coz-compatible
%R
%X In this paper, for each {it lattice-valued map} $Arightarrow L$ with some properties, a ring representation $Arightarrow mathcal{R}L$ 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 $p, qin mathbb Q$ and $ain A$, 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}:Arightarrow L $ for each $f$-ring homomorphism $phi: Arightarrow mathcal{R}L$. 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 $Arightarrow mathcal{R}L$ and the lattice-valued maps $Arightarrow L$, Where the mapping $c^r:Arightarrow L$ 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.
%U http://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf