Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58537Special Issue on the Occasion of Banaschewski's 90th Birthday (II)20170701$mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps14116338548ENAbolghasemKarimi FeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.Ali AkbarEstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.BatoolEmamverdiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.Journal Article20160728In 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.<br /> We use the notation $delta_{pq}^{a}=(a -p)^{+}wedge (q-a)^{+}$,<br /> where $p, qin mathbb Q$ and $ain A$, that is nominated as {it interval projection}.<br /> 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 $c$,<br /> 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 $Arightarrow L$,<br /> Where the mapping $c^r:Arightarrow L$ is called a {it realization} of $c$. It is shown that $tau_{c^r}=tau_c$ and $c^{rr}=c^r$.<br /> <br /> Finally, we describe how $tau_c$ can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.<br /> http://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf