TY - JOUR ID - 38548 TI - $\mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps JO - Categories and General Algebraic Structures with Applications JA - CGASA LA - en SN - 2345-5853 AU - Karimi Feizabadi, Abolghasem AU - Estaji, Ali Akbar AU - Emamverdi, Batool AD - Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran. AD - Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Y1 - 2017 PY - 2017 VL - 7 IS - Special Issue on the Occasion of Banaschewski's 90th Birthday (II) SP - 141 EP - 163 KW - frame KW - cozero lattice-valued map KW - strong $f$-ring KW - interval projection KW - bounded KW - continuous KW - $mathbb{Q}$-compatible KW - coz-compatible DO - N2 - 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.   UR - https://cgasa.sbu.ac.ir/article_38548.html L1 - https://cgasa.sbu.ac.ir/article_38548_d61135e6f18b53e9ac1eb29192263dbc.pdf ER -