$\mathcal{R}L$- valued $f$-ring homomorphisms and lattice-valued maps

Document Type: Research Paper


1 Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.


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.


Dedicated to Professor Bernhard Banaschewski on the occasion of his 90th Birthday


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