$\mathcal{R}L$- 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


[1] Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered algebraic structures (Curacao, 1995), Kluwer Acad. Publ., Dordrecht, (1997), 123-148.
[2] Banaschewski, B., The real numbers in pointfree topology, Texts in Mathematics (Series B), 12, University of Coimbra, 1997.
[3] Bigard, A., K. Keimel, and S. Wolfenstein, Groups et anneaux reticules, Lecture Notes in Math. 608, Springer-Verlag, 1977.
[4] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291-299.
[5] Gillman, L. and M. Jerison, "Rings of Continuous Function", Graduate Texts in Mathematics 43, Springer-Verlag, 1979.
[6] Karimi Feizabadi, A., Representation of slim algebraic regular cozero maps, Quaest. Math. 29 (2006), 383-394.
[7] Karimi Feizabadi, A., Free lattice-valued functions, reticulation of rings and modules, submitted.
[8] Picado, J. and A. Pultr, "Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, Basel, 2012.