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

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.  

کلیدواژه ها:

frame ، cozero lattice-valued map ، strong f-ring ، interval projection ، bounded ، continuous ، mathbb{Q}-compatible ، coz-compatible