RL- valued f-ring homomorphisms and lattice-valued maps

In this paper, for each lattice-valued map A ! L with some properties, a ring representation A ! RL is constructed. This representation is denoted by c which is an f-ring homomorphism and a Q-linear map, where its index c, mentions to a lattice-valued map. We use the notation a pq = (a − p) ^ (q − a), where p, q 2 Q and a 2 A, that is nominated as interval projection. To get a well-defined f-ring homomorphism c, we need such concepts as bounded, continuous, and Q-compatible for c, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued map cϕ : A ! L for each f-ring homomorphism ϕ : A ! RL. It is proved that cc = cr and cϕ = ϕ, which they make a kind of correspondence relation between ring representations A ! RL and the lattice-valued maps A ! L, where the mapping cr : A ! L is called a realization of c. It is shown that cr = c and crr = cr. Finally, we describe how c can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.