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 τc 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 δapq=(a−p)+∧(q−a)+, where p,q∈Q and a∈A, that is nominated as interval projection. To get a well-defined f-ring homomorphism τc, we need such concepts as bounded, continuous, and Q-{\it 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 cτc=cr and τcϕ=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 τcr=τc and crr=cr. Finally, we describe how τc can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.