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.