Throughout the thesis, let X be any non-empty set. We denote by F(X) the algebra of all real-valued mappings on X. We denote by (X) the subalgebra of F(X) consisting of all bounded elements of F(X). For all f,g ? F(X), the mappings (f ?g) : X ? R and (f ?g) : X ? R are defined by )f ? g)(x) = max{f(x),g(x)} and (f ? g)(x) = min{f(x),g(x)} for ever x ? X, respectively. By a function lattice on X we mean a vector suace F of F(X( such that F contians the constant mappings and f ?g ? F and f ?g ? F for all f,g ? F. A filter on X is a non-empty family ? of subsets of X with the following properies: .1 If A,B ? ?, then A ? B ? ? .2 If A ? ? and A ? B ? X, then B ? ?. 3. ? ?. A F-family on X is a non-empty family A of non-empty subsets of X sush that,for every A ? A with A X, there exist some B ? A and a function f ? F such that f(B) = {0} and f(X\\A) ={1} . An F-filter on X is a filter ? on X which is also an F-family on X. Suppose that F is a lattice consisting of real-valued mappings on a non-empty setX which contains the Costant mappings. We use certain filters on X determined by F, to construct a compact Hausdorff space X such that bounded elements of F extendable continuously over ?X . These extended mappings form a dense suace of C(?X). It is remarkable that we not need the Stone-Weierstrass Theorem to prove the density of these extensions. For a large part of the theory developed in this thesis, it is the lattice structure of real-valued mappings that is important for our development. Therefore, we work with a lattice of real-valued mappings(which might contain unbounded mappings). The organization of thesis is as follows: In chapter1, after recalling some back ground and notations and definitions, we discuss some basic properties of F-filters that we shall need in establishing our main results. In chapter 2, we introduce the main object of thesis, namely F-filters and F-ultrafilters, and we study some their basic properties. We define a topology on the set of all F-ultrafilters and we show that the resulting space ?X is a compact Hausdorff space. Furthermore, we show that the F-filters describe the topology of ?X. In chapter 3, we assume that F is a mappings lattice on X. We study continuous mappings on ?X. We show that every bounded elements of F extends to ?X and that these extenstions form a dense suace of the algebra of all continuous, real-valued mappings on ?X. It is remarkable that we not need the Stone-Weierstrass Theorem to prove the density of these extensions. Our main results concern closed subalgebras of the algebra of all bounded real-valued mappings on ?X. In chapter 4, we study some relationships between function lattices. We establish a correspondence between F-filters and closed, proper ideals of F. Roughly speaking, we show how the ideals of F can be used to generate F-filters on ?X. We apply the following convention for the rest of this thesis: By an ideal of F, we always mean a closed, proper ideal of F. In the previous chapter, we made no assumption about algebraic or topological structure on the set X. In chapter 5, we study a treatment of F-filters on a Hausdorff topological space X in the case that every member of F is a continuous mappings of X.