Order Continuous and Topological Representations of Archimedean Vector Lattices via $S(X)$-spaces

For an arbitrary topological space $X$, assume that $S(X)$ is the vector lattice of all equivalence classes of real-valued continuous functions on open dense subsets of $X$; it is a laterally complete vector lattice but not a normed lattice, certainly. Nevertheless, we can have the extended unbounded norm topology ($un$-topology) on it. On the other hand, by a remarkable result of Wickstead, there exists a representation approach for every Archimedean vector lattice $E$ in terms of $S(X)$-spaces. In this paper, we show that this representation is order continuous and when $E$ is order complete, it coincides with the known Maeda-Ogasawara representation. Moreover, when $E$ is a Banach lattice, by consideration of the $un$-topology on $E$ and the extended $un$-topology on $S(X)$, we show that this representation is, in fact, a homeomorphism. With the aid of this topological attitude, we establish a representation theorem (in fact a homeomorphism) for the Fremlin projective tensor product between Banach lattices, in terms of $S(X)$-spaces, as well.