Fremlin tensor product behaves well with the unbounded order convergence

Suppose $\mathrm{\Sigma }$ is a topological space and $S(\mathrm{\Sigma })$ is the vector lattice of all equivalent classes of continuous real-valued functions defined on open dense subsets of $\mathrm{\Sigma }$. In this paper, we establish some lattice and topological aspects of $S(\mathrm{\Sigma })$. In particular, as an application, we show that the unbounded order convergence and the order convergence are stable under passing to the Fremlin tensor product of two Archimedean vector lattices.