Fremlin tensor product behaves well with the unbounded order convergence

TL;DR

This paper investigates the properties of the Fremlin tensor product in the context of vector lattices, demonstrating that certain convergence behaviors are preserved under this tensor product.

Contribution

It establishes that unbounded order convergence and order convergence are stable under the Fremlin tensor product of Archimedean vector lattices.

Findings

Unbounded order convergence is preserved under Fremlin tensor product.

Order convergence remains stable in the tensor product setting.

Provides lattice and topological insights into the structure of $S(al)$.

Abstract

Suppose $\Sigma$ is a topological space and $S(\Sigma)$ is the vector lattice of all equivalent classes of continuous real-valued functions defined on open dense subsets of $\Sigma$. In this paper, we establish some lattice and topological aspects of $S(\Sigma)$. 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.