Horizontal Egorov property of Riesz spaces

We say that a Riesz space $E$ has the horizontal Egorov property if for every net $(f_\alpha )$ in $E$, order convergent to $f \in E$ with $\vert f_\alpha \vert + \vert f\vert \le e \in E^+$ for all $\alpha$, there exists a net $(e_\beta )$ of fragments of $e$ laterally convergent to $e$ such that for every $\beta$, the net $\bigl (\vert f - f_\alpha \vert \wedge e_\beta \bigr )_\alpha$ $e$-uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space $E$ is said to satisfy the weak distributive law if for every $e \in E^+ \setminus \{0\}$ the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ satisfies the weak distributive law; that is, whenever $(\Pi _n)_{n \in \mathbb{N}}$ is a sequence of partitions of $\mathfrak{F}_e$, there is a partition $\Pi$ of $\mathfrak{F}_e$ such that every element of $\Pi$ is finitely covered by each of $\Pi _n$ (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net $(f_\alpha )$ order convergent to $f$ in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net $(v_\beta )$ and a net $(u_{\alpha , \beta })_{(\alpha , \beta ) \in A \times B}$, which uniformly tends to zero for every fixed $\beta$ such that $\vert f - f_\alpha \vert \le u_{\alpha , \beta } + v_\beta$ for all $\alpha , \beta$.