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