Two different definitions of solid subspaces of a Riesz space

Question

In D.H. Fremlin, Topological Riesz Spaces and Measure Theory, [14C], a subset $F$ of a Riesz space $E$ is defined as solid if

$x \in F$ whenever there is an $y \in F$ such that $|x| \le |y|$.

In G. Letta, Argomenti scelti di Teoria della Misura, Chap. VII, Def. (2.3), a linear subspace $F$ of a Riesz space $E$ is defined as solid if

for every couple $(x,y)$ of elements of $E$, the following implication holds: $$0 \le x \le y \in F \Rightarrow x \in F. $$

It is quite obvious that Fremlin’s definition implies Letta’s, for Fremlin’s condition coincides with Letta’s when $x$ and $y$ are positive.

Questions.

1. Is Letta’s condition equivalent to Fremlin’s, at least when the set is $F$ is a linear subspace of $E$?
2. Is it true in general that Letta’s condition implies Fremlin’s?

Answer 1

Letta's condition does not necessarily imply Fremlins's condition even if $F$ is a linear subspace of $E$. Let $E = \mathbb R^2$ with $(x_1,x_2) \le (y_1, y_2) \Leftrightarrow (x_1 \le y_1 \text{ and } x_2 \le y_2 )$. Let $F = \{(t,-t) : t \in \mathbb R\}$. See that Letta's condition is satisfied (because if $0 \le y \in F$, then $y = 0$). But Fremlin's condition fails, because $|(1,0)| \le |(1,-1)|$.