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July 19th, 2014, 11:56 PM   #1
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Saturated subgroup

Hi,
let $\displaystyle G$ be an l-group and $\displaystyle g\in G$. $\displaystyle S(g)$ denotes the minimal saturated subgroup that contains $\displaystyle g$.

Proposition: $\displaystyle S(g)$ is the subgroup of $\displaystyle G$ generated by the components of $\displaystyle g$.
Proof: Let $\displaystyle A(g)$ be the subgroup of $\displaystyle G$ generated by the components of $\displaystyle g$. WLOG $\displaystyle 1\leq g$ ($\displaystyle 1$ denotes the neutral element of the group). And so on.

My problem: I don't understand why I can write the WLOG part. While I know that $\displaystyle S(g)=S(|g|)$, I am not sure if $\displaystyle A(g)\supseteq A(|g|)$. For this to be true, it would have to be possible to express any component of $\displaystyle |g|$ in terms of components of $\displaystyle g$ and the group (not lattice) operations.
The inclusion $\displaystyle A(g)\subseteq A(|g|)$ is clear to me.

Thank you.
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