July 19th, 2014, 11:56 PM  #1 
Newbie Joined: Jul 2014 From: Czech republic Posts: 1 Thanks: 0  Saturated subgroup
Hi, let $\displaystyle G$ be an lgroup 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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saturated, subgroup 
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