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I'm new to lattice groups and I'm stuck in the very first proposition (2.1) of Paul Conrad's paper "Characteristic Subgroups of Lattice-Ordered Groups" (http://www.jstor.org/stable/1995910). Part (c) of the statement says that

"If $A$ is an atom of the lattice of $\ell$-ideals, and it is a summand of $G$, then it is simple; moreover if $G$ is representable, then $A$ is linearly ordered."

When I look at the proof, I don't see where $A$ being a summand or where $G$ being representable comes in. Aren't atoms, almost by definition, simple?

Chrystomath
  • 10,798
  • I now understand why atoms need not be simple: An atom $H$ cannot contain subgroups that are normal in $G$, by definition, but it may contain subgroups that are normal in $H$. – Chrystomath Feb 13 '15 at 11:32
  • If there are no (proper) disjoint elements, then every convex lattice subgroup is prime, including $0$ by itself. But ${0}$ is also an $\ell$-ideal, hence $G$ is isomorphic to $G/{0}$ which is totally ordered. – Chrystomath Feb 13 '15 at 11:47
  • $G$ has to be representable for the polar $a^\perp$ to be normal. – Chrystomath Feb 13 '15 at 11:58

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