Normal subgroups and elementary theories of lattice-ordered groups

✍ Scribed by Manfred Droste

• Publisher: Springer Netherlands
• Year: 1988
• Tongue: English
• Weight: 741 KB
• Volume: 5
• Category: Article
• ISSN: 0167-8094
• DOI: 10.1007/bf00354894

✦ Synopsis

We show that any lattice-ordered group (/-group) G can be/-embedded into continuously many/-groups 11, which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups Hz can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically 'nice' properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S,) of infinite linearly ordered sets (S,,~<) such that each group A(S,) has only inner automorphisms. We also show that any countable/-group G can be/-embedded into a countable/-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.