Abstract
Paul Conrad has asked whether for every lattice-ordered group G, there exists an Abelian N sharing its lattice C(G) of convex l-subgroups, i.e., such that C(N)≃C(G). Several counter-examples are exhibited, including all free l-groups of uncountably infinite rank. Also, for A(ℝ) the l-group of order-automorphisms of the real line ℝ, it is shown that any N sharing its C would have to be an l-subgroup of the l-group C(ℝ) of continuous real-valued functions-and that C(C(ℝ))≄C(A(ℝ)).
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References
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Communicated by A. M. W. Glass
This work was done while the author was spending a most enjoyable year at Boise State University.
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McCleary, S.H. Lattice-ordered groups whose lattices of convex l-subgroups guarantee noncommutativity. Order 3, 307–315 (1986). https://doi.org/10.1007/BF00400294
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DOI: https://doi.org/10.1007/BF00400294