Ordering groups and validity in lattice-ordered groups |
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Authors: | Almudena Colacito George Metcalfe |
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Affiliation: | Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland |
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Abstract: | An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order. |
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Keywords: | Corresponding author. 03B25 03C05 06F05 06F15 20E05 Ordered groups Free groups Lattice-ordered groups Decidability |
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