Ordering Free Groups and Validity in Lattice-Ordered Groups

Colacito, Almudena; Metcalfe, George (2019). Ordering Free Groups and Validity in Lattice-Ordered Groups. Journal of pure and applied algebra, 223(12), pp. 5163-5175. Elsevier 10.1016/j.jpaa.2019.03.015

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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 (l-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 l-groups and generation of the variety of l-groups by the l-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 l-groups to subsets of relatively free groups that extend to the positive cone of an order.

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