Fully Preorderable Groups

Efe A. Ok, Gil Riella

    Research output: Contribution to journalArticlepeer-review


    We say that a group is fully preorderable if every (left- and right-) translation invariant preorder on it can be extended to a translation invariant total preorder. Such groups arise naturally in applications, and relate closely to orderable and fully orderable groups (which were studied extensively since the seminal works of Philip Hall and A. I. Mal’cev in the 1950s). Our first main result provides a purely group-theoretic characterization of fully preorderable groups by means of a condition that goes back to Ohnishi (Osaka Math. J. 2, 161–164 16). In particular, this result implies that every fully orderable group is fully preorderable, but not conversely. Our second main result shows that every locally nilpotent group is fully preorderable, but a solvable group need not be fully preorderable. Several applications of these results concerning the inheritance of full preorderability, connections between full preorderability and full orderability, vector preordered groups, and total extensions of translation invariant binary relations on a group, are provided.

    Original languageEnglish (US)
    Pages (from-to)127-142
    Number of pages16
    Issue number1
    StatePublished - Apr 2021


    • Bi-invariant orders
    • Nilpotent groups
    • Orderable groups
    • Primary 06F15
    • Secondary 20F60, 20F18

    ASJC Scopus subject areas

    • Algebra and Number Theory
    • Geometry and Topology
    • Computational Theory and Mathematics


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