Abstract
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.
Similar content being viewed by others
References
Arrow, K.: Social Choice and Individual Values. Wiley, New York (1951)
Aumann, R.: Utility theory without the completeness axiom. Econometrica 30, 445–462 (1962)
Everett, C.: Note on a result of L. Fuchs on ordered groups. Amer. J. Math. 72, 216 (1950)
Evren, O., Ok, E.A.: On the multi-utility representation of preference relations. J. Math. Econ. 47, 554–563 (2011)
Fuchs, L.: On the extension of the partial order of groups. Amer. J. Math. 72, 191–194 (1950)
Holland, C.H.W.: Extensions of Ordered Algebraic Structures, Doctoral Thesis, Tulane University (1961)
Kargapolov, M.: Fully orderable groups (Russian). Algebra i Logika 2, 5–14 (1963)
Kargapolov, M., Kokorin, A., Kopytov, V.: On the theory of orderable groups (Russian). Algebra i Logika 6, 21–27 (1965)
Kokorin, A.: On fully orderable groups (Russian). Dokl. Akad. Nauk SSSR 151, 31–33 (1963)
Kokorin, A., Kopytov, V.M.: Fully Ordered Groups, Halsted Press. John Wiley & Sons, New York (1974)
Kopytov, V.M.: On the theory of fully orderable groups (Russian). Algebra i Logica 5, 27–31 (1966)
Kopytov, V.M., Medvedev, N.Y.A.: The theory of lattice-ordered groups. Kluwer Academic Publishers, Dordrecht (2010)
Mal’cev, A.: On the full ordering of groups (Russian). Trudy. Mat. Inst. Steklov. 38, 173–175 (1951)
Mal’cev, A.: On partially ordered nilpotent groups (Russian). Algebra i Logika 2, 5–9 (1962)
Minassian, D.: Types of fully ordered groups. Amer. Math. Monthly 80, 159–169 (1973)
Ohnishi, M: On linearization of ordered groups. Osaka Math. J. 2, 161–164 (1950)
Ok, E.A., Riella, G.: Topological closure of translation invariant preorders. Math. Oper. Res. 39, 737–745 (2014)
Ok, E.A., Ortoleva, P., Riella, G.: Incomplete preferences under uncertainty: indecisiveness in beliefs vs. tastes. Econometrica 80, 1791–1808 (2012)
Robinson, D.: A Course in the Theory of Groups. Springer, New York (1996)
Seidenfeld, T., Schervish, M., Kadane, J.: A representation of partially ordered preferences. Ann. Statist. 23, 2168–2217 (1995)
Suzumura, K.: Remarks on the theory of collective choice. Economica 43, 381–390 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The comments of Andrew Glass, Andrés Navas, Cristóbal Rivas and an anonymous referee of this journal have improved the content of this work; we gratefully acknowledge our intellectual debt to them. This paper is in final form and no version of it will be submitted for publication elsewhere.
Rights and permissions
About this article
Cite this article
Ok, E.A., Riella, G. Fully Preorderable Groups. Order 38, 127–142 (2021). https://doi.org/10.1007/s11083-020-09532-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-020-09532-5