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The Class of Orderable Groups Is a Quasi-Variety with Undecidable Theory
Department of Mathematics, Fairfield University, Fairfield, Connecticut, USA
Department of Mathematics, United States Naval Academy, Annapolis, Maryland, USA
Fachbereich Mathematik, University of Hamburg, Bundestrasse, Hamburg, Germany
Department of Statistics, Temple University, Philadelphia, Pennsylvania, USA
- 1 Department of Mathematics, Fairfield University, Fairfield, Connecticut, USA
- 2 Department of Mathematics, United States Naval Academy, Annapolis, Maryland, USA
- 3 Fachbereich Mathematik, University of Hamburg, Bundestrasse, Hamburg, Germany
- 4 Department of Statistics, Temple University, Philadelphia, Pennsylvania, USA
Advances in Pure Mathematics·Volume 11 (2021)·Pages 180–186·Published 12 March 2021·DOI10.4236/apm.2021.113013
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Abstract
Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg 1 ≤ hg 2 whenever g 1 ≤ g 2 . G is orderable provided it admits a total order ≤ satisfying both: hg 1 ≤ hg 2 whenever g 1 ≤ g 2 and g 1 h ≤ g 2 h whenever g 1 ≤ g 2 . A classical result shows that free groups are orderable. In this paper, we prove that left-orderable groups and orderable groups are quasivarieties of groups both with undecidable theory. For orderable groups, we find an explicit set of universal axioms.
KeywordsOrderable GroupLeft Orderable GroupQuasi-Variety
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