TITLE:
The Class of Orderable Groups Is a Quasi-Variety with Undecidable Theory
AUTHORS:
Benjamin Fine, Anthony Gaglione, Gerhard Rosenberger, Dennis Spellman
KEYWORDS:
Orderable Group, Left Orderable Group, Quasi-Variety
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.11 No.3,
March
29,
2021
ABSTRACT: Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg1 ≤ hg2 whenever g1 ≤ g2. G is orderable provided it admits a total order ≤ satisfying both: hg1 ≤ hg2 whenever g1 ≤ g2 and g1h ≤ g2h whenever g1 ≤ g2. 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.