Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg<sub>1</sub> <span style="white-space:normal;">≤ <span style="white-space:normal;">h...Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg<sub>1</sub> <span style="white-space:normal;">≤ <span style="white-space:normal;">hg<sub>2 </sub>whenever <em style="white-space:normal;">g<sub style="white-space:normal;">1</sub><span style="white-space:normal;"> <span style="white-space:normal;">≤ g<sub>2</sub>. G is orderable provided it admits a total order ≤ satisfying both: <em style="white-space:normal;">hg<sub style="white-space:normal;">1</sub><span style="white-space:normal;"> <span style="white-space:normal;">≤ hg<sub>2</sub> whenever <span style="white-space:nowrap;">g<sub>1</sub> ≤ g<sub>2</sub> and <em style="white-space:normal;">g<sub style="white-space:normal;">1</sub><em style="white-space:normal;">h<span style="white-space:normal;"> ≤ <em style="white-space:normal;">g<sub style="white-space:normal;">2</sub><em style="white-space:normal;">h whenever g<sub>1</sub> ≤ g<sub>2</sub>. 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.展开更多
文摘Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg<sub>1</sub> <span style="white-space:normal;">≤ <span style="white-space:normal;">hg<sub>2 </sub>whenever <em style="white-space:normal;">g<sub style="white-space:normal;">1</sub><span style="white-space:normal;"> <span style="white-space:normal;">≤ g<sub>2</sub>. G is orderable provided it admits a total order ≤ satisfying both: <em style="white-space:normal;">hg<sub style="white-space:normal;">1</sub><span style="white-space:normal;"> <span style="white-space:normal;">≤ hg<sub>2</sub> whenever <span style="white-space:nowrap;">g<sub>1</sub> ≤ g<sub>2</sub> and <em style="white-space:normal;">g<sub style="white-space:normal;">1</sub><em style="white-space:normal;">h<span style="white-space:normal;"> ≤ <em style="white-space:normal;">g<sub style="white-space:normal;">2</sub><em style="white-space:normal;">h whenever g<sub>1</sub> ≤ g<sub>2</sub>. 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.
