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Keywords:
$\ell $-cyclically ordered set; completeness; monotone permutation; half cyclically ordered group
$\ell $-cyclically ordered set; completeness; monotone permutation; half cyclically ordered group
Summary:
For an $\ell $-cyclically ordered set $M$ with the $\ell $-cyclic order $C$ let $P(M)$ be the set of all monotone permutations on $M$. We define a ternary relation $\overline{C}$ on the set $P(M)$. Further, we define in a natural way a group operation (denoted by $\cdot $) on $P(M)$. We prove that if the $\ell $-cyclic order $C$ is complete and $\overline{C}\ne \emptyset $, then $(P(M), \cdot ,\overline{C})$ is a half cyclically ordered group.
References:
[1] Š. Černák: On the maximal Dedekind completion of a half partially ordered group. Math. Slovaca 46 (1996), 379–390. MR 1472632
[2] Š. Černák: Cantor extension of a half lattice ordered group. Math. Slovaca 48 (1998), 221–231. MR 1647682
[3] Š. Černák: Maximal Dedekind completion of a half lattice ordered group. Math. Slovaca 49 (1999), 403–416. MR 1719680
[4] M. Droste, M. Giraudet, and D. Macpherson: Periodic ordered permutation groups and cyclic orderings. J. Combinatorial Theory, Series B 63 (1995), 310–321. DOI 10.1006/jctb.1995.1022 | MR 1320173
[5] P. C. Fishburn, D. R. Woodall: Cyclic orders. Order 16 (1999), 149–164. DOI 10.1023/A:1006381208272 | MR 1752172
[6] M. Giraudet, F. Lucas: Groupes à moitié ordonnés. Fundamenta Math. 139 (1991), 75–89. DOI 10.4064/fm-139-2-75-89 | MR 1150592
[7] M. Giraudet, J. Rachůnek: Varieties of half lattice ordered groups of monotonic permutations in chains. Prepublication No 57, Université Paris 7, CNRS Logique (1996).
[8] J. Jakubík: On half lattice ordered groups. Czechoslovak Math. J. 46 (1996), 745–767. MR 1414606
[9] J. Jakubík: Lexicographic products of half linearly ordered groups. Czechoslovak Math. J. 51 (2001), 127–138. DOI 10.1023/A:1013761906636 | MR 1814638
[10] J. Jakubík: On half cyclically ordered groups. Czechoslovak Math. J. 52 (2002), 275–294. DOI 10.1023/A:1021718426347 | MR 1905435
[11] J. Jakubík, Š. Černák: On convex linearly ordered subgroups of a $h\ell $-group. Math. Slovaca 50 (2000), 127–133. MR 1763115
[12] V. Novák: Cyclically ordered sets. Czechoslovak Math. J. 32 (1982), 460–473. MR 0669787
[13] V. Novák, M. Novotný: Universal cyclically ordered sets. Czechoslovak Math. J. 35 (1985), 158–161. MR 0779343
[14] V. Novák, M. Novotný: On representations of cyclically ordered sets. Czechoslovak Math. J. 39 (1989), 127–132. MR 0669787
[15] A. Quilliot: Cyclic orders. European J. Combin. 10 (1989), 477–488. DOI 10.1016/S0195-6698(89)80022-8 | MR 1014556 | Zbl 0692.05059
[16] Dao Rong Ton: Torsion classes and torsion prime selectors of $h\ell $-groups. Math. Slovaca 50 (2000), 31–40. MR 1764343
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