Partially ordered permutation groups
A partially ordered permutation group is a permutation group (G,\(\Omega)\) together with a G-partial order on \(\Omega\). Such G-partial orders correspond in a natural way to certain subsemigroups of G. The author gives necessary and sufficient conditions for (1) each G-partial order to extend to a G-total order, (2) in addition, there are only two total orders (necessarily inverses of each other), (3) there are only two non- trivial G-partial orders and that these are total (4) the existence of an isolated G-partial order \((\alpha g^ n=\alpha\) implies \(\alpha g=\alpha)\) and (5) the existence of an upper directed G order on \(\Omega\). All these straightfoward results are analogues of Ohnishi type results for groups [\textit{R. B. Mura} and \textit{A. H. Rhemtulla}, Orderable Groups, Lect. Notes Pure Appl. Math. 27 (1977; Zbl 0452.06011)]. The author also shows that any 0-2 transitive permutation group satisfies (3). Note: In Lemma 7.2, f should be replaced by \(g^{n_{\beta}- n_{\alpha}}\cdot f\) where \((\gamma g)_ k=\gamma_{k-1}\) and \(\ell_ 1<...<\ell_ m\) are the places where \(\beta\) and \(\alpha g^{n_{\beta}-n_{\alpha}}\) disagree. This paper is taken from the author's Ph. D. thesis [Orderable Permutation Groups, Bowling Green State Univ. (1977)] under W. C. Holland; for some reason this is not mentioned in the paper.
scientific article; zbMATH DE number 3880781
scientific article