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a-configuration of G a-full o-block a-full point a-full set a,ag a.ag aa aa acting on ft block chapter cofinal configuration Conv convex l-subgroup Corollary Dedekind complete Dedekind cuts define FxGa G acting G is depressible G is o-primitive G is regular G which contains group isomorphism h c G h c H Hence integers interval of support inversely well ordered l-subgroup of A(ft Lemma Let G LnGa locally o-primitive MidGa negative orbits non-trivial o-block o-block of G o-group o-isomorphism o-permutation group o-set orbit of G ordered set permutation Pick g c G positive long orbit positive orbits primitive segment Proof properly contained Proposition regular representation representing subgroup right ray set of orbits SFxG SFxGa SFxGa)g SLnG SLnGa SLnGa)g strongly long orbit subgroup of G support h totally ordered transitive l-subgroup translate weakly o-primitive WFxG WFxGa WLnG WLnGa wreath product