An Even Better Representation for Free Lattice - Ordered Groups
چکیده
The free lattice-ordered group Fn (of rank r¡) has been studied in two ways: via the Conrad representation on the various right orderings of the free group Gv (sharpened by Kopytov's observation that some one right ordering must by itself give a faithful representation), and via the Glass-McCleary representation as a pathologically o-2-transitive /-permutation group. Each kind of representation yields some results which cannot be obtained from the other. Here we construct a representation giving the best of both worlds—a right ordering (Gn, vp^ i j ' ' j We shall refer to this as the natural action of Fn on (G^, <). When this action is a representation (i.e., faithful), we shall call (Gv, <) a representing right ordering. Another kind of action of Fv is usual transitive action on the chain F /P of right cosets of a prime subgroup P (namely (Pf)w = P(fw)). When this action is a representation, P is called a representing subgroup of F . Received by the editors February 21, 1984. 1980 Mathematics Subject Classification. Primary 06F15.