Some sharply transitive partially ordered sets

A partially ordered set (X, is called sharply transitive if its automorphism group is sharply transitive on X, that is, it is transitive and the stabilizer of every element is triviaL It is shown that every free group is the automorphism group of a sharply transitive partially ordered set. It is also shown that there exists a sharply transitive partially ordered set (-,Y, :::;) having some maximal chains isomorphic to the rationals and automorphism group isomorphic to the additive group of a vector space of dimension two over the rationals. The automorphism group A'u,t( .. Y,:::;) of a partially ordered set (X,:::;) is the group of all permutations 9 of X such that x :::; y if and only if xg S; yg for all x, y E X. The partially ordered set (X,:::;) is called sharply transitive if Aut(X,:::;) is sharply transitive on X, that is, it is transitive and the stabilizer of every element is trivial. Sharply transitive linearly ordered sets were first studied by Tadashi Ohkuma [5], [6], and later by A.M.W. Glass, Yuri Gurevich, W. Charles Holland and Saharon Shelah [4] (see also [3],[7]). The author gave some constructions and non-existence results for sharply transitive partially ordered sets in [1] and [2]. Australasian Journal of Combinatorics 4( 1991) I PP 269-275 If is the (full) aUl:;,om.OITm group of n-::'lr'r.I-.:r transitive partially n1',nI01'C,r! set then either G has at most or G contains this condition is not sufficient ...,n'-lr"nIIH transitive paran element of infinite order. 2.1 in [1]). All 'V.., .. "",' ..... IJ' ... ,~u tially ordered sets [2] contain an infinite group in their centre, We shall show in this paper that this is not a necessary has a trivial more than one is to the ","U'IiVJ.JiJ.'JJ. transitive partially nT"nlL-H'L."> set. Another comInon i-D~i-ll1"C> sets in [1] [2] is that nlaxirnal chains are "' .... r"to ..... 'e,l, ..... ' We shall construct a v\J'-'n the additive group a vector space "1,.,(\,o1'lc"",, two over Theorem 1. Let F be ~oopoo m m~y there exists on F such that Aut(F, is transitive on F and is x < aiao:r closure of we have to to .) be by {aili E I} where I ,n} for some n E and let I' 1\ {a}. For x F and i E l' define x aox, < ao Let ::; be the transitive relation. Tn order to show that it is a partial order, that it is antisymmetric. Suppose x, y E F with x < y and y < x. Then there exist Cl,'" ,Cr , Cr+l,'" ,Cs E {ao,aiao, ao i+l ai -Iii E I'} such that x = Cl ... CrY and y = Cr+l ... CsX. Hence