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By Manfred Droste

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Let b £ 9, \ M. We distinguish between three cases. Observe a A.. and B £ S (M) i ^ j . g. let 6 a - a a is an isomorphism, we can assume b £ A. v. {a Clearly M - a < C. a,b £ M. We have to show that = cp. i with w£t/z l £ z \^J (A. \ {a }) , we can define i €* X We claim 6 € A(M) . , where M M = Let exists \I^M a. a chain i*ui+r £ Zj . T/zen there % A^ i £ 1} c Q a £ M, £ M b || M, hence = M and b || a and a || a . b < a. and for some b < a. Thus i £ Z. Then b a < M a and b < a. b a for all j £ Z, a < a .

Ft = C x r For any two elements (b) two holds: and contains and at least 2-transitive3 some finite (c) ft - ft' 0 IP(Q) r £ M is conditions (b) ft - ft? 0 ft„ such r £M in ft containing {a^^b^} is chain. exists a £ ft with c £ ft with c £ Q with chain. a . II b . 2). < c p. Let ft be non-trivially ordered. 7 implies (1). Hence we may assume that there exists an infinite chain in ft. 7(e), and all other compo- nents of ft are isomorphic to if C c N. 7) (d). 3, A(C) is k-transitive and hence also 2-transitive.

B •> C isomorphisms 45 TT>, with TT0 a ' = d, b z = c; here put TTL = TT. , T T L = TT-. In any of these cases, TT is an isomorphism as desired. tyi By assumption (2i) , there exists an isomorphism by (2ii) we have either obtain a € A(ft) x,y £ ft or with a| The result follows. 20 we za = z = TT and x -> , and for all z £ ft with e ^ z. 17. 2. 23. The a. < a. ) with Let A = { a . ; i £ Z } c C c f t = a. L with 2-homogeneous. an isomorphism a| tree 2-transitive. below in ft. Then there Proof.

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