PARTIALLY ORDERED SETS WITH TRANSITIVE AUTOMORPHISM GROUPS MANFRED DROSTE [Received 5 September 1985] ABSTRACT In this paper, we study the structure of infinite partially ordered sets (Q, ==) under suitable transitivity assumptions on their group A(£l) of all order-automorphisms of (Q, =s). Let us call A(Q) ife-transitive (Jfc-homogeneous) if whenever A, B are two isomorphic subsets of Q each with k elements, then some (any) isomorphism from (A, =£) onto (JB, «s) extends to an automorphism of Q, respectively. We show that if it & 4 (k = 3), there are precisely k (5) non-isomorphic countable partially ordered sets (Q, =£) not containing the pentagon such that A(Q) is A:-transitive but not ^-homogeneous; if k = 2, there are a unique countable, and many different uncountable sets (Q, «s) of this type. We also give necessary and sufficient conditions for two partially ordered sets (Q, =s) not containing the pentagon and with Jt-transitive automorphism group (fc>2) to be L^-equivalent. 1. Introduction and results In this paper let (Q, ^) always be an infinite partially ordered set (p.o. set) and A(Q) = Aut((Q, «s)) be the group of all order-automorphisms of Q. Let k e fol. We call A(Q) k-transitive (k-homogeneous), if whenever A, B c Q each have k elements and cp: A^>B is an isomorphism, then there exists a eA(Q) with A a = B (a\ A = cp), respectively. Further A(Q) is co-transitive (co-homogeneous), if A(Q) is ^-transitive (^-homogeneous) for each k e M, respectively. Trivially, ^-homogeneity implies k -transitivity of A(Q) for each k e N, and to-homogeneity implies co-transitivity. Linearly ordered sets (chains) (Q, ^ ) with 2-transitive automorphism groups have been extensively studied. Their automorphism groups A(Q) have been used for example for the construction of certain infinite simple torsion-free groups (Higman [23]), or, in the theory of lattice-ordered groups, in dealing with embeddings of arbitrary lattice-ordered groups into simple divisible ones (Hol- land [26]). The interplay between the structure of these chains and the structure of the normal subgroup lattices of their automorphism groups was studied in [1, 2, 11, 13, 15, 26]. Obviously, all linearly ordered fields are examples of such chains. For a variety of further results in this area see Glass [20]. Henson [22] showed that there are 2*° non-isomorphic countable binary relational structures with co -homogeneous automorphism group; precisely countably many of these are graphs (Lachlan and Woodrow [29]), see also [17-19, 28, 34]. Schmerl [35] characterized all countable p.o. sets with co-homogeneous automorphism group. Further related work is listed in the references. In this paper we examine the structure of infinite p.o. sets (Q, «s) of arbitrary cardinality under the assumption that A(Q) is ^-transitive for some 2 «£ k e N. A study of this kind was proposed by Wielandt [39] and begun in [11,12], where we obtained a classification of these partial orders and in almost all cases a characterization of the condition that A(Q) is A;-transitive for some k 3= 2 by the A.M.S. (1985) subject classification: primary 06A10; secondary 20B22, 20B27, 03C35. Proc. London Math. Soc. (3) 54 (1987) 517-543.