Structure of partially ordered sets with transitive automorphism groups
Manfred Droste · Memoirs of the American Mathematical Society · 1985
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