Ordered sets with no infinite antichains

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One approach to analyzing the structure of an ordered set has been to characterize its structure in terms of that of related objects, and vice versa. For example results in [2] characterize in terms of the ordered set P when the lattice Z(P) of lower sets (also called initial segments) of P and the ordered set Id(P) of order ideals (or simply ideals) of P have infinite antichains. The results in [2] are that Z(P) has an infinite antichain if and only if P has an infinite antichain or P has a copy of one of three specific ordered sets; likewise, it is shown that Id(P) has an infinite antichain if and only if P has one or P has a copy of a certain ordered set. In this paper we ‘dismantle’ these results by characterizing separately when P has an infinite antichain or a copy of each of the ordered sets which gives an infinite antichain in Z(P) or Id(P). We begin by describing the ordered sets which play a role in these results. We use w to denote the natural numbers in their usual order, 1 G 2 c 3, . . . , and md denotes the natural numbers in the opposite order, 12 223, . . . . By o @ md, we mean the disjoint union of w and md, where each is given the order indicated, and there is no comparison between elements of w and those of gd. Thus, an ordered set contains a copy of o @ md if and only if it has an infinite ascending chain and an infinite descending chain such that any pair of elements, one from each chain, is incomparable. Let K = {(i, j) E N X N 1 i < j}, and define an order on K by (i, j) s (r, s) if and only if i = r and j c s, or j < r. The Hasse diagram of K is given in Fig. 1, along with that of Kd, the set K with the opposite order.