The ANTI-order for caccc posets — part I

AbstractThis paper deals with a generalization of the following simple observation. Suppose there are distinct elements a, b of the chain complete poset (P, ⩽) such that P(< a) ⊆ P(< b) and P(> a) ⊆ P(> b); if P(< a) and P(> a) are both fixed point free (fpf), then P is also fpf (we say P is trivially fpf), otherwise, P has the fixed point property (fpp) if and only if P{a} has this property. We introduce a new quasi-order on a poset (P, ⩽), called the ANTI-order denoted by , where holds if and only if every element strictly comparable with x is also strictly comparable with y. A set X ⊆ P is an ANTI-good subset of P, if X is maximal (with respect to inclusion) and its elements are -maximal and pairwise -incomparable. A poset (P, ⩽) is caccc if it is chain complete and every countably infinite antichain has a supremum (infimum) whenever the antichain is bounded above (below). The caccc property is preserved by retracts and the intersection of a decreasing chain of caccc subposets also has this property. We show that for a caccc poset (P, ⩽) an ANTI-good subset is a retract and it is uniquely determined up to isomorphism. Moreover, if P is not trivially fpf, then P has the fpp if and only if an ANTI-good subset has the fpp. A strictly decreasing sequence, Π = (Pξ : ξ ⩽ λ), of subsets of a caccc poset P is called an ANTI-perfect sequence of P, if P − P0 and, for each ξ < λ, Pξ+1, is a ξ-good subset of Pξ, where ξ is the ANTI-order on Pξ, and Pξ = ⋒{Pη:η < ξ} when ξ is a limit ordinal, and Pλ is a λ-good subset of itself. We call Pλ an ANTI-core of P. Our main result is that an ANTI-core of a caccc poset is a retract. The proof of this will be given separately in the second part of the paper [5]. In this part we establish the existence of ANTI-perfect sequences

The ANTI-order for caccc posets — Part II

AbstractIn Part I we defined the ANTI-order, ANTI-good subsets, ANTI-perfect sequences and ANTI-cores for caccc posets. In this part we prove the main result: If Π − (Pξ : ξ ⩽ λ) is an ANTI-perfect sequence of a connected caccc poset P which does not contain a one-way infinite fence, then Pξ is a retract of P for all ξ ⩽ λ