The ANTI-order for caccc posets — part I

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This 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 = ( P ξ : ξ ⩽ λ), of subsets of a caccc poset P is called an ANTI-perfect sequence of P, if P − P 0 and, for each ξ < λ, P ξ+1 , is a ▪ ξ-good subset of P ξ , where ▪ ξ is the ANTI-order on P ξ , and P ξ = ⋒{P η:η < ξ} when ξ is a limit ordinal, and Pλ is a ▪ λ-good subset of itself. We call P λ 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.