Chain-complete Poset Research Articles

On the Bourbaki–Witt principle in toposes

AbstractThe Bourbaki–Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically.We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki–Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes.

On the Existence of Optima in Complete Chains and Lattices

A necessary and sufficient condition for a preference ordering defined on a chain-complete poset to attain its maximum in every subcomplete chain is obtained. A meet-superextremal, or join-superextremal, function on a complete lattice attains its maximum in every subcomplete sublattice if and only if it attains a maximum in every subcomplete chain.

Fixpoint Theorem for Continuous Functions on Chain-Complete Posets

Fixpoint Theorem for Continuous Functions on Chain-Complete PosetsThis text includes the definition of chain-complete poset, fix-point theorem on it, and the definition of the function space of continuous functions on chain-complete posets [10].

Perfect sequences of chain-complete posets

The formation of a perfect sequence for a chain-complete poset generalizes the process of dismantling a finite poset by irreducibles. In the finite case, according to a theorem of Duffus and Rival, the end result, or ‘core’, is unique up to isomorphism, no matter how the poset is dismantled. For chain-complete posets with no infinite antichains, the core is unique up to isomorphism, finite, and every perfect sequence has finite length, by an important and difficult theorem of Li and Milner. Li has asked if cores of chain-complete posets with no one-way infinite fence Fω and no tower are all isomorphic. He has also asked if the number of steps in the dismantling process, the length of the perfect sequence, is uniquely determined. The following results are obtained. (1) An example refuting the length conjecture is presented. (2) If at least one perfect sequence of a chain-complete poset has length λ < ω 2, then they all have length less than λ + ω, and their cores are isomorphic. (3) Both the isomorphism class of the core and the length of a perfect sequence are unique for posets with no F ω and no infinite chains; at every step of a perfect sequence, the corresponding poset is unique up to isomorphism. (4) A new, quick proof, perhaps yielding new insights, is presented of the theorem of Li and Milner.

The ANTI-order for caccc posets — part I

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.

A chain complete poset with no infinite antichain has a finite core

It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order ⊴ defined onP so thata⊴b holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence Π = 〈P ξ :ξ≤λ〉 of a chain complete posetP represents a canonical procedure to produce a coreP λ. It has been proved that if the posetP contains no infinite antichain then this coreP λ is a retract ofP andP has the fixed point property iffP λ has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.

The PT-order and the fixed point property

The PT-order, or passing through order, of a poset P is a quasi order ⊴ defined on P so that a⊴b holds if and only if every maximal chain of P which passes throug a also passes through b. We show that if P is chain complete, then it contains a subset X which has the properties that (i) each element of X is ⊴-maximal, (ii) X is a ⊴-antichain, and (iii) X is ⊴-dominating; we call such a subset a ⊴-good subset of P. A ⊴-good subset is a retract of P and any two ⊴-good subsets are order isomorphic. It is also shown that if P is chain complete, then it has the fixed point property if and only if a ⊴-good subset also has the fixed point property. Since a retract of a chain complete poset is also chain complete, the construction may be iterated transfinitely. This leads to the notion of the “core” of P (a ⊴-good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.

Finite antichain cutsets in posets

In 1985 I. Rival and N. Zaguia conjectured that in a chain complete poset having the finite cutset property every element belongs to an antichain cutset iff there is no alternating-cover cycle. In this paper a counterexample is given to show that the condition that there is no alternating-cover cycle is not sufficient and other two conditions on generalized alternating-cover paths are added to complete the characterization.

A chain decomposition theorem

It is proved that every chain-complete poset with the finite cutset property is the union of countably many chains.

Chain-complete posets and directed sets with applications

Let a poset P be called chain-complete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets satisfying some sort of chain-completeness condition (see [,11, [-31, [6], I-71, [17], [,181, ['191, [,211, [,221), and used them in a variety of applications. In this paper we study the notion of chain-completeness and demonstrate its usefulness for various applications. Chain-complete posets behave in many respects like complete lattices; in fact, a chaincomplete lattice is a complete lattice. But in many cases it is the existence of sup's of chains, and not the existence of arbitrary sup's, that is crucial. More generally, let P be called chain s-complete when every chain of cardinality not greater than ~ has a sup. We first show that if a poset P is chain s-complete, then every directed subset of P with cardinality not exceeding ct has a sup in P. This sharpens the known result ([,8], [,181) that in any chain-complete poset, every directed set has a sup. Often a property holds for every directed set i f and only if it holds for every chain. We show that direct (inverse) limits exist in a category if and only if 'chain colimits' ('chain limits') exist. Since every chain has a well-ordered cofinal subset [11, p. 681, one need only work with well-ordered collections of objects in a category to establish or disprove the existence of direct and inverse limits. Similarly, a topological space is compact if and only if every 'chain of points' has a cluster point. A 'chain of points' is a generalization of a sequence. Chain-complete posers, like complete lattices, arise from closure operators in a fairly direct manner. Using closure operators we show how to form the chaincompletion P of any poset P. The chain-completion/~ of a poset P is a chain-complete poset with the property that any chain-continuous map from a poser P into a chain-complete poset Q extends uniquely to a chain-continuous map from the completion/~ into Q, where by a chaincontinuous map we mean one that preserves sup's of chains. If P is already chaincomplete, then/~ is naturally isomorphic to P. This completion is not the MacNeille