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Cores

Having such a dismantling process available it is reasonable to ask when it might stop (compare with the removal step for Xia's reduction algorithm and Proposition 2.21). The problems encountered at the limit ordinal are quite subtle (for example in their worst form encountered when trying to infinite-dismantle the one-way infinite fence) and are dealt with in [70]-[74], [76] and [77], where conditions on the ordered set are given that ensure a smooth transition past the limit ordinal. We will consider the other possible stopping point here, namely the situation in which at a certain stage the set does not have a suitable retraction any more. (In finite ordered sets this is the only way the dismantling process stops.)

define4042

Easy examples show that a set can have many cores (every singleton subset of a finite fence is a tex2html_wrap_inline9130 -core of that fence), but one can show the tex2html_wrap_inline9130 -core of a finite set is unique up to isomorphism (cf. [31], [39]).

define4052

  lem4062

Proof: Let tex2html_wrap_inline7890 be such that tex2html_wrap_inline9174 . Then since the set of points that are not fixed by f has tex2html_wrap_inline9178 elements, there are distinct tex2html_wrap_inline9180 such that tex2html_wrap_inline9182 . Choose i<j both minimal such that tex2html_wrap_inline9182 . Then tex2html_wrap_inline9188 and tex2html_wrap_inline9190 . Moreover for all tex2html_wrap_inline9192 we have

displaymath9194

Thus

displaymath9196

Since tex2html_wrap_inline9198 for all fixed points of f, we are done \

  lem4082

Proof: For tex2html_wrap_inline9218 let tex2html_wrap_inline9220 be the smallest ordinal such that tex2html_wrap_inline9222 for all tex2html_wrap_inline9224 . (Such tex2html_wrap_inline9226 exists since the family tex2html_wrap_inline9228 is infinitely composable.) Now since tex2html_wrap_inline9230 , we have tex2html_wrap_inline9232 \

        theorem4095

Proof: Part 1a is proved in [119], the finite case being proved in [31] and in [39]. The proof is similar as the proof for part 2 that we are about to present. Parts 1b and 1c are easy as there is a largest up-retraction, resp. a smallest down-retraction on every chain-complete ordered set.
For part 2 let tex2html_wrap_inline9262 be an tex2html_wrap_inline9264 -core such that there is an infinite tex2html_wrap_inline9264 -dismantling S of P onto C. Let tex2html_wrap_inline9274 be an infinite tex2html_wrap_inline9264 -dismantling, which is the infinite composition of the tex2html_wrap_inline9264 -retractions tex2html_wrap_inline9280 . We will prove inductively that for all tex2html_wrap_inline9282 we have tex2html_wrap_inline9284 . For tex2html_wrap_inline9286 this is trivial and for tex2html_wrap_inline9282 a limit ordinal this follows from Lemma 3.13. If tex2html_wrap_inline9282 is a successor ordinal, note that tex2html_wrap_inline9292 . Since tex2html_wrap_inline9294 is an tex2html_wrap_inline9264 -retraction and since tex2html_wrap_inline9298 is injective, we have that tex2html_wrap_inline9300 . Thus if tex2html_wrap_inline9302 , then tex2html_wrap_inline9304 is a retraction by Lemma 3.12 and tex2html_wrap_inline9306 . Since C is an tex2html_wrap_inline9264 -core, this means that tex2html_wrap_inline9312 , i.e., that tex2html_wrap_inline9314 .
Thus tex2html_wrap_inline9316 . Now suppose tex2html_wrap_inline9318 are tex2html_wrap_inline9264 -cores of P with tex2html_wrap_inline9324 being an infinite tex2html_wrap_inline9264 -dismantling to tex2html_wrap_inline9328 (i=1,2). Then by the above tex2html_wrap_inline9332 and tex2html_wrap_inline9334 , and hence tex2html_wrap_inline9336 and tex2html_wrap_inline9338 are isomorphic.
For part 3 let P be finite and let tex2html_wrap_inline9262 be an tex2html_wrap_inline9344 -core such that there is an tex2html_wrap_inline9344 -dismantling S of P onto C. Let tex2html_wrap_inline9274 be an tex2html_wrap_inline9344 -dismantling which is the composition of the tex2html_wrap_inline9344 -retractions tex2html_wrap_inline9280 , where tex2html_wrap_inline9362 is finite. We will prove inductively that for all tex2html_wrap_inline9282 there is an order-preserving mapping tex2html_wrap_inline9366 such that tex2html_wrap_inline9368 . For tex2html_wrap_inline9286 this is trivial (choose tex2html_wrap_inline9372 ) and since tex2html_wrap_inline9362 is finite we do not need to consider limit ordinals. If tex2html_wrap_inline9282 is a successor ordinal, note that tex2html_wrap_inline9378 . Since tex2html_wrap_inline9294 is an tex2html_wrap_inline9344 -retraction and since tex2html_wrap_inline9298 is injective, we have that tex2html_wrap_inline9386 . Thus if tex2html_wrap_inline9388 , then tex2html_wrap_inline9390 is a retraction by Lemma 3.12 and tex2html_wrap_inline9392 . Since C is an tex2html_wrap_inline9344 -core, this means that tex2html_wrap_inline9398 . Thus tex2html_wrap_inline9400 is the map with tex2html_wrap_inline9402 .
Since the dismantling of P stops after finitely many steps, there is an order-preserving map R such that tex2html_wrap_inline9408 . Now suppose tex2html_wrap_inline9318 are tex2html_wrap_inline9344 -cores of P with tex2html_wrap_inline9324 being a finite dismantling to tex2html_wrap_inline9328 (i=1,2). Then by the above there are order-preserving maps tex2html_wrap_inline9422 such that tex2html_wrap_inline9424 and tex2html_wrap_inline9426 . Hence tex2html_wrap_inline9336 and tex2html_wrap_inline9338 have the same number of elements. But then tex2html_wrap_inline9432 is an isomorphism from tex2html_wrap_inline9338 onto tex2html_wrap_inline9436 . \

cor4191

Proof: Existence follows from the main theorem in [70], as for ordered sets with no one-way infinite fence and no infinite chains infinite tex2html_wrap_inline9130 -dismantlability is equivalent to the notion of dismantlability used in [70]. \

define4199


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Next: Fixed Point Theorems/Reflection Conditions Up: Reduction Theorems Previous: Dismantlability

Bernd.S.W.Schroder