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p-Chain – Road To Zorn’s Lemma Part 1

Hello, infinite-set reader. Set theory is back.

Starting this post, we’ve entered about Zorn’s Lemma. This discussion will start from several definitions and theorems that later lead to Zorn’s Lemma. Happy reading.

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For any poset it is not guaranteed that every non-empty well ordered subset has an upper bound. It automatically does not have the smallest upper bound too. As an example \left\{ \left(a_{1};a_{2};a_{3};\ldots\right),\left(b_{1};b_{2};b_{3};\ldots\right)\right\}. Any non-empty well ordered subset of the poset does not have the smallest upper bound.

In this discussion, we will discuss about posets that each of the non-empty well ordered subset have the smallest upper bound.

Definition :

Given any poset \left(P,\leq\right) with each of the non-empty well ordered subset having the smallest upper bound on P. Given function f:P\rightarrow P with x\leq f\left(x\right) for all x\in P. For any p\in P defined p-chain that denoted byC\left(p,r\right) under the condition :

(i) C\left(p,r\right) is well ordered subset with p being the smallest element and r is the largest element.

(ii) For all x^{+}\in C\left(p,r\right) apply x^{+}=f\left(x\right)

(iii) The smallest upper bound of any non-empty subset of C\left(p,r\right) is in C\left(p,r\right)

In the above definition, the condition (ii), such terms clearly apply only tox\in C\left(p,r\right)-\left\{ r\right\}. It because, if x=r is always true that r\leq r^{+}. On the other hand, r^{+}\in C\left(p,r\right). Then there is a contradiction with the definition of p-chain conditions (i) that r is the largest element of C\left(p,r\right). Next defined

    \[W\left(p\right)=\left\{ r\in P|\exists p\in P\mbox{ such that }C\left(p,r\right)\right\}\]

Next will be given properties about p-chain C\left(p,r\right) and W\left(p\right) which is presented in the theorems below.

Theorem : (more…)