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Chapter: Order Relations

A natural ability of the human mind is to compare the size of different things or order them by size. Also in elementary school, we learn how to compare numbers. For instance, any two integer numbers $a\in\mathbb Z$ and $b\in\mathbb Z$ can be compared to each other with the common notations "$a<b$" expressing "$a$ is smaller than $b$", or "$a\ge b$" expressing "$a$ is greater than or equal to $b$".

Let us take a more general look at what happens here and free ourselves from thinking about $a$ and $b$ as numbers. Suppose, $a$ and $b$ could be anything, cars, houses, or mathematical objects which are not necessarily numbers.

• We can always compare $a$ with itself (Reflexivity).
• If $a$ can be compared to $b$ and $b$ can be compared to $c$, then we can compare $a$ with $c$ (Transitivity).
• Suppose, $a$ and $b$ can be compared with each other. Imagine, we want to put $a$ and $b$ into a list and order them by our comparison criteria. There are situations, in which we will have still problems in deciding whether we should put $a$ before $b$ or vice versa. Those situations are exactly when we think $a=b$ with respect to our comparison criteria (Antisymmetry).
• There are situations, in which $a$ and $b$ are not comparable at all.

These properties are strongly related to the properties of binary relations and allow us to define a concept of a generalized order relation, which is applicable to all kinds of mathematical objects and not only to numbers.

Table of Contents

Explanations: 1 2 3

1. Definition: Preorder, Partial Order and Poset
2. Definition: Total Order and Chain
3. Definition: Comparing the Elements of Posets and Chains
4. Definition: Strict Total Order, Strictly-ordered Set
5. Lemma: Comparing the Elements of Strictly Ordered Sets
6. Definition: Special Elements of Ordered Sets
7. Definition: Bounded Subsets of Ordered Sets
8. Definition: Bounded Subsets of Unordered Sets
9. Lemma: Zorn's Lemma
10. Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice
11. Definition: Well-order, Well-ordered Set
12. Definition: Order Embedding

Mentioned in:

Branches: 1
Chapters: 2
Definitions: 3
Parts: 4

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References

Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition