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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.

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.

| | | | created: 2014-02-20 23:35:39 | modified: 2019-02-01 00:05:18 | by: bookofproofs | references: [979]

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.Explanation: Summary of Different Order Relations

7.Explanation: Hasse Diagram

8.Definition: Special Elements of Ordered Sets

9.Explanation: Notes on Special Elements of Posets

10.Definition: Bounded Subsets of Ordered Sets

11.Definition: Bounded Subsets of Unordered Sets

12.Lemma: Zorn's Lemma

13.Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice

14.Definition: Well-order, Well-ordered Set

15.Definition: Order Embedding

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[979] Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10