310: Linearly-Ordered Set

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A definition of linearly-ordered set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of set.
• The reader knows a definition of relation.

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Target Context

• The reader will have a definition of linearly-ordered set.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

Any set with any relation, $⟨S,R⟩$, such that 1) $R$ is a trichotomy for any element, $p_1\in S$: for any element, $p_2,\in S$, exclusively $p_{1}R{p}_2$, $p_1=p_2$, or $p_{2}R{p}_1$; 2) $R$ is transitive: for any elements, $p_1,p_2,p_3\in S$, such that $p_{1}R{p}_2$ and $p_{2}R{p}_3$, $p_{1}R{p}_3$

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2: Note

Any linearly-ordered set is a kind of partially-ordered set.

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References

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