311: Maximal Element of Set

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A definition of maximal element of 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 partially-ordered set.

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Target Context

• The reader will have a definition of maximal element of 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

For any partially-ordered set, $⟨S,R⟩$, any element, $p\in S$, such that there is no element, $p^{\prime }\in S$, such that $pR{p}^{\prime }$

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2: Note

There may be multiple maximal elements when $R$ is properly partial (meaning non-linear), because 2 maximal elements, $p_1,p_2\in S$, may be just not related.

For any linearly-ordered set, which is a kind of partially-ordered set, there can be no multiple maximal elements, because if $p_1,p_2\in S$ were maximal, exclusively $p_{1}R{p}_2$, $p_1=p_2$, or $p_{2}R{p}_1$, but the 1st and the 3rd cases are impossible, because $p_1$ or $p_2$ respectively would not be maximal.

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References

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