307: Unique Existence of Monoid Identity Element

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A description/proof of unique existence of monoid identity element

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Topics

About: monoid

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of monoid.

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Target Context

• The reader will have a description and a proof of the proposition that there is only one identity element in any monoid.

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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: Description

For any monoid, $(S,\bullet )$, there is only one identity element, $i\in S$.

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2: Proof

For any element, $p_1\in S$, such that for any element, $p_2\in S$, $p_1\bullet p_2=p_2\bullet p_1=p_2$, $p_2$ can be i, so, $p_1\bullet i=i$, but as $p_1\bullet i=p_1$, $p_1=i$.

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References

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