2022-06-19

307: Unique Existence of Monoid Identity Element

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A description/proof of unique existence of monoid identity element

Topics


About: monoid

The table of contents of this article


Starting Context



Target Context


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

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.


Main Body


1: Description


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


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 \(p_1 \bullet i = p_1\), \(p_1 = i\).


References


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