A description/proof of unique existence of monoid identity element
Topics
About: monoid
The table of contents of this article
Starting Context
- The reader knows a definition of monoid.
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 \(p_1 \bullet i = p_1\), \(p_1 = i\).
