Difference between revisions of "Monoid"
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* Has identity element - that is <math>\exists e\in S\forall x\in S[ex=xe=x]</math> | * Has identity element - that is <math>\exists e\in S\forall x\in S[ex=xe=x]</math> | ||
(Here {{M|xy}} denotes {{M|\times_S(x,y)}} which being an operator would be written {{M|x\times_S y}}) | (Here {{M|xy}} denotes {{M|\times_S(x,y)}} which being an operator would be written {{M|x\times_S y}}) | ||
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| + | ===Abelian monoid=== | ||
| + | A monoid is '''Abelian''' or '''commutative''' if: | ||
| + | * {{M|1=\forall x,y\in S[xy=yx]}} | ||
==See also== | ==See also== | ||
Latest revision as of 07:48, 27 April 2015
Not to be confused with group
Definition
A monoid[1] is a set [ilmath]S[/ilmath] and a function [ilmath]\times_S:S\times S\rightarrow S[/ilmath] (called the operation) such that [ilmath]\times_S[/ilmath] is:
- Associative - that is [math]\forall x,y,z\in S[(xy)z=x(yz)][/math]
- Has identity element - that is [math]\exists e\in S\forall x\in S[ex=xe=x][/math]
(Here [ilmath]xy[/ilmath] denotes [ilmath]\times_S(x,y)[/ilmath] which being an operator would be written [ilmath]x\times_S y[/ilmath])
Abelian monoid
A monoid is Abelian or commutative if:
- [ilmath]\forall x,y\in S[xy=yx][/ilmath]
See also
References
- ↑ Algebra - Serge Lang - Revised Third Edition - Graduate Texts In Mathematics
