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
Contents
[hide]Definition
A monoid[1] is a set S and a function ×S:S×S→S (called the operation) such that ×S is:
- Associative - that is ∀x,y,z∈S[(xy)z=x(yz)]
- Has identity element - that is ∃e∈S∀x∈S[ex=xe=x]
(Here xy denotes ×S(x,y) which being an operator would be written x×Sy)
Abelian monoid
A monoid is Abelian or commutative if:
- ∀x,y∈S[xy=yx]
See also
References
- Jump up ↑ Algebra - Serge Lang - Revised Third Edition - Graduate Texts In Mathematics
