Difference between revisions of "Monoid"
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| − | A '''monoid''' is a set <math>S</math> | + | A '''monoid''' is a set <math>S</math> closed under an [[operation]] <math>\times</math> which is defined everywhere on <math>S</math>, is [[associative]], and has an [[identity]] in <math>S</math>. That is, <math>M = (S, \times)</math> is a monoid if and only if |
| + | * There is a well-defined element <math>a \times b</math> of <math>S</math>, for all <math>a,b \in S</math>; | ||
| + | * <math>a \times (b \times c) = (a \times b)\times c</math> for all <math>a, b, c \in S</math>; | ||
| + | * There is an element <math>e \in S</math> such that <math>e\times a = a \times e = a</math> for all <math>a \in S</math>. | ||
| + | Alternatively, a monoid can be thought of as a [[group]] without [[inverse with respect to an operation | inverses]], or as an associative [[magma]] with an identity. | ||
| + | |||
| + | By abuse of notation, we often identity a monoid with its underlying set. That is, we often refer to a monoid <math>(S,\times)</math> simply as the monoid <math>S</math>, when there is no risk of confusion. | ||
| + | |||
| + | Because the conditions on monoids are so weak, there are very few theorems of "monoid theory." However, monoids do arise from time to time in the study of [[abstract algebra]], and many objects (such as all groups, as well as any [[ring]] with respect to either of its operations) are in fact monoids. | ||
| + | |||
| + | == Monoid Operating on a Set == | ||
| − | + | Let <math>M</math> be a monoid whose law of composition is written multiplicatively and whose identity is <math>e</math>, and let <math>S</math> be a set. Let <math>S^S</math> be the set of [[function]]s on <math>S</math>. We call a mapping <math>a \mapsto f_a</math> from <math>M</math> to <math>S^S</math> a ''left operation of <math>M</math> on <math>S</math>'' if <math>f_e</math> is the identity map on <math>S</math> and for all <math>a,b</math> in <math>M</math>, | |
| + | <cmath> f_{ab} = f_a \circ f_b . </cmath> | ||
| + | (A right operation is defined similarly, except that <math>f_{ab} = f_b \circ f_a</math>.) | ||
| + | In other words, a left operation of <math>M</math> on <math>S</math> is a [[homomorphism]] from the monoid <math>M</math> to the monoid <math>S^S</math>; a right operation is a homomorphism into the opposite monoid of <math>S^S</math>. | ||
| − | + | We may also say that <math>M</math> acts on <math>S</math>. A set <math>S</math> with an action of a monoid <math>M</math> on <math>S</math> is called an <math>M</math>-set. | |
| + | Every monoid acts on the set of its elements. | ||
| − | + | Often one speaks of groups acting on sets. Since elements groups must have unique inverses, for every <math>a</math> in a group <math>G</math> acting on a set <math>S</math>, the function <math>f_a</math> must be a [[bijection]]. | |
| + | If <math>x</math> is an element of <math>S</math>, and <math>a</math> is an element of a monoid <math>M</math> with a left operation on <math>S</math>, we often write <math>f_a(x)</math> simply as <math>ax</math>, when there is no risk of confusion. Then we may rewrite our criteria thus, for <math>a,b</math> in <math>M</math> and <math>x</math> in <math>S</math>. | ||
| + | * <math>ex = x</math>; | ||
| + | * <math>(ab)x = a(bx)</math>. | ||
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{{stub}} | {{stub}} | ||
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| + | [[Category:Abstract algebra]] | ||
Revision as of 21:09, 13 May 2008
A monoid is a set 
closed under an operation 
which is defined everywhere on , is associative, and has an identity in . That is, 
is a monoid if and only if
- There is a well-defined element
of 
, for all 
; 
for all 
;- There is an element
such that 
for all 
.
of 
;
for all 
;
such that 
for all 
.Alternatively, a monoid can be thought of as a group without inverses, or as an associative magma with an identity.
By abuse of notation, we often identity a monoid with its underlying set. That is, we often refer to a monoid 
simply as the monoid , when there is no risk of confusion.
Because the conditions on monoids are so weak, there are very few theorems of "monoid theory." However, monoids do arise from time to time in the study of abstract algebra, and many objects (such as all groups, as well as any ring with respect to either of its operations) are in fact monoids.
Monoid Operating on a Set
Let 
be a monoid whose law of composition is written multiplicatively and whose identity is 
, and let be a set. Let 
be the set of functions on . We call a mapping 
from to a left operation of on if 
is the identity map on and for all 
in ,
![\[f_{ab} = f_a \circ f_b .\]](http://latex.artofproblemsolving.com/e/9/3/e9319a5f24d0c6ee336081be0306ee8191e28d79.png)
(A right operation is defined similarly, except that 
.)
In other words, a left operation of on is a homomorphism from the monoid to the monoid ; a right operation is a homomorphism into the opposite monoid of .
We may also say that acts on . A set with an action of a monoid on is called an -set.
Every monoid acts on the set of its elements.
Often one speaks of groups acting on sets. Since elements groups must have unique inverses, for every 
in a group 
acting on a set , the function 
must be a bijection.
If 
is an element of , and is an element of a monoid with a left operation on , we often write 
simply as 
, when there is no risk of confusion. Then we may rewrite our criteria thus, for in and in .
;
.
;
.
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