Difference between revisions of "Monoid"
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* 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>. | * 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>. | ||
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| + | Alternatively, a monoid is a [[group]] without [[inverse with respect to an operation | inverses]]. | ||
Revision as of 15:50, 16 October 2006
A monoid is a set 
with an operation 
which is associative and has an identity. That is, 
is a monoid if and only if
for all 
for all 
- There is an element
such that 
for all 
.
such that 
for all 
.
Alternatively, a monoid is a group without inverses.
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.
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