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 .
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 , (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.
We say that a monoid's action on is faithful if the mapping is injective, i.e., for any distinct , there exists some for which .
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 .
We may also identify the function with , thus writing instead of .
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