This is a variation of group|Find other variations of group | Read a survey article on varying group
This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
View other such properties
This forms a variety of algebras
This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Monoid, all facts related to Monoid) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki
QUICK PHRASES: group without inverses, set with associative binary operation with identity element, semigroup with identity element (neutral element)
Definition with symbols
A monoid is a set together witha binary operation and an element such that:
Note that monoid differs from the group in the sense of there being no guarantee for the existence of inverses. Thus, every group can be viewed as a monoid but not vice versa.
Facts in the definition
The neutral element (also called identity element) in a monoid is unique. For full proof, refer: Neutral element
Examples and properties
Occurrence of monoids
Monoids typically occur as collections of transformations which are closed under composition. For instance, the endomorphisms of a group form a monoid under composition: the identity element here is the identity map and the multiplication is the usual composition.
The invertible elements within the monoid form a group. If we are looking at a monoid of transformations, the invertible elements sitting there give transformations that have inverse transformations, or are reversible.
Relation with other structures
- Group is a monoid where every element has an inverse with respect to the identity element
- Inverse monoid is a monoid where every element has an inverse in a somewhat weaker sense
- Semigroup is a set with an associative binary operation -- there may or may not be a neutral element
- Magma is simply a set with a binary operation
Notation for the monoid operations
We typically omit the multiplication symbol when referring to the monoid operation. We also omit parentheses on account of associativity. (refer associative binary operation#Parenthesization can be dropped).
The identity element is usually denoted as .
- Group Theory: A First Journey is an article making a reference to monoids in an attempt to motivate and build up to the definition of group