# Monoid

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This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.

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*This forms a variety of algebras*

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QUICK PHRASES: group without inverses, set with associative binary operation with identity element, semigroup with identity element (neutral element)

## Definition

### Symbol-free definition

A **monoid** is a set equipped with a binary operation that is associative and has a neutral element (or identity 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

### Stronger 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

### Weaker structures

- 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

### 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 .

## Mathjourneys links

- 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