# 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 $S$ together witha binary operation $*$ and an element $e \in S$ 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 $e$.