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.

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