Editing Inverse element (section)

=== In a unital magma ===
Let <math>S</math> be a unital [[Magma (algebra)|magma]], that is, a [[Set (mathematics)|set]] with a [[binary operation]] <math>*</math> and an [[identity element]] <math>e\in S</math>. If, for <math>a,b\in S</math>, we have <math>a*b=e</math>, then <math>a</math> is called a '''left inverse''' of <math>b</math> and <math>b</math> is called a '''right inverse''' of <math>a</math>. If an element <math>x</math> is both a left inverse and a right inverse of <math>y</math>, then <math>x</math> is called a '''two-sided inverse''', or simply an '''inverse''', of <math>y</math>. An element with a two-sided inverse in <math>S</math> is called '''invertible''' in <math>S</math>. An element with an inverse element only on one side is '''left invertible''' or '''right invertible'''.

Elements of a unital magma <math>(S,*)</math> may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table
{|class="wikitable" style="text-align:center"
|-
!width=15|*
!width=15|1
!width=15|2
!width=15|3
|-
!1
|1
|2
|3
|-
!2
|2
|1
|1
|-
!3
|3
|1
|1
|}
the elements 2 and 3 each have two two-sided inverses.

A unital magma in which all elements are invertible need not be a [[loop (algebra)|loop]]. For example, in the magma <math>(S,*)</math> given by the [[Cayley table]]
{|class="wikitable" style="text-align:center"
|-
!width=15|*
!width=15|1
!width=15|2
!width=15|3
|-
!1
|1
|2
|3
|-
!2
|2
|1
|2
|-
!3
|3
|2
|1
|}
every element has a unique two-sided inverse (namely itself), but <math>(S,*)</math> is not a loop because the Cayley table is not a [[Latin square]].

Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table
{|class="wikitable" style="text-align:center"
|-
!width=15|*
!width=15|1
!width=15|2
!width=15|3
!width=15|4
!width=15|5
|-
!1
|1
|2
|3
|4
|5
|-
!2
|2
|3
|1
|5
|4
|-
!3
|3
|4
|5
|1
|2
|-
!4
|4
|5
|2
|3
|1
|-
!5
|5
|1
|4
|2
|3
|}
the only element with a two-sided inverse is the identity element 1.

If the operation <math>*</math> is [[associative]] then if an element has both a left inverse and a right inverse, they are equal. In other words, in a [[monoid]] (an associative unital magma) every element has at most one inverse (as defined in this section). In a monoid, the set of invertible elements is a [[group (mathematics)|group]], called the [[group of units]] of <math>S</math>, and denoted by <math>U(S)</math> or ''H''<sub>1</sub>.