Definition:Identity (Abstract Algebra)/Two-Sided Identity

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Definition

Let $\struct {S, \circ}$ be an algebraic structure.

An element $e \in S$ is called an identity (element) if and only if it is both a left identity and a right identity:

$\forall x \in S: x \circ e = x = e \circ x$


In Identity is Unique it is established that an identity element, if it exists, is unique within $\struct {S, \circ}$.

Thus it is justified to refer to it as the identity (of a given algebraic structure).


This identity is often denoted $e_S$, or $e$ if it is clearly understood what structure is being discussed.


Also known as

Other terms which are seen that mean the same as identity are:

  • Two-sided identity, to reflect the fact that it is both a left identity and a right identity.
  • Neutral element, which is perfectly okay, but considered slightly old-fashioned.
  • Unit element, but this is not recommended as it is too easy to confuse it with a unit of a ring.
  • Unity, but this is generally reserved for a ring unity or unity of field.
  • Zero, but it is best to reserve that term for a zero element.
  • The trivial element, in the context of a group.

The symbols used for an identity element are often found to include $0$ and $1$. In the context of the general algebraic structure, these are not recommended, as this can cause confusion.

Some sources use $I$ for the identity.


Examples

Symmetry Group of Square

Consider the symmetry group of the square:


Let $\mathcal S = ABCD$ be a square.

SymmetryGroupSquare.png

The various symmetry mappings of $\mathcal S$ are:

The identity mapping $e$
The rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ counterclockwise respectively about the center of $\mathcal S$.
The reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively.
The reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$.
The reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square.


The mapping $e$ which leaves $\SS$ unchanged is the identity element.


Also see

  • Results about identity elements can be found here.


Sources