# Category:Identity Elements

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This category contains results about Identity Elements in the context of Abstract Algebra.

Definitions specific to this category can be found in Definitions/Identity Elements.

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.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

### G

### I

### U

## Pages in category "Identity Elements"

The following 35 pages are in this category, out of 35 total.

### E

### I

- Identities are Idempotent
- Identity Element is Idempotent
- Identity Element of Addition on Numbers
- Identity Element of Multiplication on Numbers
- Identity is Only Group Element of Order 1
- Identity is only Idempotent Cancellable Element
- Identity is only Idempotent Element in Group
- Identity is Unique
- Identity of Algebraic Structure is Preserved in Substructure
- Identity of Cancellable Monoid is Identity of Submonoid
- Identity of Group is in Center
- Identity of Group is in Singleton Conjugacy Class
- Identity of Group is Unique
- Identity of Power Set with Intersection
- Identity of Power Set with Union
- Identity of Subgroup
- Identity of Subsemigroup of Group
- Identity Property in Semigroup
- Induced Structure Identity
- Inverse of Identity Element is Itself