# Category:Identity Elements

Jump to navigation
Jump to search

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 4 subcategories, out of 4 total.

### G

### I

## Pages in category "Identity Elements"

The following 32 pages are in this category, out of 32 total.

### E

### I

- Identities are Idempotent
- Identity Element is Idempotent
- 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
- Inverse of Identity Element is Itself