Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

This article defines a property of elements or tuples of elements with respect to a binary operation

Contents 1 Definition 1.1 Definition with symbols 2 Facts 2.1 Any left neutral and right neutral element are equal 2.2 Some easy corollaries 3 Relation with other properties 3.1 Weaker properties

Definition

Definition with symbols

Given a binary operation * on a set S (i.e., a magma (S, * )), an element e in S is termed:

• left neutral or a left identity if e * a = a for any a in S
• right neutral or a right identity if a * e = a for any a in S
• neutral if it is both left and right neutral

A neutral element is also termed an identity element.

Facts

Any left neutral and right neutral element are equal

The proof of this fact goes as follows: let e₁ be a left neutral element and e₂ be a right neutral element. Then, the product e₁ * e₂ is equal to e₁ (because e₂ is right neutral) and is also equal to e₂ (because e₁ is left neutral). Hence, e₁ = e₂.

For full proof, refer: Equality of left and right neutral element

Some easy corollaries

• If there exists a left neutral element, there can exist at most one right neutral element; moreover, if it exists, then it is the same as the left neutral element and is hence a neutral element
• If there exists a right neutral element, there can exist at most one left neutral element; moreover, if it exists, then it is the same as the right neutral element and is hence a neutral element
• There can exist at most one neutral element. Thus, if a neutral element exists, it is unique

Relation with other properties

Weaker properties

• Idempotent element
• Cancellative element: Any left neutral element is left cancellative, and any right neutral element is right cancellative. Hence, any neutral element is cancellative