Identity element

An identity element for a binary operation leaves every element of its set unchanged when used with it on the appropriate side. A two-sided identity does so on both sides.

Binary operation: A rule that combines each pair of elements from a set to produce an element. An identity element is defined relative to a particular binary operation.
Identity function: A function that maps every element of its domain to itself. Its unchanged-output rule is the function analogue of an identity element.
Zero (number): The additive identity of the integers, rational numbers, real numbers, and related number systems. Adding zero leaves every number unchanged.
Semigroup: A set equipped with an associative binary operation. Adding an identity to a semigroup produces a monoid.
Left zero: An element that absorbs every element when it appears on the left of a binary operation. It produces itself rather than preserving the other input.
Set: A collection of distinct objects treated as a single mathematical object. The identity and the elements it leaves unchanged belong to the operation's set.
Uniqueness of identity elements: The theorem that a binary operation has at most one two-sided identity element. Associativity is not needed: any two-sided identity must equal every other one.
One (number): The multiplicative identity of the integers, rational numbers, real numbers, and related number systems. Multiplying by one leaves every number unchanged.
Monoid: A set with an associative binary operation and a two-sided identity element. A monoid makes the identity element part of its defining structure.
Right zero: An element that absorbs every element when it appears on the right of a binary operation. It is a one-sided absorbing element, not a one-sided identity.

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