Identity element

In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as in the case of additive identity for 0 and multiplicative identity for 1) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.

Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an .

An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a group for example, the identity element is sometimes simply denoted by the symbol . The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit. The symbol was chosen because it is the first letter in Einheit, the German for "unit" or "unity."

Examples Set Operation Identity Real numbers, complex numbers + (addition) 0 Real numbers, complex numbers, excluding 0· (multiplication) 1 Positive integers Least common multiple 1 Non-negative integers Greatest common divisor 0 (under most definitions of GCD) Vectors Vector addition Zero vectorScalar multiplication 1 -by- matrices Matrix addition Zero matrix -by- square matrices Matrix multiplication In (identity matrix) -by- matrices ○ (Hadamard product) (matrix of ones) All functions from a set, , to itself ∘ (function composition) Identity function All distributions on a group, ∗ (convolution) (Dirac delta) Extended real numbers Minimum/infimum +∞ Maximum/supremum −∞ Subsets of a set ∩ (intersection) ∪ (union) ∅ (empty set) Strings, lists Concatenation Empty string, empty list A Boolean algebra (conjunction) (truth) (equivalence) (truth) (disjunction) (falsity) (nonequivalence) (falsity) Knots Knot sum Unknot Compact surfaces # (connected sum) S2 Abstract groups Direct product Trivial group Two elements, ∗ defined by and Both and are left identities, but there is no right identity and no two-sided identity Homogeneous relations on a set X Relative product Identity relation Relational algebra Natural join (⨝) The unique relation degree zero and cardinality one A unital magma Its operation Its identity element

Properties In the example S = {e,f} with the equalities given, S is a semigroup. It demonstrates the possibility for to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity.

To see this, note that if is a left identity and is a right identity, then . In particular, there can never be more than one two-sided identity: if there were two, say and , then would have to be equal to both and .

It is also quite possible for to have no identity element, such as the case of even integers under the multiplication operation. Another common example is the cross product of vectors, where the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additive semigroup of positive natural numbers.

Identity element

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