Identity element

Inmathematics,anidentity elementorneutral elementof abinary operationis an element that leaves unchanged every element when the operation is applied.^[1]^[2]For example, 0 is an identity element of theadditionofreal numbers.This concept is used inalgebraic structuressuch asgroupsandrings.The termidentity elementis often shortened toidentity(as in the case of additive identity and multiplicative identity)^[3]when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.

Definitions[edit]

Let $(S, *)$ be a set $S$ equipped with abinary operation∗. Then an element $e$ of $S$ is called aleftidentityif $e * s = s$ for all $s$ in $S$ ,and arightidentityif $s * e = s$ for all $s$ in $S$ .^[4]If $e$ is both a left identity and a right identity, then it is called atwo-sided identity,or simply anidentity.^[5]^[6]^[7]^[8]^[9]

An identity with respect to addition is called anadditive identity(often denoted as 0) and an identity with respect to multiplication is called amultiplicative identity(often denoted as 1).^[3]These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of agroupfor example, the identity element is sometimes simply denoted by the symbol $e$ .The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such asrings,integral domains,andfields.The multiplicative identity is often calledunityin the latter context (a ring with unity).^[10]^[11]^[12]This should not be confused with aunitin ring theory, which is any element having amultiplicative inverse.By its own definition, unity itself is necessarily a unit.^[13]^[14]

Examples[edit]

Set	Operation	Identity
Real numbers	+ (addition)	0
Real numbers	· (multiplication)	1
Complex numbers	+ (addition)	0
Complex numbers	· (multiplication)	1
Positive integers	Least common multiple	1
Non-negative integers	Greatest common divisor	0 (under most definitions of GCD)
Vectors	Vector addition	Zero vector
$m$ -by- $n$ matrices	Matrix addition	Zero matrix
$n$ -by- $n$ square matrices	Matrix multiplication	I_n(identity matrix)
$m$ -by- $n$ matrices	○ (Hadamard product)	$J m, n$ (matrix of ones)
Allfunctionsfrom a set, $M$ ,to itself	∘ (function composition)	Identity function
Alldistributionson agroup, $G$	∗ (convolution)	$δ$ (Dirac delta)
Extended real numbers	Minimum/infimum	+∞
Extended real numbers	Maximum/supremum	−∞
Subsets of aset $M$	∩ (intersection)	$M$
Subsets of aset $M$	∪ (union)	∅ (empty set)
Strings,lists	Concatenation	Empty string,empty list
ABoolean algebra	∧ (logical and)	⊤ (truth)
	↔ (logical biconditional)	⊤ (truth)
	∨ (logical or)	⊥ (falsity)
	⊕ (exclusive or)	⊥ (falsity)
Knots	Knot sum	Unknot
Compact surfaces	# (connected sum)	S²
Groups	Direct product	Trivial group
Two elements, ${e, f}$	∗ defined by $e * e = f * e = e$ and $f * f = e * f = f$	Both $e$ and $f$ are left identities, but there is no right identity and no two-sided identity
Homogeneous relationson a setX	Relative product	Identity relation
Relational algebra	Natural join(⨝)	The unique relationdegree zeroand cardinality one

Properties[edit]

In the exampleS= {e,f} with the equalities given,Sis asemigroup.It demonstrates the possibility for $(S, *)$ 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 $l$ is a left identity and $r$ is a right identity, then $l = l * r = r$ .In particular, there can never be more than one two-sided identity: if there were two, say $e$ and $f$ ,then $e * f$ would have to be equal to both $e$ and $f$ .

It is also quite possible for $(S, *)$ to havenoidentity element,^[15]such as the case of even integers under the multiplication operation.^[3]Another common example is thecross productofvectors,where the absence of an identity element is related to the fact that thedirectionof any nonzero cross product is alwaysorthogonalto 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 additivesemigroupofpositive natural numbers.

Notes and references[edit]

^Weisstein, Eric W."Identity Element".mathworld.wolfram.Retrieved2019-12-01.
^"Definition of IDENTITY ELEMENT".merriam-webster.Retrieved2019-12-01.
^^a ^b ^c"Identity Element".encyclopedia.Retrieved2019-12-01.
^Fraleigh (1976,p. 21)
^Beauregard & Fraleigh (1973,p. 96)
^Fraleigh (1976,p. 18)
^Herstein (1964,p. 26)
^McCoy (1973,p. 17)
^"Identity Element | Brilliant Math & Science Wiki".brilliant.org.Retrieved2019-12-01.
^Beauregard & Fraleigh (1973,p. 135)
^Fraleigh (1976,p. 198)
^McCoy (1973,p. 22)
^Fraleigh (1976,pp. 198, 266)
^Herstein (1964,p. 106)
^McCoy (1973,p. 22)

Bibliography[edit]

Beauregard, Raymond A.; Fraleigh, John B. (1973),A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields,Boston:Houghton Mifflin Company,ISBN 0-395-14017-X
Fraleigh, John B. (1976),A First Course In Abstract Algebra(2nd ed.), Reading:Addison-Wesley,ISBN 0-201-01984-1
Herstein, I. N.(1964),Topics In Algebra,Waltham:Blaisdell Publishing Company,ISBN 978-1114541016
McCoy, Neal H. (1973),Introduction To Modern Algebra, Revised Edition,Boston:Allyn and Bacon,LCCN 68015225