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.[1][2] 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 and multiplicative identity)[3] when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Let (S, ∗) be a set S equipped with a binary operation ∗. Then an element e of S is called a left identity if e ∗ s = s for all s in S, and a right identity if s ∗ e = s for all s in S.[4] If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.[5][6][7][8][9]
An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative 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 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 unity in the latter context (a ring with unity).[10][11][12] 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.[13][14]
The unique relation degree zero and cardinality one
Propertiesedit
In the example S = {e,f} with the equalities given, S is a semigroup. 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 have no identity element,[15] such as the case of even integers under the multiplication operation.[3] 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 positivenatural numbers.
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, ISBN0-395-14017-X
McCoy, Neal H. (1973), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225
Further readingedit
M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN3-11-015248-7, p. 14–15
December 08, 2023
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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 1 2 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 and multiplicative identity 3 when there is no possibility of confusion but the identity implicitly depends on the binary operation it is associated with Contents 1 Definitions 2 Examples 3 Properties 4 See also 5 Notes and references 6 Bibliography 7 Further readingDefinitions editLet S be a set S equipped with a binary operation Then an element e of S is called a left identity if e s s for all s in S and a right identity if s e s for all s in S 4 If e is both a left identity and a right identity then it is called a two sided identity or simply an identity 5 6 7 8 9 An identity with respect to addition is called an additive identity often denoted as 0 and an identity with respect to multiplication is called a multiplicative 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 a group for example the identity element is sometimes simply denoted by the symbol e displaystyle e nbsp 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 unity in the latter context a ring with unity 10 11 12 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 13 14 Examples editSet Operation IdentityReal numbers addition 0 multiplication 1Complex numbers addition 0 multiplication 1Positive integers Least common multiple 1Non negative integers Greatest common divisor 0 under most definitions of GCD Vectors Vector addition Zero vectorm by n matrices Matrix addition Zero matrixn by n square matrices Matrix multiplication In identity matrix m by n matrices Hadamard product Jm n matrix of ones All functions from a set M to itself function composition Identity functionAll distributions on a group G convolution d Dirac delta Extended real numbers Minimum infimum Maximum supremum Subsets of a set M intersection M union empty set Strings lists Concatenation Empty string empty listA Boolean algebra logical and truth logical biconditional truth logical or falsity exclusive or falsity Knots Knot sum UnknotCompact surfaces connected sum S2Groups Direct product Trivial groupTwo 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 identityHomogeneous relations on a set X Relative product Identity relationRelational algebra Natural join The unique relation degree zero and cardinality oneProperties editIn the example S e f with the equalities given S is a semigroup 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 have no identity element 15 such as the case of even integers under the multiplication operation 3 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 See also editAbsorbing element Additive inverse Generalized inverse Identity equation Identity function Inverse element Monoid Pseudo ring Quasigroup Unital disambiguation Notes and references edit Weisstein Eric W Identity Element mathworld wolfram com Retrieved 2019 12 01 Definition of IDENTITY ELEMENT www merriam webster com Retrieved 2019 12 01 a b c Identity Element www encyclopedia com Retrieved 2019 12 01 Fraleigh 1976 p 21 Beauregard amp Fraleigh 1973 p 96 Fraleigh 1976 p 18 Herstein 1964 p 26 McCoy 1973 p 17 Identity Element Brilliant Math amp Science Wiki brilliant org Retrieved 2019 12 01 Beauregard amp 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 editBeauregard 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 68015225Further reading editM Kilp U Knauer A V Mikhalev Monoids Acts and Categories with Applications to Wreath Products and Graphs De Gruyter Expositions in Mathematics vol 29 Walter de Gruyter 2000 ISBN 3 11 015248 7 p 14 15 Retrieved from https en wikipedia org w index php title Identity element amp oldid 1178592339, wikipedia, wiki, book, books, library,
