Identity element
Encyclopedia
In mathematics
, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation
on that set. It leaves other elements unchanged when combined with them. This is used for group
s and related concepts
.
The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion.
Let (S,*) be a set S with a binary operation * on it (known as a magma
). Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.
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). The distinction is used most often for sets that support both binary operations, such as ring
s. The multiplicative identity is often called the unit in the latter context, where, unfortunately, a unit
is also sometimes used to mean an element with a multiplicative inverse.
It is also quite possible for (S, *) to have no identity element. The most common example of this is the cross product
of vectors. 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 – so that it is not possible to obtain a non-zero vector in the same direction as the original. Another example would be the additive semigroup of positive natural number
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
on that set. It leaves other elements unchanged when combined with them. This is used for group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s and related concepts
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
.
The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion.
Let (S,*) be a set S with a binary operation * on it (known as a magma
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
). Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.
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). The distinction is used most often for sets that support both binary operations, such as ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
s. The multiplicative identity is often called the unit in the latter context, where, unfortunately, a unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
is also sometimes used to mean an element with a multiplicative inverse.
Examples
| set | operation | identity |
|---|---|---|
| real number Real number In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π... s |
0 0 (number) 0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems... |
|
| real number Real number In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π... s |
· (multiplication) | 1 |
| real number Real number In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π... s |
ab (exponentiation) | 1 (right identity only) |
| positive integers | least common multiple Least common multiple In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b... |
1 |
| nonnegative integers | greatest common divisor Greatest common divisor In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to... |
0 (under most definitions of GCD) |
| m-by-n matrices Matrix (mathematics) In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element... |
+ (addition) | matrix of all zeroes |
| n-by-n square matrices Matrix (mathematics) In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element... |
· (multiplication) | In (matrix with 1 on diagonal and 0 elsewhere Identity matrix In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context... ) |
| all functions Function (mathematics) In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can... from a set M to itself |
∘ (function composition) | identity function Identity function In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument... |
| all functions Function (mathematics) In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can... from a set M to itself |
* (convolution Convolution In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation... ) |
δ (Dirac delta) |
| character strings, lists | concatenation | empty string Empty string In computer science and formal language theory, the empty string is the unique string of length zero. It is denoted with λ or sometimes Λ or ε.... , empty list |
| extended real numbers | minimum/infimum | +∞ |
| extended real numbers | maximum/supremum | −∞ |
| subsets of a set M | ∩ (intersection) | M |
| sets | ∪ (union) | { } (empty set Empty set In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced... ) |
| boolean logic Boolean logic Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of... |
∧ (logical and) | ⊤ (truth) |
| boolean logic Boolean logic Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of... |
∨ (logical or) | ⊥ (falsity) |
| boolean logic Boolean logic Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of... |
⊕ (Exclusive or) | ⊥ (falsity) |
| compact surfaces | # (connected sum) | S² |
| only 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 |
Properties
As the last example shows, it is possible for (S, *) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just 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, 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. The most common example of this is the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
of vectors. 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 – so that it is not possible to obtain a non-zero vector in the same direction as the original. Another example would be the additive semigroup of positive natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s.
See also
- Absorbing elementAbsorbing elementIn mathematics, an absorbing element is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element...
- Inverse elementInverse elementIn abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
- Additive inverseAdditive inverseIn mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
- MonoidMonoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
- UnitalUnitalIn mathematics, a unital algebra or unitary algebra is an algebra which contains a multiplicative identity element , i.e. an element 1 with the property 1x = x1 = x for all elements x of the algebra....
- QuasigroupQuasigroupIn mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...
- Properties weaker than having an identity