Identity element

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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, which leaves any element of the set unchanged when combined with it.[1] [2][3] 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),[4] when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.

Definitions

Let (S, ∗) be a set S equipped with a binary operation ∗. Then an element e of S is called a left identity if ea = a for all a in S, and a right identity if ae = a for all a in S.[5] If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.[6][7][8][9][10]

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).[4] These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. The distinction 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).[11][12][13] 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.[14][15]

Examples

Set Operation Identity
Real numbers + (addition) 0
Real numbers · (multiplication) 1
Positive integers Least common multiple 1
Non-negative integers Greatest common divisor 0 (under most definitions of GCD)
m-by-n matrices Matrix addition Zero matrix
n-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 function
All distributions on a groupG ∗ (convolution) δ (Dirac delta)
Extended real numbers Minimum/infimum +∞
Extended real numbers Maximum/supremum −∞
Subsets of a set M ∩ (intersection) M
Sets ∪ (union) ∅ (empty set)
Strings, lists Concatenation Empty string, empty list
A Boolean algebra ∧ (logical and) ⊤ (truth)
A Boolean algebra ∨ (logical or) ⊥ (falsity)
A Boolean algebra ⊕ (exclusive or) ⊥ (falsity)
Knots Knot sum Unknot
Compact surfaces # (connected sum) S2
Groups Direct product Trivial group
Two elements, {e, f}  ∗ defined by
ee = fe = e and
ff = ef = f
Both e and f are left identities,
but there is no right identity
and no two-sided identity
Homogeneous relations on a set X Relative product Identity relation

Properties

As the last example (a semigroup) shows, it is possible 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 = lr = r. In particular, there can never be more than one two-sided identity: if there were two, say e and f, then ef would have to be equal to both e and f.

It is also quite possible for (S, ∗) to have no identity element,[16] such as the case of even integers under the multiplication operation.[4] 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 group without identity element involves the additive semigroup of positive natural numbers.

Notes and references

1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Identity". Math Vault. 2019-08-01. Retrieved 2019-12-01.
2. ^ Weisstein, Eric W. "Identity Element". mathworld.wolfram.com. Retrieved 2019-12-01.
3. ^ "Definition of IDENTITY ELEMENT". www.merriam-webster.com. Retrieved 2019-12-01.
4. ^ a b c "Identity Element | Encyclopedia.com". www.encyclopedia.com. Retrieved 2019-12-01.
5. ^ Fraleigh (1976, p. 21)
6. ^ Beauregard & Fraleigh (1973, p. 96)
7. ^ Fraleigh (1976, p. 18)
8. ^ Herstein (1964, p. 26)
9. ^ McCoy (1973, p. 17)
10. ^ "Identity Element | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-01.
11. ^ Beauregard & Fraleigh (1973, p. 135)
12. ^ Fraleigh (1976, p. 198)
13. ^ McCoy (1973, p. 22)
14. ^ Fraleigh (1976, pp. 198,266)
15. ^ Herstein (1964, p. 106)
16. ^ McCoy (1973, p. 22)

Further reading

• 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, ISBN 3-11-015248-7, p. 14–15