Английская Википедия:Identity element
Шаблон:Short description 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.
Definitions
Let Шаблон:Math be a set Шаблон:Mvar equipped with a binary operation ∗. Then an element Шаблон:Mvar of Шаблон:Mvar is called a Шаблон:Visible anchor if Шаблон:Math for all Шаблон:Mvar in Шаблон:Mvar, and a Шаблон:Visible anchor if Шаблон:Math for all Шаблон:Mvar in Шаблон:Mvar.[4] If Шаблон:Mvar is both a left identity and a right identity, then it is called a Шаблон:Visible anchor, or simply an Шаблон:Visible anchor.[5][6][7][8][9]
An identity with respect to addition is called an [[Additive identity|Шаблон:Visible anchor]] (often denoted as 0) and an identity with respect to multiplication is called a Шаблон:Visible anchor (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 <math>e</math>. 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 Шаблон:Visible anchor 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
Properties
In the example S = {e,f} with the equalities given, S is a semigroup. It demonstrates the possibility for Шаблон:Math 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 Шаблон:Mvar is a left identity and Шаблон:Mvar is a right identity, then Шаблон:Math. In particular, there can never be more than one two-sided identity: if there were two, say Шаблон:Mvar and Шаблон:Mvar, then Шаблон:Math would have to be equal to both Шаблон:Mvar and Шаблон:Mvar.
It is also quite possible for Шаблон:Math 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
- Absorbing element
- Additive inverse
- Generalized inverse
- Identity (equation)
- Identity function
- Inverse element
- Monoid
- Pseudo-ring
- Quasigroup
- Unital (disambiguation)
Notes and references
Bibliography
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, Шаблон:ISBN, p. 14–15
