Identity element

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. 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), when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Definitions.
Let be a set  equipped with a binary operation ∗. Then an element  of  is called a left identity if for all  in , and a right identity if for all  in . If 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). 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 formula_1. 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). 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.
Properties.
In the example "S" = {"e,f"} with the equalities given, "S" is a semigroup. It demonstrates the possibility for 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 is a left identity and is a right identity, then . In particular, there can never be more than one two-sided identity: if there were two, say and , then would have to be equal to both and .
It is also quite possible for to have "no" identity element, such as the case of even integers under the multiplication operation. 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.