The identity element of a semigroup (S,•) is an element e in the set S such that for all elements a in S, e•a = a•e = a. Such a semigroup is also a monoid.


Uniqueness of the identity element

An important fact in mathematics is that whenever a binary operation on a set has an identity, the identity is unique; no other element as the set serves as the identity. This ensures that zero and one are unique within the number system. We can refer to the identity of a set as opposed to an identity of a set.

Theorem. (Uniqueness of an identity element) Let (S,·) be a semigroup. If there exists an identity element with respect to ·, then that identity element is unique.
Proof. (By contradiction) Suppose and are distinct identity elements with respect to ·. Since is an identity element, , and since is also an identity element, . Thus, must be equal to , so and cannot be distinct. We have a contradiction, therefore the identity element must be unique.
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