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 e and f are distinct identity elements with respect to ·. Since e is an identity element, f\cdot e=f, and since f is also an identity element, f \cdot e =e. Thus, e must be equal to f, so e and f cannot be distinct. We have a contradiction, therefore the identity element must be unique.

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