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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.

Examples

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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