An identity element for a binary operation ∗ on a set S is an element e ∈ S such that e ∗ a = a and a ∗ e = a for all a ∈ S. The identity element “does nothing” — combining any element with it returns that element unchanged.
If an identity element exists, it is unique: if e and e’ are both identity elements, then e = e ∗ e’ = e’. A magma with an identity element is called unital; a semigroup with an identity element is a monoid.
Familiar examples: 0 is the identity for addition (a + 0 = a), 1 is the identity for multiplication (a × 1 = a), the empty string is the identity for concatenation, and the identity function id is the identity for composition. In a category, every object has an identity morphism — this is one of the axioms of a category. In the semiotic universe, the identity operator on the Heyting algebra H maps every semantic value to itself: it is the trivial closure that stabilizes nothing.