identitetsegenskapen

Identitetsegenskapen, or the identity property, refers to the presence of an identity element with respect to a binary operation on a set. An element e is called an identity (or neutral element) if for every element a in the set, the operation yields a: a ∘ e = a and e ∘ a = a. If a structure has such an element, we say the operation has a neutral element; generally the element is unique, and its existence often characterizes algebraic structures such as monoids and groups.

Common examples: In the real numbers under addition, 0 is the additive identity because a + 0 = a for all a. In the real numbers under multiplication, 1 is the multiplicative identity because a × 1 = a for all a. In linear algebra, the identity matrix I acts as the identity for matrix multiplication: AI = IA = A for any matrix A. In boolean algebra, different identities exist depending on the operation: for AND, true (1) is the identity since a ∧ true = a; for OR, false (0) is the identity since a ∨ false = a.

Notes: The identity element is central in the definition of richer structures. A monoid requires a neutral element for its binary operation; a group also requires that every element has an inverse with respect to the operation. If a left and a right identity exist, and they are equal, they constitute the two-sided identity element.