# Idempotence

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Idempotence (pronounced /ˌaɪdɨmˈpoʊtəns/ EYE-dəm-POH-təns) is the property of certain operations in mathematics and computer science, that they can be applied multiple times without changing the result. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency).

The term was introduced by Benjamin Peirce[1] in the context of elements of an algebra that remain invariant when raised to a power.

There are several meanings of idempotence, depending on what the concept is applied to:

• A unary operation (or function) is idempotent if, whenever it is applied twice to any value, it gives the same result as if it were applied once. For example, the absolute value function is idempotent as a function from the set of real numbers to the set of real numbers: abs(abs (x)) = abs(x).
• A binary operation is idempotent if, whenever it is applied to two equal values, it gives that value as the result. For example, the operation giving the maximum value of two values is idempotent: max(x, x) = x.
• Given a binary operation, an idempotent element (or simply an idempotent) for the operation is a value for which the operation, when given that value for both of its operands, gives the value as the result. For example, the number 1 is an idempotent of multiplication: 1 × 1 = 1.