Idempotence

Author: Leandro Alegsa Creation date: November 13, 2022

Idempotence is a property that an operation in mathematics or computer science may have. It roughly means that the operation can be carried out again and again without changing the result.

The word idempotence was made by Benjamin Pierce because he saw the concept when studying algebra.

The meaning is different if we are talking about different kinds of operations. It can also be used to describe elements than an operation can take:

For a unary operation (or function), that we label f, we say that f is idempotent if for any x in the domain of f it is true that: f(f(x)) = f(x). For example, the absolute value: abs(abs(x)) = abs(x).

We say that an element c in the domain of f is an idempotent element if f(f(c)) = f(c). This means that f is idempotent if every element of its domain is an idempotent element.

For a binary operation, that we label *, we say that * is idempotent if for any x which the binary operation can take the following is true: x * x = x.

We say that an element c which * can take is an idempotent element for * if c * c = c. For example, the number 1 is an idempotent element for multiplication because 1 times 1 is 1.

Questions and Answers

Q: What is idempotence?

A: Idempotence is a property that an operation in mathematics or computer science may have, which means that the operation can be carried out again and again without changing the result.

Q: Who coined the term "idempotence"?

A: The term "idempotence" was made by Benjamin Pierce.

Q: How does idempotence differ for different kinds of operations?

A: The meaning of idempotence differs depending on the type of operation being discussed.

Q: What is true for a unary operation to be considered idempotent?

A: For a unary operation (or function) to be considered idempotent, it must be true that f(f(x)) = f(x) for any x in its domain.

Q: What is an example of an element that can take a unary operation and still be considered idempotent?

A: An example of an element that can take a unary operation and still be considered idempotent would be the absolute value; abs(abs(x)) = abs(x).

Q: What must hold true for a binary operation to be considered idempotent? A: For a binary operation to be considered idempotent, it must hold true that x * x = x for any x which the binary operation can take.

Q: Can you give an example of an element which meets this criteria? A: An example of an element which meets this criteria would be the number 1; 1 times 1 is 1.