Definition:Idempotence/Element

Definition

Let $S$ be a set.

Let $\circ: S \times S \to S$ be a binary operation on $S$.

Let $x \in S$ have the property that $x \circ x = x$.

Then $x \in S$ is described as idempotent under the operation $\circ$.

Examples

Zero is Idempotent for Addition

$0$ is idempotent under the operation of addition in the set of integers $\Z$, but no other element of $\Z$ is so.

One is Idempotent for Multiplication

$1$ is idempotent under the operation of multiplication in the set of real numbers $\R$.

Unit Matrix

Let $\mathbf I_n$ be the unit matrix of order $n$.

Then $\mathbf I_n$ is idempotent under the operation of (conventional) matrix multiplication.

Also known as

The concept of idempotence can also be referred to as idempotency.

Also see

• Results about idempotence can be found here.

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.4: \ 12$
• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: $(1.14)$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.17$
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Semigroups: Exercise $4$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): idempotent
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): idempotent