# Idempotence

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In mathematics and computer science idempotence is the property of an operation that repeated application has no further effect.

## In mathematics

A binary operation ${\displaystyle \star }$ is idempotent if

${\displaystyle x\star x=x}$ for all x:

equivalently, every element is an idempotent element for ${\displaystyle \star }$.

Examples of idempotent binary operations include join and meet in a lattice; union and intersection on sets; disjunction and conjunction in propositional logic.

A unary operation (a function from a set to itself) π is idempotent if it is an idempotent element for function composition, ${\displaystyle \pi \circ \pi =\pi }$.

## In computing

In applications such as databases and transaction processing, idempotent operations are those for which the intended effect is that repeated application should have no effect, such as inserting a record into a file, an element into a set, or sending a message. Implementations must therefore be constructed in such a way that the intended effect is actually carried into practice. For example, messages might have unique sequence numbers with duplicates being discarded on receipt; a set might be implemented as a bit vector, and member insertion implemented by an idempotent mathematical operation such as inclusive or with a bit mask.

When a particular unit of work (i.e., transaction), has the idempotent property, relaxation of the ACID properties usually required for reliable transaction processing, can be relaxed.