idempotent

Latin roots, idem (“same”) +‎ potent (“having power”) – literally, “having the same power”.

Coined 1870 by American mathematician Benjamin Peirce in context of algebra.[1]

• (US) IPA^(key): /aɪdəmˈpoʊtənt/, /ɪdəmˈpoʊtənt/

idempotent (not comparable)

1. (mathematics, computing) Said of a function: describing an action which, when performed multiple times on the same subject, has no further effect on its subject after the first time it is performed.

   A projection operator is idempotent.

2. (mathematics) Said of an element of an algebraic structure with a binary operation (such as a group or semigroup): when the element operates on itself, the result is equal to itself.

   Every finite semigroup has an idempotent element.

   Every group has a unique idempotent element: namely, its identity element.

3. (mathematics) Said of a binary operation: that all of the distinct elements it can operate on are idempotent (in the sense given just above).

   Since the AND logical operator is commutative, associative, and idempotent, then it distributes with respect to itself.

4. (mathematics) Said of an algebraic structure: having an idempotent operation (in the sense above).

See the Usage notes section of nullipotent.

• nilpotent
• nullipotent

• idempotence
• nilpotent
• nullipotent
• unipotent

idempotent (plural idempotents)

1. (mathematics) An idempotent element.
2. (mathematics) An idempotent structure.

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