# idempotent

## English

### Etymology

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

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

### Pronunciation

• (US) IPA(key): /aɪ.dəmˈpoʊ.tənt/, /ɪ.dəmˈpoʊ.tənt/, /aɪˈdem.pə.tənt/

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idempotent (not comparable)

1. (said of a function) Such that, when performed multiple times on the same subject, it has no further effect on its subject after the first time it is performed.
A projection operator is idempotent.
2. (said of an element of an algebraic structure with a binary operation, such as a group or semigroup) Such that, when it 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. (said of a binary operation) Such 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, it distributes with respect to itself.
4. (said of an algebraic structure) Having an idempotent operation (in the sense given above).

#### Usage notes

See the Usage notes section of nullipotent.

### Noun

idempotent (plural idempotents)

1. An idempotent element.
2. An idempotent structure.

### References

1. ^ Polcino & Sehgal (2002), p. 127

## French

### Pronunciation

•  Audio (file)

idempotent (feminine idempotente, masculine plural idempotents, feminine plural idempotentes)

1. idempotent

## German

### Pronunciation

•  Audio (file)

idempotent (strong nominative masculine singular idempotenter, not comparable)

1. idempotent