Idempotence

Idempotence (UK: /ˌɪdɛmˈpoʊtəns/,^[1] US: /ˈaɪdəm-/)^[2] is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency).

The term was introduced by American mathematician Benjamin Peirce in 1870^[3]^[4] in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from idem + potence (same + power).

^ "idempotence". Oxford English Dictionary (3rd ed.). Oxford University Press. 2010.
^ "idempotent". Merriam-Webster. Archived from the original on 2016-10-19.
^
Original manuscript of 1870 lecture before National Academy of Sciences (Washington, DC, USA): Peirce, Benjamin (1870) "Linear associative algebra" From pages 16-17: "When an expression which is raised to the square or any higher power vanishes, it may be called nilpotent; but when raised to a square or higher power it gives itself as the result, it may be called idempotent.
The defining equation of nilpotent and idempotent expressions are respectively Aⁿ = 0 and Aⁿ = A; but with reference to idempotent expressions, it will always be assumed that they are of the form Aⁿ = A unless it be otherwise distinctly stated."
- Printed: Peirce, Benjamin (1881). "Linear associative algebra". American Journal of Mathematics. 4 (1): 97–229. See p. 104.
- Reprinted: Peirce, Benjamin (1882). Linear Associative Algebra (PDF). New York, New York, USA: D. Van Nostrand. p. 8.
^ Polcino & Sehgal 2002, p. 127.

[1] "idempotence". Oxford English Dictionary (3rd ed.). Oxford University Press. 2010.

[2] "idempotent". Merriam-Webster. Archived from the original on 2016-10-19.

[3] Original manuscript of 1870 lecture before National Academy of Sciences (Washington, DC, USA): Peirce, Benjamin (1870) "Linear associative algebra" From pages 16-17: "When an expression which is raised to the square or any higher power vanishes, it may be called nilpotent; but when raised to a square or higher power it gives itself as the result, it may be called idempotent.
The defining equation of nilpotent and idempotent expressions are respectively Aⁿ = 0 and Aⁿ = A; but with reference to idempotent expressions, it will always be assumed that they are of the form Aⁿ = A unless it be otherwise distinctly stated."
Printed: Peirce, Benjamin (1881). "Linear associative algebra". American Journal of Mathematics. 4 (1): 97–229. See p. 104.

Reprinted: Peirce, Benjamin (1882). Linear Associative Algebra (PDF). New York, New York, USA: D. Van Nostrand. p. 8.

[4] Printed: Peirce, Benjamin (1881). "Linear associative algebra". American Journal of Mathematics. 4 (1): 97–229. See p. 104.

[5] Reprinted: Peirce, Benjamin (1882). Linear Associative Algebra (PDF). New York, New York, USA: D. Van Nostrand. p. 8.

[FOOTNOTEPolcinoSehgal2002[httpsbooksgooglecombooksid7m9P9hM4pCQCqIdempotencepgPA127_127]-4] Polcino & Sehgal 2002, p. 127.

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