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I'm not sure quite how to phrase this naive question, but what is the term for an operator that does the same thing no matter how many times it is applied?

For example a projection operator $P$ has the defining property that $$P^2=P$$ Is there a name for this property or a term that characterizes this "finality"?

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    Some additional terminology you might find helpful: as the accepted answer correctly notes, an operator with that property is said to be "idempotent". If an operation gives the same value when repeated but only on a certain set of values then those values are called "fixed points"; for example, squaring is not idempotent in general, but zero and one are fixed points; you can keep squaring them and you always get the same result. If an operation on vectors gives you back a vector that points in the same direction when iterated then those vectors are called "eigenvectors". – Eric Lippert Mar 26 '13 at 05:06

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Maybe you're thinking of Idempotence.?

Idempotence (pron.: /ˌaɪdɨmˈpoʊtəns/ EYE-dəm-POH-təns) is the property of certain operations in mathematics and computer science, that 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).

-Wikipedia

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