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From what I can gather from the Wikipedia articles on idempotent function and projection, both terms refer to a function $f:S \to S$ such that $f \circ f \equiv f$, i.e. $f(x) = x$ for all $x$ in the image of $f$.

Is there any difference between these two terms?

(I know that there are some similar terms like "projection operator" that have narrower meanings, but I'm just asking about the term "projection" itself.)

tparker
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The term "projection" has two distinct, but related, meanings:

  1. A "projection" can refer to an idempotent function $f:S\rightarrow S$ that preserves some additional property of the set. For instance: In linear algebra, a projection is an idempotent linear map. In group theory, a projection is an idempotent homomorphism. In topology, the word "retraction" is used instead, but this refers to a continuous idempotent function, so it could be considered a topological projection.

  2. "Projection" can also refer to a sort of inverse of a cartesian product, i.e. $f:A\times B \rightarrow A$ is a projection if $f(a,b) = a$. A function is called a projection if it can be written in such a form. However, this is not an idempotent function because the image $A$ is not a subset of $A\times B$.

These two meanings are closely related in many contexts. For example, in linear algebra a projection operator is an idempotent function on a vector space, i.e. $f:V\rightarrow V$ is linear and $f\circ f = f$. If the image of $f$ is a subspace $U$, it is also a fact that $V$ can be written as $W\times U$ in such a way that $f$ is a projection onto $U$ in the second sense (this is the inner direct sum, and $W$ is the nullspace of $f$).

Dark Malthorp
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    The "in such a way" is just the (inner) direct sum $V = W \oplus U$, right? – tparker Mar 31 '25 at 01:17
  • @tparker Yes, it is. – Dark Malthorp Mar 31 '25 at 01:31
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    Similarly, if you have an equivalence relation $R$ on a set $X$, then the induced map $X \to X / R$ is called a projection. – Daniel Schepler Mar 31 '25 at 14:05
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    (the common visual metaphor in all these cases is of a thing in three dimensions mapping to its shadow in two, hence the desirability of the imprecise language. if precision is important [when isn't it?] you can use idempotent when you want to emphasize the 'e^2 = e' property for formal manipulation and 'projection' when you want to emphasize the map) – hunter Mar 31 '25 at 14:09
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    Re (2): A projection in sense (2) can still look idempotent if there is some natural way or "obvious" of going from A back to $A \times B$. For example, the real part function can be read as a projection of $\mathbb{C} \rightarrow \mathbb{R}$, assuming you take the usual construction of $\mathbb{C} = \mathbb{R} \times \mathbb{R}$... but it can also be read as a function on $\mathbb{C} \rightarrow \mathbb{C}$, and that function is idempotent. This works because every real number a directly corresponds to the complex number a + 0i, so we can treat them as if they are interchangeable. – Kevin Mar 31 '25 at 19:22
  • @Kevin: More generally, sense (2) can be thought of as mapping an ordered pair $(a,b) \in (A \times B)$ to $(a, b_0)$, where $b_0$ is some fixed element of set $B$. In this sense, the map "trivializes" the set $B$ by reducing it to a single known element. If the set $B$ has some algebraic structure, then $b_0$ is often chosen to be the "zero" or "trivial" element (e.g. the identity element if $B$ is a group). – tparker Apr 01 '25 at 12:53