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Consider two sets $A$ and $B$, and two functions $f: A \rightarrow B$ and $g: B \rightarrow A$.

Assuming that $g$ is not the inverse of $f$, how should I call elements $a$ of $A$ such that $g(f(a)) = a$?

It's not exactly a fixed point, and a quick google search for "bi-fixed point" didn't return anything useful.

Abdallah
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    If $f$ and $g$ form a Galois correspondence, then $g \circ f$ will either be a closure operator or an interior operator, depending your conventions. Hence the fixed points of $g \circ f$ could reasonably be called "closed elements" or "open elements" in this case. – goblin GONE Feb 12 '15 at 13:53
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    "fixed points of $g\circ f$"? – Clement C. Feb 12 '15 at 14:07
  • @ClementC. Turn your comment into an answer? I realize how simple it is now, but I was puzzled indeed. – Abdallah Feb 16 '15 at 07:13
  • @Abdallah: sure (just did it). – Clement C. Feb 16 '15 at 12:19

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To reproduce the comment above: the best and simplest term I can think of is fixed point of $g\circ f$.

Clement C.
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