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I have a function $f : A \to B$ and an inverse $f^{-1} : B \to A$, and the only property of the inverse is that $(f \circ f^{-1} \circ f)(x) = f(x)$. In particular, it is not necessarily true that $(f^{-1} \circ f)(x) = x$. I normally associate this property with inverses, so what should I call $f^{-1}$?

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    $\begingroup$ Are you trying to say that $f^{-1}$ is a right-inverse, but not a left-inverse? $\endgroup$ Sep 21, 2012 at 21:24
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    $\begingroup$ It certainly appears that that is the case and here for a bit more reading $\endgroup$
    – Ghost
    Sep 21, 2012 at 21:39

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Such things come up in the theory of von Neumann regular rings. The object is sometimes called a pseudoinverse, but several different items are called by that name. It is called a pseudoinverse in the theory of regular semigroups.

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