A very simple question, I just totally forgot how it was called, and Google is not helping.


There's a pair of functions $f:X\to Y$, $g:Y\to X$.


$fgf = f$, $gfg = g$, but $fg$ and $gf$ don't need to be identities (and usually are not in interesting cases).


A simple example would be $f(a,b,c)=(a,b)$, $g(a,b)=(a,b,0)$


What were $f$ and $g$ called?



$fgf = f$, $gfg = g$, $fg$ not necessarily identity, what is this called?
It is called "generalized inverse". In that case $fg$ and $gf$ are idempotents. In particular, if you have a semigroup of maps $X\to X$ (i.e. a set of maps closed under composition) such that every $f$ has a generalized inverse, the semigroup is called  *regular*. If the generalized inverse is unique, the semigroup is called  *inverse*. See Clifford and Preston "Algebraic theory of semigroups". 


