In "A Newton Method For Systems of $m$ Equations in $n$ Variables" by Yuri Levin and Adi Ben-Israel, authors use this notation $\{2\}$-inverse, however I am not familiar with it and thus I cannot fully understand the whole paper.
For example they say: "More generally, any $\{2\}$–inverse of the Jacobian can be used ...", "We say that $X$ is a low rank [high rank] $\{2\}$-inverse of $A$ if its rank is near $0$ [near rank $A$], respectively.".
And also: "Recall that a $\{2\}$-inverse (also outer inverse) of $A\in B(m\times n)$ is a matrix $X \in R(n\times n)$ satisfying $XAX = X$ , in which case $\operatorname{rank} X \leq \operatorname{rank} A$ , with equality if $X = A^†$." $A^†$ - from what I understand is pseudoinverse of matrix $A$ (ex. Moore–Penrose pseudoinverse).
So my question is, what is $\{2\}$-inverse, is it outer inverse? Googling "matrix outer inverse" gives no sensible results? Anyone can explain this concept to me?
Thanks in advance.