Let $x$ and $y$ be elements of an inverse semigroup $S$. Then the following are equivalent:
(1) $xy^{-1}x = x$
(2) $x^{-1}yx^{-1}=x^{-1}$
I'm not sure in my solution, please check it for me.
For implication $(1) \Rightarrow (2)$, let $a,b \in S$ such that $ab^{-1}a=a$. Then we have $a^{-1}(b)a^{-1}=(a^{-1}ba^{-1})ba^{-1}=a^{-1}ba^{-1}= a^{-1}$. And for $(2) \Rightarrow (1)$ is similarly.
Let $xy^{-1}x = x$. To prove (2) you should use the identity $(ab)^{-1}=b^{-1}a^{-1}$. Then $$ x^{-1}=(xy^{-1}x)^{-1}=x^{-1}yx^{-1} $$