I'm trying to prove the following for inverse semigroups
$\bf Def:$ an inverse semigroup $S$is a semigruop such that for each $x\in S$ the exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$.
An involution $*$ on $S$ can be defined as follows for $x\in S$ define $x^* = y.$
I'm trying to show that this is an involution and then If $S$ is a topological inverse semigroup this involution is continuous. I searched around and found this sketch of the proof:
1) Let $E(S)=\{x\in S : x^2 =x \}$ then for every $x\in E(S)$ we have $x^* =x$, no problem here it is easy to check.
2) $E(S)$ is a commutative subsemigroup of $S$ and for $s\in S$ and $e\in E(S)$ we have $ses^*\in E(S)$, again no problem here I checked it.
3) from 1 and 2 one can ched directly that for $s,t\in S$ we have $(s^*)^*=s$ and $(st)^*=t^*s^*$ and I'm stuck here.
and I have no Idea how to check if the involution is continuous
any ideas.
Thank you