For a commutative semigroup $S$, define the relation $\theta_n^S\ \ (n\geqslant 1)$ by $$a\theta_n^Sb\text{ if and only if } (\forall x\in S^n) xa = xb. $$ (a) Show that $\theta_n^S$ is a congruence on $S$, and that $$\theta_1^S\subseteq\theta_2^S\subseteq\cdots$$ (b) Show that $\theta_n^S=1_S$ for all $n$ if $S$ is a monoid.
(c)For $n=1,2,\ldots,$ denote $S/\theta_n^S$ by $S_n$. Show that, for all $n\geqslant 2$, $$S_n\simeq S_{n-1}/\theta_1^{S_{n-1}}. $$
How can i solve the second portion of part (a)?First portion of part (a) is easy one for me but i stuck on the second portion. Any help in this regard.