I have the homomorphism $h$ of monoid $ \lbrace a,b \rbrace ^* $ which is defined as $h(a) = b , h(b) = aba $.
I know how this is working for example I have in book question is it true that
$h(b^+) \subseteq (a+ba)^* $
and I know that is right answer becouse $h(b^+) = (aba)^+ \subseteq (a+ba)^* $.
But here is another question: is it true that
$h^3(b^+) = (h(b)bh(b)bh(b))^+$?
The answer is yes and I don't know why. I would be very thankful if somebody could tell me how it was done. I don't know why we have here letter $b$ outside the bracket. I have tried some of ways of doing this example but I can't get into this answer $(h(b)bh(b)bh(b))^+$.
$h(b^+)=(aba)^+$
$h^2(b^+)=h(h(b^+))=h((aba)^+)=(h(a)h(b)h(a))^+=(bh(b)b)^+$
Therefore $h^3(b^+)=h(h^2(b^+))=h((bh(b)b)^+)=(h(b)h^2(b)h(b))^+ =(h(b)bh(b)bh(b))^+$