I don’t know what was going through your head, but you really picked the wrong paradoxical decomposition, somehow you were able to make a new hamster but it wasn’t right at all, you made Dini near sighted! Wow, that’s awful! He’s gonna have to wear glasses his whole life! Wana try another decomposition?
A1=S(a)M, A2=S(b^−1)MUM\B , A3=S(a^−1)M, A4=S(b)BUMUB
A1=S(a^−1)BUM, A2=S(b^−1)M\B, A3=S(a)MUMUB, A4=S(b)M