Ya this hierarchy was already considered.
The main observation is that the real numbers with operations \( a+b \) and \( a*b \) are isomorphic to the positive real numbers with \( a*b \) and \( a^{\log(b)} \) (The isomorphism is \( \exp \)). I.e. we dont add really something new. Each two consecutive operations are isomorphic (i.e. behave completely the same as) to + and *.
The main observation is that the real numbers with operations \( a+b \) and \( a*b \) are isomorphic to the positive real numbers with \( a*b \) and \( a^{\log(b)} \) (The isomorphism is \( \exp \)). I.e. we dont add really something new. Each two consecutive operations are isomorphic (i.e. behave completely the same as) to + and *.
