Consider the axioms of real numbers https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach and suppose we remove the multiplication operation and its properties. Do we loose something?
I have the impression that multiplication can be constructed in a unique way to recover all the axioms (define multiplication by natural numbers as repeated addition, then define multiplication on rational numbers then extend by continuity on all real numbers).
The question is motivated by the fact the the axioms comprising addition, ordering and completeness are very intuitive facts about the geometric line. Multiplication, instead, is not as obvious and maybe would be nice to build it.
Yes, this has been done by Tarski.
https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals