Let us consider the first-order theory of Peano arithmetic (from now on PA) formulated in the vocabulary with just $+$ (for addition) and $\cdot$ (for multiplication). This vocabulary restriction is not important at all since $0$, succesor $S$, and the order $\leq$ can be defined using first-order formulas.
I am curious what it is known about the dependency, in non standard models, of each one of these basic operations with respect to the other. To be more precise, I am interested on any known answer to the following two questions:
Are there two different (i.e., non-isomorphic) PA models with the same universe and the same interpretation of multiplication?
Are there two different (i.e., non-isomorphic) PA models with the same universe and the same interpretation of addition?