Inspired by reading through this page: https://golem.ph.utexas.edu/category/2019/08/turing_categories.html
https://en.wikipedia.org/wiki/Primitive_recursive_function
Among the primitive recursive functions one can define are addition and multiplication, as derived functions from zero, successor, projection, composition, and bounded recursion.
https://en.wikipedia.org/wiki/Presburger_arithmetic
The signature of Presburger arithmetic lacks the multiplication operator, but explicitly includes addition.
Question: How can the axioms of the primitive recursion functions be modified to only generate those functions of Presburger arithmetic?
For example, a modification might be the replacement of the primitive recursion operator with a different kind of recursion.
Trivially, the axioms of primitive recursion could be "modified" by replacing them wholecloth with those of Presburger arithmetic, but the focus of this question is on reconciling two points:
while lacking explicit mention of either addition or multiplication in the axioms of the primitive recursive functions, both functions are definable.
while explicitly including addition, Presburger arithmetic cannot define multiplication, or any equivalent function.
Modifications of PR which retain the non-explicit nature of its axioms, while preventing the implicit construction of multiplication or its equivalents, would be very interesting.
