There are four elementary arithmetic operators. Are all operations in mathematics derived from the four elementary arithmetic operators?
I'm studying linear algebra and noticed that some exercises define a new operation such as "adition between (a,b) and (c,d) is (a+d, c+b)". Is there a freedom in mathematics to define new operations at wild?
I'm not an expert in this field, although I don't think all of the operations in the integers can be expressed in terms of those.
For example consider the operation $a\star b$ which is defined to be the $a$'th prime larger than $b$.