On the least prime in an arithmetic progression $a + nb$ where $a,b$ are distinct primes. Dirichlet's Theorem: there are infinitely many primes in every arithmetic progression $\{a + nd: n \geq 0\}$ for coprime $a, d$.  Consider just the case where $a, d$ can take on prime values or $0$.
Then define $a \oplus a = 0$, $a \oplus 0 = 0 \oplus a = a$ and $a \oplus b$ for $a \neq b$ two prime numbers to be the smallest number in the above progression ($a + nb$) greater than $a$ that is a prime.
For instance $3 \oplus 5 = 3 + 2\cdot 5 = 13$
By Dirichlet's theorem there are infinitely many primes, so choose the smallest one greater than $a$!
Associativity:
$$
(3 \oplus 5) \oplus 7 = 13 \oplus 7 = 13 + 4 \cdot 7 = 41 \\
3 \oplus (5 \oplus 7) = 3 \oplus 19 = 41
$$
Does associativity hold at least when $a, b, c$ are distinct?

Take $7\oplus 11=29$, so that $(7\oplus 11)\oplus 13=29\oplus 13=107$.
On the other hand we have $11\oplus 13=37$, and $7\oplus(11\oplus 13)=7\oplus 37=229$. So
$$
(7\oplus 11)\oplus 13=107\neq 229=7\oplus(11\oplus 13)
$$
