Let us call any commutative semigroup $(S, +)$ strongly homogeneous if it satisfies the following three properties:
P-1. Every endomorphism on $S$ is a bijection.
P-2. Any two endomorphisms on $S$ commute.
P-3. For every $x, y \in S$ there exist one and only one endomorphism $\psi$ on $S$ such that $\psi(x) = y$.
Is the following true?
Corresponding to any element $1 \in S$ there exist one a̶n̶d̶ ̶o̶n̶l̶y̶ ̶o̶n̶e̶ associative binary operation $\times$ satisfying, for all $x,y,z \in S$,
$$\tag 1 1 \times 1 = 1$$
$$\tag 2 x \times y = y \times x$$
$$\tag 3 x \times (y+z) = x \times y + x \times z$$
Moreover, for every $x \ S$ there exists $y \in S$ with $x \times y = 1$.
I sketched out a proof for a special case here, and I think the argument can be lifted, but I'm not completely confident in using this logic and would like verification/confirmation.
If the theory is false, a counter-example would be appreciated.