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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 $x \times y = 1$.

My Work

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.

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  • $\begingroup$ Apart from the empty semigroup and the one-element semigroup, do you know any example of a strongly homogeneous commutative semigroup? $\endgroup$ – J.-E. Pin Jun 19 at 6:01
  • $\begingroup$ @J.-E.Pin Yes - $(\Bbb R^{\gt 0}, 1, +)$. $\endgroup$ – CopyPasteIt Jun 19 at 11:30

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