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Is there a two-variable (real) polynomial that is associative, not commutative, and not a projection?

That is: Does there exist a polynomial $p$ such that

  1. $\forall x, y, z. p(p(x, y), z) = p(x, p(y, z))$

  2. $\exists x, y. p(x, y) \neq p(y, x)$

  3. $\exists x, y. p(x, y) \neq x$

  4. $\exists x, y. p(x, y) \neq y$

Or is there a proof that no such polynomial exists?

This was inspired by this question and this question.

user76284
  • 6,408

2 Answers2

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Using essentially the same technique as done here (or just use their theorem), one can show that the only two-variable real associative polynomials are of the form $$c, x,\ y,\ x+y+c,\ a(x+b)(y+b)-b$$ The first is commutative, the next two are projections, the last two are also commutative.

So there is no two-variable real polynomial that is associative, not commutative, and not a projection.

HackR
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This is just an addendum to the other answer.

The more general problem that replaces polynomials with rational functions is also solved.

From the paper Associative rational functions in two variables (2001):

We say that $R(x, y)$ is associative if the equation $$R(R(x, y), z) = R(x, R(y, z))$$ is valid in $F(x, y, z)$, the rational function field with distinct variables $x$, $y$, $z$.

A rational function $R(x, y)$ is said to be equivalent to $R(x, y)$ if there is a linear fractional map $f(x) = (a x + b) / (c x + d) \in \overline{F}(x)$, $ad - bc \neq 0$, such that $$R_1(x, y) = f^{-1}(R(f(x), f(y)))$$ where $\overline{F}$ denotes the algebraic closure of $F$ and $f^{-1}(x) = (-dx + b) / (cx - a)$ is the inverse of $f$.

Theorem 1. Let $F$ be any field. Then any nonconstant associative rational function in $F(x, y)$ is equivalent to exactly one of $x$, $y$, $x + y$ and $x + y + xy$.

Thus there is no two-variable real rational function that is associative, not commutative, and not a projection.

Further related work suggested in the comments: Associative formal power series in two indeterminates (2013).

user76284
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  • A generalisation is proved in "ASSOCIATIVE FORMAL POWER SERIES IN TWO INDETERMINATES" prop 3.10: any associative formal power series in two variables is commutative. This generalises the fact that every one-dimensional formal group law over $\Bbb R$ is commutative, which is essentially because every one-dimensional Lie group is abelian. This suggests looking at higher dimensions: eg $(a, b) \cdot (c, d) = (ac, ad + b)$ is an assoc non-comm polynomial operation on $\Bbb R^2$ ("the monoid of affine transformations of $\Bbb R$") – Izaak van Dongen Jun 08 '25 at 09:54