Q:

Comparing "axiomatized function spaces"

This was previously asked and bountied at math.stackexchange with no response.

Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$ with the compact-open topology. I'm interested in analyzing subspaces of $C(\mathbb{R}^2,\mathbb{R})$ determined by first-order theories, especially finite equational theories. Specifically, for $T$ a set of first-order sentences in the language of a single binary operation, let $T_\mathbb{R}$ be the subspace of continuous $f$ such that $(\mathbb{R};f)\models T$. I'm curious about how $T$ affects the purely topological properties of $T_\mathbb{R}$. For example, when is $T_\mathbb{R}$ connected? etc. 
However, even in very concrete cases, I'm having trouble understanding what $T_\mathbb{R}$ looks like. Letting $C$ and $A$ be the usual statements of commutativity and associativity, I think an answer to the following question would clear things up immensely:

Question: is $\{C\}_\mathbb{R}\cong\{A\}_\mathbb{R}$?

$\{C\}_\mathbb{R}$ is pretty tame since as Eric Wofsey observed it's a vector subspace of $C(\mathbb{R}^2,\mathbb{R})$. However, $\{A\}_\mathbb{R}$ seems much weirder. For example, I don't even know whether $\{A\}_\mathbb{R}$ is connected.

A:

The following result from Section 6.2 of János Aczél's classic book Lectures on Functional Equations and their Applications seems to be closely related, and the long list of references on the solutions of the associativity equation (p.253/4) may provide further entry points into the literature. (Although that list is way out of date, with the book being from 1966.)

Theorem: Let $f \in C(\mathbb{R}^2,\mathbb{R})$ satisfy the associativity equation
$$f(x,f(y,z)) = f(f(x,y),z).$$
Suppose moreover that $f$ is cancellative, i.e. that
$$
f(x,y) = f(x,z) \quad \Rightarrow \quad y = z,
$$
and similarly in the other argument. Then $f$ is conjugate to addition $+ : \mathbb{R}^2 \to \mathbb{R}$ via a monotone homeomorphism. In particular, $f$ is commutative.

It seems plausible that this result can be used to prove that if your theory $T$ consists of associativity and the cancellativity laws, then $T_{\mathbb{R}}$ is homemorphic to the space of monotone homeomorphisms $\mathbb{R} \to \mathbb{R}$. But I haven't worked out the details.

