Given a set $S$, a function $M: S\times S \rightarrow S$ is a **mean** if it satisfies the properties:

1. $M(a,a)=a$

2. $M(a,b)=M(b,a)$.

and possibly

3. $M(M(a,b),M(a,c))=M(a,M(b,c))\qquad$ **(weak associativity)**

4. $M(M(a,b),M(c,d))=M(M(a,c),M(b,d))\qquad$ (**strong associativity**)

5. $a\ne b \implies a\ne M(a,b)\ne b\qquad$ (**non-singularity, separation**?).

When $S$ is an abelian groupoid or an ordered set or a topological space, $M$ can have additional specific requirements, such as:

6. $M(ac,bc)=cM(a,b)\qquad$  (**homogeneousness**)

7. $a < b \implies a\le M(a,b) \le b\qquad$ (**order preservation**)

8. **continuity**.


In general (3) does not imply (4) as can be seen in this example for $S=\{a,b,c,d,e\}$:

$$
\begin{array}{c|ccccc}
M & a & b & c & d & e\\
\hline
a & a & a & a & a & a\\
b & a & b & d & c & a\\
c & a & d & c & b & a\\
d & a & c & b & d & e\\
e & a & a & a & e & e\\
\end{array}$$

where $M(M(b,c),M(d,e))  \ne M(M(b,d),M(c,e))$.

Here are some of the questions that come to mind.

**Q1**.  Is there a finite example where (3) and (5) hold, but not (4)? I know that $S$ will need to have at least 6 elements.

**Q2**. Does $M$ in the above example naturally extend to a mean in $\mathbb{R}[a,b,c,d,e]$ where both (3) and (6) hold?

Another example: if $A$ and $G$ are the arithmetic and geometric means on $\mathbb{R}^+$, it's easy to check that the mean function $M(x,y)=G(A(x,y),G(x,y))$ satisfies all the properties except (3) and (4).

**Q3**. Assuming all of the above properties **except** (4) hold for $M$ on $\mathbb{R}^+$, does (4) follow?

**Q4**. My starting point leading to this post: if all the above properties, including (4), hold for $M$ on $\mathbb{R}^+$, does it follow that $M$ is equivalent to the arithmetic mean, in the sense that $M(x,y)=f^{-1}\big(\frac{f(x)+f(y)}{2}\big)$ for some continous strictly monotonic function $f: \mathbb{R}^+ \to \mathbb{R}$?

I welcome suggestions for improvements to this post and references to relevant work.