Does there exist a non-commutative algebraic structure with the following properties? https://ift.tt/eA8V8J

A magnium is a set M with a binary operation $\cdot$ satisfying:

1. $|M| \ge 2$
2. For all $a$, $b$, $c$ $\in M$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
3. For all $a$, $b$ $\in M$ with $a \ne b$, exactly one of the equations $a \cdot x = b$ and $b \cdot x = a$ has a solution for $x$ in $M$.
4. For all $a$, $b$ $\in M$, the equation $a \cdot x = b$ has a solution for $x$ in $M$ if and only if the equation $y \cdot a = b$ has a solution for $y$ in $M$.

Examples of magniums are the positive real numbers and the non-negative integers under addition. Another example is the set $\{1, 2, 3, ..., 120\}$ under the operation $x \cdot y = \min\{x + y, 120\}$, which shows that magniums generally do not have the cancellation property.

So the question is, is there a non-commutative magnium? Currently I'm trying to think of some two-valued function $f(x, y)$ on $\Bbb{R}$ satisfying $f(x, y) \ge \max\{x, y\}$ that's associative but not commutative, and I'm not coming up with anything good.

from Hot Weekly Questions - Mathematics Stack Exchange
Perry Ainsworth