 preityk7t

2022-06-25

Does there exist a non-commutative algebraic structure with the following properties?
1. $|M|\ge 2$
2. For all $a$, $b$, $c$ $\in M$, $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$.
3. For all $a$, $b$ $\in M$ with a≠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 $M$. Let $M=\left({\mathbb{Q}}_{+}×\left\{0\right\}\right)\cup \left(\mathbb{Q}×\left\{1\right\}\right)\cup \left\{\mathrm{\infty }\right\}$ and consider the binary operation on $M$ defined as follows:
- $\left(q,0\right)\cdot \left(r,0\right)=\left(q+r,0\right)$ for all $q,r\in {\mathbb{Q}}_{+}$
- $\left(q,0\right)\cdot \left(r,1\right)=\left(q+r,1\right)$ for all $q\in {\mathbb{Q}}_{+},r\in \mathbb{Q}$
- $\left(r,1\right)\cdot \left(q,0\right)=\left(2q+r,1\right)$ for all $q\in {\mathbb{Q}}_{+},r\in \mathbb{Q}$
- $\left(q,1\right)\cdot \left(r,1\right)=\mathrm{\infty }$ for all $q,r\in \mathbb{Q}$
- $\mathrm{\infty }\cdot x=x\cdot \mathrm{\infty }=\mathrm{\infty }$ for all $x\in M$
A bit of casework shows that this is associative. It also has the property that $a\cdot x=b$ and $x\cdot a=b$ each have a solution (for $a\ne b$) iff $a, where $<$ is the total order on $M$ defined by ordering each of ${\mathbb{Q}}_{+}×\left\{0\right\}$ and $\mathbb{Q}×\left\{1\right\}$ according to their first coordinate and saying that every element of ${\mathbb{Q}}_{+}×\left\{0\right\}$ is less than every element of $\mathbb{Q}×\left\{1\right\}$ and that $\mathrm{\infty }$ is the greatest element. It follows that your properties (3) and (4) hold, so $M$ is a magnium. However, it is not commutative. opepayflarpws

As another way to get counterexamples, let $G$ be any totally ordered nonabelian group, and let $M$ be the monoid of nonnegative elements of $G$. Properties (3) and (4) follow from the fact that ${a}^{-1}b$ and $b{a}^{-1}$ are each nonnegative iff $a\le b$. An explicit example of such a $G$ is the group of affine maps $K\to K$ of positive slope for any ordered field $K$. The subset $M$ can then be explicitly described as the set of maps of the form $x↦ax+b$ where $a\ge 1$ and if $a=1$ then $b\ge 0$. (When $K=\mathbb{Q}$, this is closely related to the first example above, identifying $\left(q,0\right)$ with $x↦x+q$ and $\left(r,1\right)$ with $x↦2x+r$.)