A left shelf $(S, \rhd)$ is a magma with the left self-distributive law:
$$
\forall x, y, z \in S: x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z).
$$
Shelves are generalization of [racks and quandles](https://en.wikipedia.org/wiki/Racks_and_quandles) from the knot theory.


I am looking for examples of shelves with the following additional axiom:
$$
\forall x, y \in S: x \neq y \implies (x \rhd y = y \iff y \rhd x \neq x).
$$
Can a left shelf satisfy this property?


(See also [the follow-up question](https://mathoverflow.net/questions/328423) about *racks*.)



Shelves with "trichotomy"
Linearly ordered sets when considered as lattices satisfy this property. Suppose that $(X,\wedge)$ is a meet-semilattice with corresponding partial ordering $\leq$. Then $\wedge$ is a self-distributive operation. Furthermore, $x\wedge y=y$ if and only if $y\leq x$. Therefore, the meet-semilattice $(X,\wedge)$ satisfies $x\neq y\rightarrow(x\wedge y=y\leftrightarrow y\wedge x\neq x)$ precisely when $\leq$ is a linear ordering.


**Steps towards a characterization**


The following results will establish that all left shelves that satisfy trichotomy must satisfy a weak version of idempotence. This result is a first step towards classifying shelves that satisfy tricotomy.


**Theorem:** Suppose that $(X,\*)$ is a left shelf that satisfies the trichotomy property and which is generated by $x$. Then $X=\{x,x\*x\}$. Furthermore, $r\*s=x\*x$ whenever $r,s\in X$.


A proof of the above theorem is given in [my answer to the follow up question](https://mathoverflow.net/a/328487/22277).


**Theorem:** Suppose that $(X,\*)$ is a left shelf that satisfies the trichonomy property. Then whenever $x\in X$, we have $|\{y\in X\mid y\*y=x\}|\leq 2$.


**Proof:** Suppose that $x\in X$ and $$|\{y\in X\mid y\*y=x\}|\geq 2.$$ Now let $$A=\{y\in X\mid y\*y=x\}\setminus\{x\}.$$ If $a,b\in A,a\neq b$, then
$a\*b=b$ or $b\*a=a$. Suppose therefore that $a\*b=b$. Then
$a\*(a\*b)=a\*b=b$. However, $$a\*(a\*b)=(a\*a)\*(a\*b)=x\*(a\*b)$$
$$=x\*b=(b\*b)\*b=x$$ which is a contradiction. **Q.E.D.**


