Shelves with "trichotomy" 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 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 about racks.)
 A: 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.
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.
