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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.)

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  • $\begingroup$ I suggest calling this property the "trichonomy property" or "shelf trichonomy property." $\endgroup$
    – Joseph Van Name
    Apr 18, 2019 at 23:09
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    $\begingroup$ One follow-up question is whether it always follows from these properties that $x>y$ defined by ($x\rhd y=y$ and $x\neq y$) defines a total order (i.e., it is transitive)? $\endgroup$
    – YCor
    Apr 19, 2019 at 22:07
  • $\begingroup$ If we extend the question to left shelves $(X,*,1)$ that satisfy the identities $x*1=1,1*x=x$ and where the non-unital trichotomy states that $\forall x,y\in X,|\{x,y,1\}|=3\Rightarrow(x*y=y\Leftrightarrow y*x\neq x)$, then whenever $X$ is a linear ordering with greatest element $1$ and $\rightarrow$ is the Heyting operation on $X$, then $(X,\rightarrow,1)$ satisfies the non-unital trichotomy where if $x,y\in X\setminus\{1\}$, then $x\rightarrow y=y$ iff $y<x$. $\endgroup$
    – Joseph Van Name
    Apr 19, 2019 at 22:34
  • $\begingroup$ The relation that YCor is proposing resembles the total ordering on the critical points in the algebras of rank-into-rank embeddings. If $X$ is a finite linear ordering, then $(X,\rightarrow,1)$ satisfies the non-unital trichotomy, and there some embedding $\phi:(X,\rightarrow,1)\rightarrow\mathcal{E}_{\lambda}/\equiv^{\gamma}$. However, $x<y$ if and only if $\mathrm{crit}(\phi(x))<\mathrm{crit}(\phi(y))$. $\endgroup$
    – Joseph Van Name
    Apr 19, 2019 at 22:36

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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.

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