nLab shelf

Idea

A shelf is a set with a binary operation that distributes over itself. The most widely studied class of shelves are the racks, and among those the quandles, but the axioms for shelves are simpler.

Definitions

A left shelf is a set with a binary operation $\triangleright$ obeying the left self-distributive law

$$a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright c) .$$

Similarly a set with a binary operation $\triangleleft$ obeying the right self-distributive law is called a right shelf.

A unital left shelf (meaning a left shelf together with an element that acts as an identity on both the left and the right) is the same as a graphic monoid: for a proof see graphic category.

Examples

Of course all the usual examples of racks and quandles are a fortiori shelves. But there are notable examples not of this type.

Infinite braid group

Let $B_n$ be the $n^{th}$ braid group. With the usual inclusion $B_n \to B_{n+1}$ by appending a strand to the end of a braid on $n$ strands, the colimit of the chain $B_0 \to B_1 \to B_2 \to \ldots$ is the infinite braid group $B_\infty$. Let $sh: B_\infty \to B_\infty$ be the homomorphism that sends $\sigma_i$ to $\sigma_{i+1}$. Then there is a left distributive operation on $B_\infty$ where

$$a \triangleright b \coloneqq a sh(b) \sigma_1 sh(a)^{-1}.$$

One may verify the left distributivity by a string diagram calculation (which appears on page 29 of this Google book, Dehornoy3).

In set theory

Shelves make an appearance in set theory via large cardinal axioms. Let $(V, \in)$ be a model of ZFC, and let $V_\lambda \subseteq V$ be the collection of elements of rank less than an ordinal $\lambda$ of $V$. One (rather strong) large cardinal axiom on (limit ordinals) $\lambda$ is:

There exists an elementary embedding $j: V_\lambda \to V_\lambda$ on the structure $(V_\lambda, \in)$ that is not the identity.

Then, for $A \subseteq V_\lambda$, put

$$j(A) \coloneqq \bigcup_{\alpha \lt \lambda} j(A \cap V_\alpha).$$

If we regard $A$ as an unary relation on $V_\lambda$, then $j$ induces an elementary embedding $(V_\lambda, \in, A) \to (V_\lambda, \in, j(A))$. In particular, if $k$ is any elementary embedding $(V_\lambda, \in) \to (V_\lambda, \in)$, which as a set of ordered pairs we may regard as a subset of $V_\lambda$, then $j(k)$ as a set of ordered pairs is also an elementary self-embedding of $(V_\lambda, \in)$. We get in this way a binary operation $(j, k) \mapsto j(k)$ on elementary embeddings, which we denote as $(j, k) \mapsto j \cdot k$, and it is not difficult to verify that $\cdot$ is left self-distributive.

Let $F_1$ denote the free left shelf generated by 1 element. If $E_\lambda$ denotes the collection of elementary embeddings on the structure $(V_\lambda, \in)$, then the preceding observations imply that $E_\lambda$ is a left shelf, so any $j \in E_\lambda$ induces a shelf homomorphism

$$\phi_j: F_1 \to E_\lambda.$$

Theorem

(Laver) If $j \in E_\lambda$ is not the identity, then $\phi_j$ is injective.

The famous Laver tables (derived from set-theoretic considerations which we omit for now) describe certain finite quotients of $F_1$. Letting $x$ denote the generator of $F_1$, define $x_n$ by $x_1 = x$ and $x_{n+1} = x_n \cdot x$. The quotient of $F_1$ by the single relation $x_{m+1} = x$ is a shelf of cardinality $2^k$, the largest power of $2$ dividing $m$; it is denoted $A_k$. It can be described alternatively as the unique left shelf on the set $\{1, 2, \ldots, 2^k\}$ such that $p \cdot 1 = p + 1 \mod 2^k$ (here $p$ represents the image of $x_p$ under the quotient $F_1 \to A_k$).

The “multiplication table” of an $A_k$ is called a Laver table. The behavior of Laver tables is largely not understood, but we mention a few facts. The first row consisting of entries $1 \cdot p$ is periodic (of some order dividing $2^k$). Under the large cardinal assumption that a nontrivial elementary self-embedding on a $V_\lambda$ exists, this period $f(k)$ tends to $\infty$ as $k$ does, but whether it does as a consequence of ZFC is not known. What is known is that this period, even if it increases to $\infty$, does so quite slowly: if we define $g(m)$ to be the smallest $k$ such that $f(k) \geq m$, then $g$ grows more quickly than say the Ackermann function.

Action of positive braid monoid

Let $B_n^+$ be the monoid of positive braids, which as a monoid is presented by generators $\sigma_1, \ldots, \sigma_{n-1}$ subject to the braid relations

$$\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i\; if \; {|i-j|} \gt 1.$$

Proposition

If $(X, \triangleright)$ is a shelf, then there is a monoid homomorphism $B_n^+ \to \hom(X^n, X^n)$ whose transform to an action $B_n^+ \times X^n \to X^n$ is described by the equations

$$\sigma_i(x_1, \ldots, x_i, x_{i+1}, \ldots, x_n) = (x_1, \ldots, x_i \triangleright x_{i+1}, x_i \ldots, x_n);$$

conversely, if $\triangleright$ is any binary operation, then these equations describe an action of $B_n^+$ only if $\triangleright$ is left distributive.

Related concepts

• rack
• symmetric space

References

These are some general references:

• Alissa Crans, Lie 2-algebras, Chapter 3.1: Shelves, Racks, Spindles and Quandles, Ph.D. thesis, U.C. Riverside, 2004. (pdf).

• Patrick Dehornoy, Braids and self-distributivity, Progress in Mathematics 192, Birkhäuser Verlag, 2000.

These develop the connection between the free shelf on one generator and elementary embeddings in set theory:

• Richard Laver, The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math. 91 (1992), 209–231.

• Richard Laver, On the algebra of elementary embeddings of a rank into itself, Adv. Math. 110 (1995), 334–346.

• Randall Dougherty and Thomas Jech, Finite left distributive algebras and embedding algebras, Adv. Math. 130 (1997), 201–241.

• Randall Dougherty, Critical points in an algebra of elementary embeddings, Ann. Pure Appl. Logic 65 (1993), 211–241.

• Randall Dougherty, Critical points in an algebra of elementary embeddings, II

For a popularized account of this material, see:

• John Baez, Shelves and the infinite.

category: algebra