A shelf is a set with a binary operation that distributes over itself. Shelves are similar to racks (and there are forgetful functors from racks to shelves), but shelves are axiomatically simpler.
A left shelf is a set with a binary operation $\triangleright$ obeying the left self-distributive law
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.
Of course all the usual examples of racks and quandles are a fortiori shelves. But there are notable examples not of this type.
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
One may verify the left distributivity by a string diagram calculation (which appears on page 29 of this Google book, Dehornoy3).
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
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
(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.
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
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
conversely, if $\triangleright$ is any binary operation, then these equations describe an action of $B_n^+$ only if $\triangleright$ is left distributive.
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:
Last revised on May 7, 2016 at 13:29:42. See the history of this page for a list of all contributions to it.