nLab
shelf

Contents

Contents

Idea

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.

Definitions

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

a(bc)=(ab)(ac). 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 nB_n be the n thn^{th} braid group. With the usual inclusion B nB n+1B_n \to B_{n+1} by appending a strand to the end of a braid on nn strands, the colimit of the chain B 0B 1B 2B_0 \to B_1 \to B_2 \to \ldots is the infinite braid group B B_\infty. Let sh:B B sh: B_\infty \to B_\infty be the homomorphism that sends σ i\sigma_i to σ i+1\sigma_{i+1}. Then there is a left distributive operation on B B_\infty where

abash(b)σ 1sh(a) 1.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,)(V, \in) be a model of ZFC, and let V λVV_\lambda \subseteq V be the collection of elements of rank less than an ordinal λ\lambda of VV. One (rather strong) large cardinal axiom on (limit ordinals) λ\lambda is:

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

Then, for AV λA \subseteq V_\lambda, put

j(A) α<λj(AV α).j(A) \coloneqq \bigcup_{\alpha \lt \lambda} j(A \cap V_\alpha).

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

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

ϕ j:F 1E λ.\phi_j: F_1 \to E_\lambda.
Theorem

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

The famous Laver tables (derived from set-theoretic considerations which we omit for now) describe certain finite quotients of F 1F_1. Letting xx denote the generator of F 1F_1, define x nx_n by x 1=xx_1 = x and x n+1=x nxx_{n+1} = x_n \cdot x. The quotient of F 1F_1 by the single relation x m+1=xx_{m+1} = x is a shelf of cardinality 2 k2^k, the largest power of 22 dividing mm; it is denoted A kA_k. It can be described alternatively as the unique left shelf on the set {1,2,,2 k}\{1, 2, \ldots, 2^k\} such that p1=p+1mod2 kp \cdot 1 = p + 1 \mod 2^k (here pp represents the image of x px_p under the quotient F 1A kF_1 \to A_k).

The “multiplication table” of an A kA_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 1p1 \cdot p is periodic (of some order dividing 2 k2^k). Under the large cardinal assumption that a nontrivial elementary self-embedding on a V λV_\lambda exists, this period f(k)f(k) tends to \infty as kk 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)g(m) to be the smallest kk such that f(k)mf(k) \geq m, then gg grows more quickly than say the Ackermann function.

Action of positive braid monoid

Let B n +B_n^+ be the monoid of positive braids, which as a monoid is presented by generators σ 1,,σ n1\sigma_1, \ldots, \sigma_{n-1} subject to the braid relations

σ iσ i+1σ i=σ i+1σ iσ i+1,σ iσ j=σ jσ iif|ij|>1.\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,)(X, \triangleright) is a shelf, then there is a monoid homomorphism B n +hom(X n,X n)B_n^+ \to \hom(X^n, X^n) whose transform to an action B n +×X nX nB_n^+ \times X^n \to X^n is described by the equations

σ i(x 1,,x i,x i+1,,x n)=(x 1,,x ix i+1,x i,x n);\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 +B_n^+ only if \triangleright is left distributive.

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:

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.