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\begin{document}

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\section*{shelf}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{infinite_braid_group}{Infinite braid group}\dotfill \pageref*{infinite_braid_group} \linebreak
\noindent\hyperlink{in_set_theory}{In set theory}\dotfill \pageref*{in_set_theory} \linebreak
\noindent\hyperlink{action_of_positive_braid_monoid}{Action of positive braid monoid}\dotfill \pageref*{action_of_positive_braid_monoid} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{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.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

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

\begin{displaymath}
a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright  c)   .
\end{displaymath}
Similarly a set with a binary operation $\triangleleft$ obeying the right self-distributive law is called a \textbf{right shelf}.

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

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Of course all the usual examples of [[racks]] and [[quandles]] are \emph{a fortiori} shelves. But there are notable examples not of this type.

\hypertarget{infinite_braid_group}{}\subsubsection*{{Infinite braid group}}\label{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 \emph{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

\begin{displaymath}
a \triangleright b \coloneqq a sh(b) \sigma_1 sh(a)^{-1}.
\end{displaymath}
One may verify the left distributivity by a [[string diagram]] calculation (which appears on page 29 of this \href{https://books.google.com/books?id=XfsHCAAAQBAJ&pg=PA27&lpg=PA27&dq=braid+exponentiation&source=bl&ots=4ncxE2f7oT&sig=Mn-YDL97W9NWE0baexJEWrD4J4k&hl=en&sa=X&ved=0ahUKEwiB4saR3frKAhXBSiYKHbWaBEoQ6AEIHTAA#v=onepage&q=braid%20exponentiation&f=false}{Google book}, \hyperlink{Deh3}{Dehornoy3}).

\hypertarget{in_set_theory}{}\subsubsection*{{In set theory}}\label{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:

\begin{quote}%
There exists an [[elementary embedding]] $j: V_\lambda \to V_\lambda$ on the [[structure]] $(V_\lambda, \in)$ that is not the identity.


\end{quote}
Then, for $A \subseteq V_\lambda$, put

\begin{displaymath}
j(A) \coloneqq \bigcup_{\alpha \lt \lambda} j(A \cap V_\alpha).
\end{displaymath}
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 object|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

\begin{displaymath}
\phi_j: F_1 \to E_\lambda.
\end{displaymath}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Laver)} If $j \in E_\lambda$ is not the identity, then $\phi_j$ is injective.

\end{theorem}
The famous \emph{Laver tables} (derived from set-theoretic considerations which we omit for now) describe certain [[finite set|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 \emph{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.

\hypertarget{action_of_positive_braid_monoid}{}\subsection*{{Action of positive braid monoid}}\label{action_of_positive_braid_monoid}

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

\begin{displaymath}
\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.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
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

\begin{displaymath}
\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);
\end{displaymath}
conversely, if $\triangleright$ is any binary operation, then these equations describe an action of $B_n^+$ only if $\triangleright$ is left distributive.

\end{prop}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[rack]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

These are some general references:

\begin{itemize}%
\item [[Alissa Crans]], \emph{Lie 2-Algebras}, Chapter 3.1: Shelves, Racks, Spindles and Quandles, Ph.D. thesis, U.C. Riverside, 2004. (\href{https://arxiv.org/abs/math/0409602}{pdf}).


\item [[Patrick Dehornoy]], \emph{Braids and Self-Distributivity}, Progress in Mathematics 192, Birkh\"a{}user Verlag, 2000.



\end{itemize}
These develop the connection between the free shelf on one generator and elementary embeddings in set theory:

\begin{itemize}%
\item Richard Laver, The left distributive law and the freeness of an algebra of elementary embeddings,   (1992), 209--231.


\item Richard Laver, ,   (1995), 334--346.


\item Randall Dougherty and Thomas Jech, ,   (1997), 201--241.


\item Randall Dougherty, ,   (1993), 211--241.


\item Randall Dougherty, 



\end{itemize}
For a popularized account of this material, see:

\begin{itemize}%
\item John Baez, \href{https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/}{Shelves and the infinite}.

\end{itemize}
[[!redirects shelves]] [[!redirects left shelf]] [[!redirects left shelves]] [[!redirects right shelf]] [[!redirects right shelves]] [[!redirects Laver table]] [[!redirects Laver tables]]



\end{document}