5
$\begingroup$

Let $A_{n}$ denote the classical Laver table of cardinality $2^{n}$.

Let $B_{n}^{*}$ denote the positive (including the identity) braid monoid on $n$ elements generated by $\sigma_{1},...,\sigma_{n}$. Let $BW_{n}^{*}$ denote the collection of all words over the alphabet $\sigma_{1},...,\sigma_{n}$ including the empty word.

Then whenever $(X,*)$ is a self-distributive structure, then the monoid $B_{n}$ acts on $X^{n+1}$ by letting $\sigma_{i}(x_{1},...,x_{i},x_{i+1},x_{n+1})=(x_{1},...,x_{i}*x_{i+1},x_{i},...,x_{n+1})$ (in other words, we replace $(x_{i},x_{i+1})$ with $(x_{i}*x_{i+1},x_{i})$).

Suppose now that $n,m$ are natural numbers. Then let $T(n,m)$ denote the largest natural number such that there is a positive braid word $\sigma$ such that $|\sigma|\geq T(n,m)$ along with $x_{1},...,x_{n+1}\in A_{m}$ such that if $\sigma(x_{1},...,x_{n+1})=(y_{1},...,y_{n+1})$, then $y_{i}<2^{m}$ whenever $1\leq i\leq n+1$. Here $|\sigma|$ denotes the length of the braid word $\sigma$.

What are some good bounds for the size of $T(n,m)$? The function $T(n,m)$ is increasing in both variables. Furthermore, by a fairly strightforward argument, I have obtained the lower bounds $T(n,n)\geq n(n-1)$.

If for all $n$ there exists an $n$-huge cardinal, one can prove that for all $n$, we have $^{\lim}_{m\rightarrow\infty}T(n,m)\rightarrow\infty$ (this limit however seems to grow very slowly). However, is there a ZFC proof of this result? For a given $n$, how slowly does the function $m\mapsto T(n,m)$ grow if it does converge to $\infty$?

Let $L(m,n)$ be the largest natural number such that there exists $x_{1},...,x_{n+1}\in A_{m}$ where $|\{\sigma(x_{1},...,x_{n+1})|\sigma\in BW_{n}^{+}\}|=L(m,n)$. I am also interested in lower and upper bounds for $L(m,n)$.

The motivation behind this question is that I want to see how applicable the Laver tables would be to the study of braids and knots, and a good way to test the applicability of Laver tables to braids and knots is to see how non-trivial the action of positive braids on the Laver tables is. This question is a part of a long term project to make large cardinals more applicable to diverse areas of mathematics through the study of algebras of elementary embeddings.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.