All Questions
Tagged with large-cardinals self-distributivity
21 questions
11
votes
2
answers
766
views
Motivation for Laver's use of large cardinals to show finite combinatorial properties of Laver tables
Laver showed in 1995 that the period of the first row of certain Laver tables is unbounded, assuming that a rank-into-rank cardinal exists.
The most accessible proof of his result that I was able to ...
- 245
1
vote
0
answers
92
views
Why does $p_{n}(i,1)=1$ so often where the polynomials $p_{n}$ are obtained from the classical Laver tables
So I was doing some computer calculations with the classical Laver tables and I found polynomials $p_{n}(x,y)$ such that $p_{n}(i,1)=1$ for many $n$.
The $n$-th classical Laver table is the unique ...
1
vote
0
answers
61
views
Growth rate of the critical points of the Fibonacci terms $t_{n}(x,y)$ vs $t_{n}(1,1)$ in the classical Laver tables
The classical Laver table $A_{n}$ is the unique algebra $(\{1,\dots,2^{n}\},*_{n})$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ for all $x,y,z\in A_{n}$.
Define the ...
2
votes
1
answer
107
views
Attraction in Laver tables
If $X$ is a self-distributive algebra, then define $x^{[n]}$ for all $n\geq 1$ by letting $x^{[1]}=x$ and $x^{[n+1]}=x*x^{[n]}$. The motivation for this question comes from the following fact about ...
1
vote
0
answers
75
views
Multiple roots in the classical Laver tables
The classical Laver table $A_{n}$ is the unique algebraic structure $$(\{1,\dots,2^{n}\},*_{n})$$ such that $x*_{n}1=x+1\mod 2^{n}$ and
$$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$ for all $x,y,z\in\{1,...
1
vote
0
answers
59
views
Can we have $\sup\{\alpha\mid(x*x)^{\sharp}(\alpha)>x^{\sharp}(\alpha)\}=\infty$ in an algebra resembling the algebras of elementary embeddings?
A finite algebra $(X,*,1)$ is a reduced Laver-like algebra if it satisfies the identities $x*(y*z)=(x*y)*(x*z)$ and if there is a surjective function
$\mathrm{crit}:X\rightarrow n+1$ where
$\mathrm{...
1
vote
0
answers
60
views
In the classical Laver tables, do we have $o_{n}(1)<o_{n}(2)$ for any $n>8$?
The classical Laver table $A_{n}$ is the unique algebraic structure
$(\{1,\dots,2^{n}\},*_{n})$ where
$$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$
and where $$x*_{n}1=x+1\mod 2^{n}$$ for $x,y,z\in\{1,\...
2
votes
0
answers
83
views
For each $n$ is it possible to have $\mathrm{crit}(x^{[n]}*y)>\mathrm{crit}(x^{[n-1]}*y)>\dots>\mathrm{crit}(x*y)$?
Suppose that $(X,*,1)$ satisfies the following identities:
$x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$. Define the Fibonacci terms $t_{n}(x,y)$ for $n\geq 1$ by letting
$$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=...
1
vote
0
answers
44
views
Vastness of inverse systems of Laver-like algebras
Suppose that $(X,*,1)$ satisfies the identities
$x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x$. Then we say that $(X,*,1)$ is a reduced Laver-like algebra if whenever $x_{n}\in X$ for all $n\in\omega$, there is ...
1
vote
0
answers
33
views
Can we always extend a finitely generated reduced Laver-like algebra to a vast inverse system of Laver-like algebras?
An $(X,*,1)$ that satisfies the identities $x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$ is said to be a reduced Laver-like algebra if whenever $x_{n}\in X$ for $n\in\omega$, there is some $N\in\omega$ where $x_{...
3
votes
0
answers
245
views
Ordering large cardinal axioms around the level of $n$-huge by consistency strength?
So the large cardinal axioms are for the most part considered to be linearly ordered by consistency strength. For the large cardinals between extendibility and rank-into-rank (i.e. the $n$-huge ...
2
votes
0
answers
53
views
Calibrating the strength of the quotients of subalgebras of the classical Laver tables
Define an algebraic structure $A_{n}$ by letting
$$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$
where $*_{n}$ is the unique operation such that $x*_{n}1=x+1\mod 2^{n}$ for $$x\in\{1,\dots,2^{n}-1,2^{n}\}$...
1
vote
0
answers
48
views
Density of different types of critical points in an algebra of elementary embeddings
Suppose that $j,k:V_{\lambda}\rightarrow V_{\lambda}$ are elementary embeddings. Let $\mathrm{crit}_{n}(j,k)$ denote the $n$-th element in $\{\mathrm{crit}(\ell)\mid\ell\in\langle j,k\rangle\}$. ...
1
vote
0
answers
42
views
Density of critical points subalgebras of the algebras of elementary embeddings
Let $j:V_{\lambda}\rightarrow V_{\lambda}$ be an elementary embedding. Then $\{\mathrm{crit}(k)\mid k\in\langle j\rangle\}$ has order type $\omega$, so let $\mathrm{crit}_{n}(j)$ denote the $n$-th ...
1
vote
1
answer
67
views
Is every critically subsimple Laver-like algebra a quotient of a critically simple Laver-like algebra on the same number of generators?
A finite reduced Laver-like algebra is a finite algebra $(X,*,1)$ that satisfies the identities $1*x=x,x*1=1,x*(y*z)=(x*y)*(x*z)$ and where there is a natural number $n$ and a function $\mathrm{crit}:...
4
votes
1
answer
230
views
Why do highly composite rows on the bad Laver tables have longer periods?
For all natural numbers $n$, let $(B_{n},*_{n})$ be the algebraic structure with underlying set $\{1,\dots,n\}$ where
$x*_{n}1=x+1\mod n$,
$n*_{n}y=y$, and
$x*_{n}(y+1)=(x*_{n}y)*_{n}(x+1)$ for $x<...
1
vote
0
answers
76
views
Which varieties are compatible with the classical Laver tables?
Let $$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$ denote the $n$-th classical Laver table. The operation $*_{n}$ is the unique binary operation on
$\{1,\dots,2^{n}\}$ such that $$x*_{n}(y*_{n}z)=(x*_{n}...
5
votes
0
answers
94
views
The descriptive complexity and definiteness of the space of all elementary embeddings $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$
Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$.
Suppose that $(\alpha_{n})_{n}$ is an increasing cofinal sequence in $\lambda$. Give $\...
2
votes
0
answers
82
views
Can a finitely generated algebra of rank-into-rank embeddings grow at rate $O(n\cdot\log(n))$?
Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$ is a non-trivial elementary embedding, then define $\mathrm{crit}...
3
votes
1
answer
320
views
Can Laver tables go extinct?
An algebra $(X,*)$ is said to be self-distributive if it satisfies the identity $x*(y*z)=(x*y)*(x*z)$ for all $x,y,z\in X$. If $(X,*)$ is an algebra, then a subset $L\subseteq X$ is said to be a left-...
4
votes
0
answers
113
views
How many compatible linear orders exist on the classical Laver tables?
Let $A_{n}$ be the unique algebra $(\{1,\dots,2^{n}\},*_{n})$ such that
$x*_{n}1=x+1\mod 2^{n}$ and
$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ for all $x,y,z$. We say that a linear ordering $\preceq$ ...