# Questions tagged [self-distributivity]

The self-distributivity tag has no usage guidance.

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### Racks with “trichotomy”

(This is a follow-up question; the original question was about shelves.)
A rack $(R, \rhd, \lhd)$ is a set $R$ with two binary operations $\rhd$ and $\lhd$ such that for all $x, y, z \in R$:
$x \rhd ...

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### Shelves with “trichotomy”

A left shelf $(S, \rhd)$ is a magma with the left self-distributive law:
$$
\forall x, y, z \in S: x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z).
$$
Shelves are generalization of racks and quandles ...

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### Examples of nilpotent self-distributive algebras

Suppose that $(X,*,1)$ is an algebra that satisfies the identities $x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x$. Define the right powers by letting $x^{[1]}=x$ and
$x^{[n+1]}=x*x^{[n]}$. We say that $(X,*,1)$ is ...

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### Basic questions about varieties of uniformly partially permutative algebras

Define the Fibonacci terms $t_{n}(x,y)$ for all $n\geq 1$ by letting
$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$.
We say that an algebra $(X,*)$ is $N$-uniformly partially ...

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### Cardinalities of finite uniformly partially permutative algebras

Define the Fibonacci terms $t_{n}(x,y)$ for all $n\geq 1$ by letting
$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$.
We say that an algebra $(X,*)$ is $N$-uniformly partially ...

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### The varieties axiomatized by join-semilattice, self-distributivity, and Fibonacci term identities

Define the Fibonacci terms $t_{n}(x,y)$ for all $n\geq 1$ by letting
$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$.
For $N\geq 1$, let the variety $V_{N}$ consist of all algebras $(...

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### 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 ...

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### Infinite products of complex numbers or matrices arising from rank-into-rank embeddings

I wonder what kinds of closed form infinite products of matrices, elements of Banach algebras, and complex numbers arise from the rank-into-rank embeddings.
Suppose that $\lambda$ is a cardinal and $...

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### 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 ...

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### 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 ...

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### 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,...

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### 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{...

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### 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,\...

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### What possible order type can the critical points of these algebras with one generator achieve?

Suppose that $(X,*)$ is an algebra that satisfies the self-distributivity identity $x*(y*z)=(x*y)*(x*z)$. We say that an element $x\in X$ is a left-identity if $x*y=y$ for all $x\in X$. Let $\mathrm{...

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### 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)=...