# Questions tagged [self-distributivity]

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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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### 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 $x^{}=x$ and $x^{[n+1]}=x*x^{[n]}$. The motivation for this question comes from the following fact about ...
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### 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 $\{\mathrm{crit}(\ell)\mid\ell\in\langle j,k\rangle\}$. ...
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### 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 ...
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### 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}...
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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 $\... 0answers 53 views ### Can a finitely generated algebra of rank-into-rank embeddings grow at rate$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}...
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-...