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# Questions tagged [self-distributivity]

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Let $(X, {\sqsubset}, {\circ}, {\ast})$ be a set $X$ with a strict partial order $\sqsubset$ and two partial binary operations $\circ$ and $\ast$ such that for any $a, b, c \in X$: $a \circ b$ and $a ... 1answer 169 views ### Racks with “trichotomy” (This is a follow-up question; the original question was about shelves.) A rack$(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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### 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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### 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)=...

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