The final matrix

The $n$-th final matrix is a $2^{n}\times 2^{n}$ matrix that along with the classical Laver tables encodes all the information about the generalized Laver table of the form $(A^{\leq 2^{n}})^{+}$.

Suppose that $n\in\omega$ and $A$ is a set. Let $(A^{\leq 2^{n}})^{+}$ denote the set of all non-empty strings over the alphabet $A$ whose length is at most $2^{n}$. Then there is a unique binary operation $*$ on $(A^{\leq 2^{n}})^{+}$ such that whenever $\mathbf{x},\mathbf{y},\mathbf{z}\in (A^{\leq 2^{n}})^{+}$ we have

1. $\mathbf{x}*\mathbf{y}=\mathbf{y}$ whenever $|\mathbf{x}|=2^{n}$,
2. $\mathbf{x}*a=\mathbf{x}a$ whenever $a\in A,|\mathbf{x}|<2^{n}$, and
3. $\mathbf{x}*(\mathbf{y}*\mathbf{z})=(\mathbf{x}*\mathbf{y})*(\mathbf{x}*\mathbf{z})$.

The operation $*$ is called the application operation.

If $A$ is finite, then the algebras $(A^{\leq 2^{n}})^{+}$ are quite large since $|(A^{\leq 2^{n}})^{+}|=\frac{a}{a-1}\cdot(a^{2^{n}}-1)$. Nevertheless, all of the mathematical information in $(A^{\leq 2^{n}})^{+}$ is contained in the final matrix.

The function $M_{n}(x,y,\ell)$

If $\mathbf{x}$ is a string, then let $\mathbf{x}[i]$ denote the $i$-th letter of the string $\mathbf{x}$ and let $|\mathbf{x}|$ denote the length of the string $\mathbf{x}$. In the algebra $(A^{\leq 2^{n}})^{+}$, we have
$|\mathbf{x}*\mathbf{y}|=|\mathbf{x}|*|\mathbf{y}|$ whenever $\mathbf{x},\mathbf{y}\in(A^{\leq 2^{n}})^{+}$and where the second $*$ denotes the classical Laver table operation in $A_{n}$.

Let $n$ be a natural number and let $*$ denote the application operation in $A_{n}$. Then there exists a unique function $M_{n}:\{(r,s,\ell)|r,s\in\{1,…,2^{n}\},\ell\in\{1,…,r*s\}\}\rightarrow\mathbb{Z}$ such that if $|\mathbf{x}|=r,|\mathbf{y}|=s$, then

1. if $M_{n}(r,s,\ell)>0$, then $(\mathbf{x}*\mathbf{y})[\ell]=\mathbf{y}[M_{n}(r,s,\ell)]$, and
2. if $M_{n}(r,s,\ell)<0$, then $(\mathbf{x}*\mathbf{y})[\ell]=\mathbf{x}[-M_{n}(r,s,\ell)]$.

Therefore, the function $M_{n}$ encodes all the information in the application operation on $(A^{\leq 2^{n}})^{+}$.

The final matrix

While the function $M_{n}$ encodes the application operation on $(A^{\leq 2^{n}})^{+}$, the function $M_{n}$ contains quite a bit of redundant information. The final matrices are the new mathematical objects that do not contain as much redundant information.

The final matrices $FM^{-}_{n},FM_{n}^{+}:\{1,…,2^{n}\}^{2}\rightarrow\mathbb{Z}$ are the functions defined by

$$FM^{+}_{n}(x,y)=M_{n}(x,2^{n},y)\text{ and }FM_{n}^{-}(x,y)=M_{n}(x,2^{o_{n}(x)},y).$$

The functions $FM_{n}^{-}(x,y)$ and $FM_{n}^{+}(x,y)$ are interdefinable by

1. $FM_{n}^{+}(x,y)=FM_{n}^{-}(x,y)$  whenever $FM_{n}^{-}(x,y)<0$ and
2. $FM_{n}^{+}(x,y)=FM_{n}^{-}(x,y)+2^{n}-2^{o_{n}(x)}$ whenever $FM_{n}^{-}(x,y)>0$.

The function $M_{n}(x,y,\ell)$ is obtainable from $FM_{n}^{-}(x,y)$ and $FM_{n}^{+}(x,y)$ by

v

1. $M_{n}(x,y;\ell)=FM_{n}^{+}(x,\ell)$ whenever $FM^{+}_{n}(x,\ell)<0$ and
2. $M_{n}(x,y,\ell)=FM_{n}^{+}(x,\ell)-\lfloor(2^n-y)2^{-o_{n}(x)}\rfloor 2^{o_{n}(x)}.$

Therefore since $FM_{n}^{-}(x,y)$ and $FM_{n}^{+}(x,y)$ may be used to obtain the function $M_{n}(x,y,\ell)$, one can use $FM_{n}^{-}(x,y)$ and $FM_{n}^{+}(x,y)$ to express all of the combinatorial information found in the generalized Laver tables $(A^{\leq 2^{n}})^{+}$.