Def. 1. All elements in a Cayley table

Given a magma $(M,\circ)$, Let $S=\{s_1,s_2,\dots,s_n\}$ be the ordered set obtained by imposing an order $\mathcal{O}$ onto $M$. Then the cayley table $T$ is given by $T=\{\ x\lvert \forall (a,b) \in S \times S, a\circ b=x\}$. The tranpose is defined by $T^\textrm{T}=\{\ x\lvert \forall (b,a) \in S \times S, a\circ b=x\}$.


Def 2. Rows and columns and collections

A row $R_i \subset T$ is given by $R_i= \{\ x\lvert \exists i \in S,\forall (a,i) \in S \times S, a\circ i=x\}$. Similarly, a column $C_i=\{ x\lvert \exists i \in S,\forall (i,b) \in S \ti…

We have $$U_1=\begin{bmatrix} \begin{pmatrix} 3\\ 2\\ 2\\ 1 \end{pmatrix}, \begin{pmatrix} 3\\ 3\\ 2\\ 1 \end{pmatrix}, \begin{pmatrix} 2\\ 1\\ 2\\ 1 \end{pmatrix} \end{bmatrix}, \ \
U_2=\begin{bmatrix} \begin{pmatrix} 1\\ 0\\ 4\\ 0 \end{pmatrix}, \begin{pmatrix} 2\\ 3\\ 2\\ 3 \end{pmatrix}, \begin{pmatrix} 1\\ 2\\ 0\\ 2 \end{pmatrix} \end{bmatrix}$$

I have written them in the form
$$U_1=\{(x,y,2z,z) \mid x, y, z\in \mathbb{R}\}=\{x(1,0,0,0)+y(0,1,0,0)+z(0,0,2,1)\mid x, y, z\in \mathbb{R}\}$$