8. Given the set
$$
I=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid x_{i} \in\{1,2, \cdots, 11\}\right\} \text {, }
$$
$A$ is a subset of $I$ and satisfies: for any
$$
\left(x_{1}, x_{2}, x_{3}, x_{4}\right) 、\left(y_{1}, y_{2}, y_{3}, y_{4}\right) \in A \text {, }
$$

there exist $i 、 j(1 \leqslant i<j \leqslant 4)$ such that
$$
\left(x_{i}-x_{j}\right)\left(y_{i}-y_{j}\right)<0 \text {. }
$$

Determine the maximum value of $|A|$, where $|A|$ denotes the number of elements in the set $A$. (Provided by Zhang Limin)
8. First, consider the set satisfying $x_{1}+x_{2}+x_{3}+x_{4}=24$
$$
\begin{aligned}
A= & \left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid x_{i} \in\{1,2, \cdots, 11\},\right. \\
& i=1,2,3,4\},
\end{aligned}
$$

The number of its elements is
$$
\begin{array}{l}
\mathrm{C}_{23}^{3}-4\left(\mathrm{C}_{2}^{2}+\mathrm{C}_{3}^{2}+\cdots+\mathrm{C}_{12}^{2}\right) \\
=\mathrm{C}_{23}^{3}-4 \mathrm{C}_{13}^{3}=891,
\end{array}
$$

Any two different vectors in this set satisfy the problem's conditions.
Next, consider the vector set
$$
\begin{array}{c}
B=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid x_{1}, x_{2} \in\{1,2, \cdots,\right. \\
\left.11\}, x_{3}=x_{4}=a \in\{1,2, \cdots, 11\}\right\},
\end{array}
$$

For each vector $\left(x_{1}, x_{2}, a, a\right)$, define the vector group as follows:
If $a=1$, the vector group is
$$
\begin{array}{l}
\left(x_{1}, x_{2}, 1,1\right),\left(x_{1}, x_{2}, 1,2\right), \cdots, \\
\left(x_{1}, x_{2}, 1,11\right),\left(x_{1}, x_{2}, 2,11\right), \cdots, \\
\left(x_{1}, x_{2}, 11,11\right) ;
\end{array}
$$

If $a=11$, the vector group is
$$
\begin{array}{l}
\left(x_{1}, x_{2}, 1,1\right),\left(x_{1}, x_{2}, 2,1\right), \cdots, \\
\left(x_{1}, x_{2}, 11,1\right),\left(x_{1}, x_{2}, 11,2\right), \cdots, \\
\left(x_{1}, x_{2}, 11,11\right) ;
\end{array}
$$

If $a$ is odd and greater than 1 and less than 11, the vector group is
$$
\begin{array}{l}
\left(x_{1}, x_{2}, 1,1\right), \cdots,\left(x_{1}, x_{2}, 1, a-1\right), \\
\left(x_{1}, x_{2}, 1, a\right),\left(x_{1}, x_{2}, 2, a\right), \cdots, \\
\left(x_{1}, x_{2}, a-1, a\right),\left(x_{1}, x_{2}, a, a\right) \\
\left(x_{1}, x_{2}, a, a+1\right), \cdots,\left(x_{1}, x_{2}, a, 11\right), \\
\left(x_{1}, x_{2}, a+1,11\right), \cdots,\left(x_{1}, x_{2}, 11,11\right) ;
\end{array}
$$

If $a$ is even, the vector group is
$$
\begin{array}{l}
\left(x_{1}, x_{2}, 1,1\right), \cdots,\left(x_{1}, x_{2}, a-1,1\right), \\
\left(x_{1}, x_{2}, a, 1\right),\left(x_{1}, x_{2}, a, 2\right), \cdots, \\
\left(x_{1}, x_{2}, a-1, a\right),\left(x_{1}, x_{2}, a, a\right), \\
\left(x_{1}, x_{2}, a+1, a\right), \cdots,\left(x_{1}, x_{2}, 11, a\right), \\
\left(x_{1}, x_{2}, 11, a+1\right),\left(x_{1}, x_{2}, 11,11\right) .
\end{array}
$$

For the vector set
$$
\begin{array}{c}
C=\left\{\left(a, a, x_{3}, x_{4}\right) \mid x_{3}, x_{4} \in\{1,2, \cdots,\right. \\
\left.11\}, x_{1}=x_{2}=a \in\{1,2, \cdots, 11\}\right\}
\end{array}
$$

Similarly, define the vector group.
For each vector in set $A$, it belongs to exactly one vector group, and different vectors belong to different vector groups, determining these 891 vector groups. Any 892 vectors must have two belonging to the same vector group, hence no longer possessing the property set by the problem.
Therefore, the maximum value sought is 891.