  Thakur (2010) showed that, for $r,$ $s\in \mathbb{N}$, a product of two
Carlitz zeta values $\zeta_A(r)$ and $\zeta_A(s)$ can be expressed as an
$\mathbb{F}_p$-linear combination of $\zeta_A(r+s)$ and double zeta values of
weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing
$r,$ $s\in \mathbb{N}$, we construct a $t$-module $E'$. To determine whether an
$(r+s)$-tuple $\mathfrak{C}$ in $\mathbb{F}_q(\theta)^{r+s}$ gives a shuffle
relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point
$\mathbf{v}_\mathfrak{C}\in E'(\mathbb{F}_q[\theta])$ constructed with respect
to the given $(r+s)$-tuple $\mathfrak{C}$. We also provide an effective
criterion for deciding the $\mathbb{F}_q[t]$-torsion property of the point
$\mathbf{v}_\mathfrak{C}$.
