Period of an element in direct product of finite semigroups Let $S$ and $T$ be finite semigroups and let $(x,y)\in S\times T$. What is the period of $(x,y)$ ?
I know that if
$$\mathrm{index}(x)=\mathrm{index}(y)$$,
then the period of $(x,y)$ is
$$\mathrm{lcm(period}(x),\mathrm{period}(y))$$.
What can we say in case $\mathrm{index}(x)\neq \mathrm{index}(y)$?

The period is also the
$$\mathrm{lcm(period}(x),\mathrm{period}(y))$$
Indeed suppose that $x^{i+p} = x^i$ and $y^{j+q} = y^j$. Let $k = \max\{i, j\}$. Then $x^{k+p} = x^k$ and $y^{k+q} = y^k$ and you are back to your known result.
