Period of an element in direct product of finite semigroups

Question

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 $\mathrm{index}(x)\neq \mathrm{index}(y)$?

Answer 1

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.