An intersection equation in semigroups

Question

Do you have an example of a semigroup $S$ and a collection of its subsets $(A_i)_{i\in I}$ and $a\in S$ such that $$a\big(\bigcap_{i\in I}A_i\big)\ne\bigcap_{i\in I}aA_i$$ ?

Answer 1

Let $S$ have more than one element.

Let $xy=x$ be the multiplication on $S$.

If the $A_{i}$ are not empty then $\bigcap_{i\in I}aA_{i}=\left\{ a\right\} $.

The $A_{i}$ can be chosen in such a way that $\bigcap_{i\in I}A_{i}=\emptyset$ and consequently $a\bigcap_{i\in I}A_{i}=\emptyset$.