What are annihilators?

Question

I was recently listening to Automata lecture, there it was told told that an empty set is an Annihilator for concatenation just like $0$ is for multiplication. What do we mean by this statement?

Answer 1

In a semigroup $S$ (denoted multiplicatively) an absorbing element or annihilating element or zero element is an element $z$ of $S$ such that, for all $s \in S$, $zs = z = sz$.

In particular, the set of all languages is a semigroup under concatenation. This semigroup has a unique zero, namely the empty set, since the equality $L \emptyset = \emptyset =\emptyset L$ holds for each language $L$.