# What are annihilators?

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?

• In Algebra the Annihilator of some element is the set of all elements such that their product is $0$. This for sure is in integral domains a very small set but can be very large in rings which doesn't have such nice properties. In linear algebra the annihilator of some set is the set of all linearforms which evaluate constant zero on this set. – Dominic Michaelis Sep 30 '15 at 11:34
• What @DominicMichaelis said means in particular that $A \circ \emptyset = \emptyset$, where $\circ$ is concatenation. – mrp Sep 30 '15 at 11:43
• @DominicMichaelis can you please put some examples as I am not that well versed in mathematics right now. – CodeYogi Sep 30 '15 at 11:54

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 $L$.