Can I write the empty string inbetween characters forming a string?

Question

I am wondering, whether this holds:

aeb = ab, where e stands for the empty string and the alphabet is the set {e,a,b}.

Answer 1

Yes, this does hold. You can add the empty string, ε, between any characters in a word because it's already there (as "it" is nothing). In fact, there are an infinite number of ε's between every character in every word. This can be explicitly illustrated with a finite state automata, showing that, in your specific case, no matter how many ε's you take to be in the middle of the string, the accepted word is the same.