The collection of regular languages over an alphabet Σ is defined recursively as follows:
- The empty language Ø is a regular language.
- For each a ∈ Σ (a belongs to Σ), the singleton language {a} is a regular language.
- If A and B are regular languages, then A ∪ B (union), A • B (concatenation), and A* (Kleene star) are regular languages.
- No other languages over Σ are regular.
I think that if it is closed under concatenation, then it must be closed under Kleene star, because we can take B to be A. So the definition can be without being closed under Kleene star?