This is not necessarily a research question. Just a question out of curiosity:
I am trying to understand if one can define "irreducible" languages. As a first guess I call a language L "reducible" if it can be written as $L = A \cdot B$ with $A \cap B = \emptyset$ and $|A|,|B|>1$, otherwise call the language "irreducible". Is it true:
1) If P is irreducible, A,B, C are languages such that $A\cap B = \emptyset$, $P \cap C = \emptyset$ and $A\cdot B = C\cdot P$, then there exists a language $B' \cap P = \emptyset$ such that $B = B'\cdot P$? This would correspond in integers to the lemma of Euklid and would be usefull to prove uniqueness of "factorization".
2) Is it true that every language can be factored in a finite number of irreducible languages?
If someone has a better idea on how to define "irreducible" language, I would like to hear it. (Or is there maybe already a definiton of this, which I am unaware of?)