Irreducible languages 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 $|A|,|B|&gt;1$, 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?)
There is the notion of primality of a language. It asks whether L can be written as $L_1 \cdot L_2$ where neither factor contains the empty word. A language is prime if it cannot be written in this form.

For a given regular language, represented by a DFA, it is shown in [MNS] that it is PSPACE-complete to decide primality. 

[MNS] Wim Martens, Matthias Niewerth and Thomas Schwentick,
"Schema design for XML repositories: complexity and tractability", 2010. doi:10.1145/1807085.1807117