Irreducible languages
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