# is it possible to know if a language is regular if its equivalence classes are finite?

i have a theoretical questions, and was wondering if you could help me with it so i could understand the material better.

1)suppose we have some language L over $$\Sigma$$, can we know if L is regular if all of it equivalence classes of relation $$R_L$$ is finite?

2)and what about L*? is it finite? i mean the cardinality is infinity(alef), but can it be made into a finite language?

Edit: my question is about Myhill–Nerode relation

thank you very much!

• How is $R_L$ defined? Are you referring to the Myhill-Nerode relation? Dec 3, 2018 at 8:39
• yes, thank you very much. i'll state boldly that this is the relation i was refering to Dec 3, 2018 at 8:45
• I don't understand your second question. "Finite" means "finite cardinality". Dec 3, 2018 at 8:50

One of the main results of Myhill–Nerode theory is that a language is regular iff its Myhill–Nerode relation has finitely many equivalence classes. You can find a proof in many online resources. In particular, if all equivalence classes of the Myhill–Nerdoe relation of a language are finite, then it must have infinitely many equivalence classes, and so be non-regular.

The language $$L^*$$ is finite in two cases: if $$L = \emptyset$$ or if $$L = \{\epsilon\}$$. In any other case it is infinite.