I'm trying to understand how to find equivalence classes of a language to prove its regularity. I think that if I'm able to FULLY understand one example then I will get this topic right.
Let's say I have a language
$L=\{a^{m} b^{n} :1\le m \le n\}$ over alphabet $\Sigma=\{a,b\}$
My approach is following:
$K_{\epsilon} = \{\epsilon\}$ - class containing empty word. If we concatenate any word $z$ from the language, then word from this class will be in the language.
$K_{A} = \{a^{i}\} \land i \gt 1$ - class containing strings of a's. If we concatenate word $z = b^{k} \land k>i$ than all strings from this class will be in the language. There is infinitely many such classes.
$K_{AB} = \{a^{i}b^{j}\} \land i>1 \land j - class containing strings of a's and b's. If we concatenate word $z=b^{i-j+1}$ then words from this class will be in the language. There is infinitely many such classes.
What I don't understand is why we am I looking for such word $z$ and why different length of word constitutes new class. Also why words from classes do not have to be indluded in language ?
I know that I still have to show that those classes are not related but firstly I want to understand whats above.
Please give me intuition rather than solution - I want to understand it more than just do it :)