Does an irreducible Markov chain have exactly one equivalence class?

Question

I think I have a fundamental misunderstanding of equivalence classes and irreducibility of Markov chains.

Consider this simple MC where States A and B communicate ($ A \leftrightarrow B $) and States C and D can also communicate ($ C \leftrightarrow D $), but States A and C can only move one way ($ A \rightarrow C $). So we get two equivalence classes, allowing us to reduce the Markov chain to the one on the right, which shows $ \text{Class 1} \rightarrow \text{Class 2} $:

Now that we have reduced this Markov chain, can we say that it is irreducible, and if so, why does this clearly have more than one equivalence class when supposedly "All irreducible Markov chains have exactly one equivalence class"?

Answer 1

A finite state space Markov-Chain is (by definition) irreducible if the associated graph is strongly connected (we can move from any state to any other). The reduced Markov-Chain on the right is not strongly connected (you can not go from 2 to 1).