Definition

$E_{k}$ where $E = k = 1 ⨄ \infty E_{k}$ The Equivalence Class of the state space $E$ by the Communication $\leftrightarrow$

Example

where $R_{i}$ are Recurrent block matrix and $T$ is Transient block matrix

$E_{1}, E_{2}, \dots$ are pairwise coprime

$\forall k : E_{k}$ are IRR

$\forall k : E_{k}$ is PR, NR or TR

Facts

The quotient set form a Partition

Link to original
Therefore the set of communication classes^[Quotient Set] form a Partition.