Strong lumpability (${\sum }_{j\in C}{p}_{ij}={\sum }_{j\in C}{p}_{i\prime j}$)

A partition of a Markov chain state space is strongly lumpable when, for every target block $C$, the sum ${\sum }_{j\in C}{p}_{ij}$ is identical for all source states $i$ in a given source block. These sums define the transition matrix of the block-valued Markov chain, independently of the initial law. Failure of this condition means the projection is not guaranteed to be a homogeneous Markov chain for every initial law, although special initial laws may still produce a Markov process.