  We present a sufficient condition for a non-injective function of a Markov
chain to be a second-order Markov chain with the same entropy rate as the
original chain. This permits an information-preserving state space reduction by
merging states or, equivalently, lossless compression of a Markov source on a
sample-by-sample basis. The cardinality of the reduced state space is bounded
from below by the node degrees of the transition graph associated with the
original Markov chain.
  We also present an algorithm listing all possible information-preserving
state space reductions, for a given transition graph. We illustrate our results
by applying the algorithm to a bi-gram letter model of an English text.
