  In our earlier paper, a generalized Dobrushin ergodicity coefficient of
Markov operators (acting on abstract state spaces) with respect to a projection
$P$, has been introduced and studied. It turned out that the introduced
coefficient was more effective than the usual ergodicity coefficient. In the
present work, by means of a left consistent Markov projections and the
generalized Dobrushin's ergodicity coefficient, we investigate uniform and weak
$P$-ergodicities of non-homogeneous discrete Markov chains (NDMC) on abstract
state spaces. It is easy to show that uniform $P$-ergodicity implies a weak
one, but in general the reverse is not true. Therefore, some conditions are
provided together with weak $P$-ergodicity of NDMC which imply its uniform
$P$-ergodicity. Furthermore, necessary and sufficient conditions are found by
means of the Doeblin's condition for the weak $P$-ergodicity of NDMC. The weak
$P$-ergodicity is also investigated in terms of perturbations. Several
perturbative results are obtained which allow us to produce nontrivial examples
of uniform and weak $P$-ergodic NDMC. Moreover, some category results are also
obtained. We stress that all obtained results have potential applications in
the classical and non-commutative probabilities.
