  It is known that the Dobrushin's ergodicity coefficient is one of the
effective tools to study a behavior of non-homogeneous Markov chains. In the
present paper, we define such an ergodicity coefficient of a positive mapping
defined on ordered Banach space with a base (OBSB), and study its properties.
In terms of this coefficient we prove the equivalence uniform and weak
ergodicities of homogeneous Markov chains. This extends earlier results
obtained in case of von Neumann algebras. Such a result allowed to establish a
category theorem for uniformly ergodic Markov operators. We find necessary and
sufficient conditions for the weak ergodicity of nonhomogeneous discrete Markov
chains (NDMC). It is also studied $L$-weak ergodicity of NDMC defined on OBSB.
We establish that the chain satisfies $L$-weak ergodicity if and only if it
satisfies a modified Doeblin's condition ($\frak{D}_1$-condition). Moreover,
several nontrivial examples of NDMC which satisfy the $\frak{D}_1$-condition
are provided.
