  Within the decoherent histories formulation of quantum mechanics, we consider
arbitrarily long histories constructed from a fixed projective partition of a
finite-dimensional Hilbert space. We review some of the decoherence properties
of such histories including simple necessary decoherence conditions and the
dependence of decoherence on the initial state. Here we make a first step
towards generalization of our earlier results [Scherer and Soklakov, e-print:
quant-ph/0405080, (2004) and Scherer et al., Phys. Lett. A, vol. 326, 307,
(2004)] to the case of approximate decoherence.
