  The monogamous nature of entanglement has been illustrated by the derivation
of entanglement sharing inequalities - bounds on the amount of entanglement
that can be shared amongst the various parts of a multipartite system.
Motivated by recent studies of decoherence, we demonstrate an interesting
manifestation of this phenomena that arises in system-environment models where
there exists interactions between the modes or subsystems of the environment.
We investigate this phenomena in the spin-bath environment, constructing an
entanglement sharing inequality bounding the entanglement between a central
spin and the environment in terms of the pairwise entanglement between
individual bath spins. The relation of this result to decoherence will be
illustrated using simplified system-bath models of decoherence.
