We give a condition on a Gibbs measure for an attractive Markov specification,
which assures extremality and the global Markov property. As an example of
application we consider the class of attractive Markov specifications defined on
a compact configuration space over a two-dimensional lattice by the interaction
Hamiltonians (assumed to have a finite set of periodic ground configurations)
satisfying Peierl's condition. We prove that each extremal Gibbs measure for
such a specification, at sufficiently low temperature, has the global Markov
property.