Restoration of Gibbsianness forprojected and FKG renormalized measures

We restore part of the thermodynamic formalism for somerenormalized measures that are known to be non-Gibbsian. Wedetermine a necessary and sufficient condition for consistencywith a specification that is quasilocal only in a fixeddirection. This condition is then applied to models with FKGmonotonicity and to models with appropriate directionalcontinuity rates, in particular to (noisy) decimations orprojections of the Ising model. In this way we establish: (i)the validity of the second part of the variational principlefor projected and FKG block-renormalized measures, and (ii) thealmost quasilocality of FKG block-renormalized + and –measures.