Positive and negative correlations for conditional Ising distributions

In the Ising model at zero external field with ferromagnetic first neighbors interaction the Gibbs measure is investigated using the group properties of the contours configurations. Correlation inequalities expressing positive dependence among groups and comparison among groups and cosets are used. An improved version of the Griffiths' inequalities is proved for the Gibbs measure conditioned to a subgroup. Examples of positive and negative correlations among the spin variables are proved under conditioning to a contour or to a separation line.