An AZ-style identity and Bollobلs deficiency

The powerful AZ identity is a sharpening of the famous LYM-inequality. More generally, Ahlswede and Zhang discovered a generalization in which the Bollobلs inequality for two set families can be lifted to an identity. s paper, we show another generalization of the AZ identity. The new identity implies an identity which characterizes the deficiency of the Bollobلs inequality for an intersecting Sperner family. We also give some consequences relating to Helly families and LYM-style inequalities.