Betti numbers of binomial ideals

Let us consider the family of binomial ideals B=I+J, where J is lattice ideal and I is a square-free quadratic monomial ideal. We give a formula for calculating the Betti numbers of B. Moreover we bound the Green-Lazarsfeld invariant of a family of quadratic binomial ideals B using this formula. This result extends a previous result of Eisenbud et al. for square-free quadratic monomial ideals and extends completely Fröberg's theorem. We describe also a subfamily where we can calculate the Green-Lazarsfeld invariant of any ideal B and we also compute its first non-linear Betti number.