Let X be a non-degenerate, not necessarily linearly normal projective variety in P-r. Recently the generalization of property N-p to non-linearly normal projective varieties have been considered and its algebraic and geometric properties are studied extensively. One of the generalizations is the property N-d,N-p for the saturated ideal I-X (Eisenbud et al. in Compos Math 141: 1460-1478, 2005) and the other is the property N-p(s) for the graded module of the twisted global sections of O-x (1) (Kwak and Park in J Reine Angew Math 582: 87-105, 2005). In this paper, we are interested in the algebraic and geometric meaning of properties N-p(s) for every p >= 0 and the syzygetic behaviors of isomorphic projections and hyperplane sections of a given variety with property N-p(s).