In this paper it is shown that for the study of Orlicz spaces the condition that a Young's function A be convex can be replaced by the more general (and more convenient) condition that A(x)/x be non-decreasing. Some properties of the lattice of Orlicz spaces ordered by inclusion are given. belong to Lc whenever f E LA, g E LB. The condition is also sufficient when the convolution is formed over the integers or (0,2pi]. It is proven here that the condition is also necessary; all triplets A, B, C of Y-functions for which the condition holds are determined.