Let D be the system of all distributive lattices and let D0 be the system of all L ∈ D such that L possesses the least element. Further, let D1 be the system of all infinitely distributive lattices belonging to D0. In the present paper we investigate the radical classes of the systems D, D0 and D1.
We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with φ-Laplacian (φ(u ' ))' = f(t, u, u ' ), u(0) = A, u(T) = B, where φ is an increasing homeomorphism, φ(R) = R, φ(0) = 0, f satisfies the Carathéodory conditions on each set [a, b] × R 2 with [a, b] ⊂ (0, T) and f is not integrable on [0, T] for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on [0, T].