In this paper, the concepts of indecomposable matrices and fully indecomposable matrices over a distributive lattice $L$ are introduced, and some algebraic properties of them are obtained. Also, some characterizations of the set $F_n(L)$ of all $n\times n$ fully indecomposable matrices as a subsemigroup of the semigroup $H_n(L)$ of all $n\times n$ Hall matrices over the lattice $L$ are given.
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.