some properties on derivations of lattices

In this paper we consider some properties of derivations of lattices and show that (i) for a derivation d of a lattice l with the maximum element 1, it is monotone if and only if d(x) ≤ d(1) for all x ∈ l (ii) a monotone derivation d is characterized by d(x) = x ∧ d(1) and (iii) simple characterization theorems of modular lattices and of distributive lattices are given by derivations. we also show that, for a distributive lattice l and a monotone derivation d of it, the set fixd(l) of all fixed points of d is isomorphic to the lattice l/ ker(d). we provide a counter example to the result (theorem 4) proved in [3]