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]