SOME PROPERTIES ON DERIVATIONS OF LATTICES

Abstract

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 $d(x) le d(1)$ for all $xin L$ (ii) a monotone derivation $d$ is characterized by $d(x) = xwedge d(1)$ 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 ${rm Fix}_d(L)$ 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].