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 $۱$, it is monotone if and only if $d(x) \le d(۱)$ for all $x\in L$ (ii) a monotone derivation $d$ is characterized by $d(x) = x\wedge d(۱)$ 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 ۴) proved in [۳].

Keywords:

derivation , order-preserving , modular lattice , distributive lattice