Document Type : Original Manuscript

Authors

1 Graduate School of Information Science and Technology, Hokkaido University, P.O.Box 060-0814 Sapporo, Japan

2 Department of Mathematics, School of System Design and Technology, Tokyo Denki University, P.O.Box 120-8551 Tokyo, Japan

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 $x\in L$ (ii) a monotone derivation $d$ is characterized by $d(x) = x\wedge d(1)$ 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].

Keywords

References
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