On derivations of lattices

In this paper, we introduce the notion of derivation for a lattice and discuss some related properties. We give some equivalent conditions under which a derivation is isotone for lattices with a greatest element, modular lattices, and distributive lattices. We characterize modular lattices and distributive lattices by isotone derivation. Moreover, we prove that if d is an isotone derivation of a lattice L, the fixed set Fixd(L) is an ideal of L. Finally, we prove that D(L) is isomorphic to L in a distributive lattice L.