On Generalized Triple Derivations on Lattices

Abstract

In this paper, we introduce the notion of a generalized triple derivation \(f\), with an associated triple derivation \(d\), on a lattice and investigate some related results. Among some other results, we prove that: Let \((L, \wedge, \vee)\) be a distributive lattice and \(f\) be a generalized triple derivation, with associated triple derivation \(d\), on \(L\). Then the following conditions are equivalent for all \(x, y, z \in L\):

1. \(f\) is an isotone generalized triple derivation on \(L\),
2. \(f_{x \wedge y \wedge z} = f_x \wedge f_y \wedge f_z\),
3. \(f_{x \vee y \vee z} = f_x \vee f_y \vee f_z\).