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. $f_{x\vee y\vee z}=f_x\vee f_y\vee f_z$.