  We introduce and describe the class of split regular $Hom$-Leibniz color
$3$-algebras as the natural extension of the class of split Lie algebras, split
Leibniz algebras, split Lie $3$-algebras, split Lie triple systems, split
Leibniz $3$-algebras, and some other algebras.
  More precisely, we show that any of such split regular $Hom$-Leibniz color
$3$-algebras $T$ is of the form ${T}={\mathcal U} +\sum\limits_{j}I_{j}$, with
$\mathcal U$ a subspace of the $0$-root space ${T}_0$, and $I_{j}$ an ideal of
$T$ satisfying {for} $j\neq k:$ \[[{ T},I_j,I_k]+[I_j,{
T},I_k]+[I_j,I_k,T]=0.\] Moreover, if $T$ is of maximal length, we characterize
the simplicity of $T$ in terms of a connectivity property in its set of
non-zero roots.

Rings and Algebras