Non-Global Lie Higher Derivations on Triangular Algebras Without Assuming Unity

This work establishes a unified structural theory for non-global Lie higher derivations on triangular algebras $\mathcal{T}$, without assuming the existence of a unit element. The primary contribution is the introduction of extreme non-global Lie higher derivations and the proof that every non-global Lie higher derivation on $\mathcal{T}$ admits a unique decomposition into three components: a higher derivation, an extreme non-global Lie higher derivation, and a central map vanishing on all commutators $[x,y]$, where $x,y\in \mathcal{T}$ satisfy $xy=0$. This general framework is then explicitly applied to describe such derivations on two significant classes of algebras: upper triangular matrix algebras over faithful algebras and over semiprime algebras. By encompassing both unital and non-unital cases within a single characterization, the theory developed here not only generalizes numerous earlier results but also substantially expands the scope of the existing research landscape.