Let $L$ be an almost Dirac structure on a manifold $M$. In [2] Theodore James Courant defines the tangent lifting of $L$ on $TM$ and proves that: If $L$ is integrable then the tangent lift is also integrable. In this paper, we generalize this lifting to tangent bundle of higher order.

The tangent lifts of higher order of Dirac structures and some properties have been defined in [9] and studied in [11]. By the same way, the tangent lifts of higher order of Poisson structures have been studied in [10] and some applications are given. In particular, the authors have studied the nature of the Lie algebroids and singular foliations induced by these lifting. In this paper, we study the tangent lifts of higher order of multiplicative Poisson structures, multiplicative Dirac structures...

Let $\left(G,\omega \right)$ be a presymplectic groupoid. In this paper we characterize the infinitesimal counter part of the tangent presymplectic groupoid of higher order, $(T^{r}G,{\omega }^{\left(c\right)})$ where $T^{r}G$ is the tangent groupoid of higher order and ${\omega }^{\left(c\right)}$ is the complete lift of higher order of presymplectic form $\omega$.