  In this document, we study the interaction between different geometric
structures that can be defined as morphisms of sections of the generalized
tangent bundle $\mathbb TM:= TM\oplus T^*M\to M$. In particular, we show the
behaviour of various generalized polynomial structures with respect to
different metrics on $\mathbb TM$, in the frame of $(\alpha,
\varepsilon)$-metric structures; and the commutation or anti-commutation of
generalized polynomial structures, forming triple structures.
