  In this article, we extend our preceding studies on higher algebraic
structures of (co)homology theories defined by a left bialgebroid (U,A). For a
braided commutative Yetter-Drinfel'd algebra N, explicit expressions for the
canonical Gerstenhaber algebra structure on Ext_U(A,N) are given. Similarly, if
(U,A) is a left Hopf algebroid where A is an anti Yetter-Drinfel'd module over
U, it is shown that the cochain complex computing Cotor_U(A,N) defines a cyclic
operad with multiplication and hence the groups Cotor_U(A,N) form a
Batalin-Vilkovisky algebra. In the second part of this article, Poisson
structures and the Poisson bicomplex for bialgebroids are introduced, which
simultaneously generalise, for example, classical Poisson as well as cyclic
homology. In case the bialgebroid U is commutative, a Poisson structure on U
leads to a Batalin-Vilkovisky algebra structure on Tor_U(A,A). As an
illustration, we show how this generalises the classical Koszul bracket on
differential forms, and conclude by indicating how classical Lie-Rinehart
bialgebras (or, geometrically, Lie bialgebroids) arise from left bialgebroids.
