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In this article, we extend our preceding studies on higher algebraicstructures of (co)homology theories defined by a left bialgebroid (U,A). For abraided commutative Yetter-Drinfel'd algebra N, explicit expressions for thecanonical 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 overU, it is shown that the cochain complex computing Cotor_U(A,N) defines a cyclicoperad with multiplication and hence the groups Cotor_U(A,N) form aBatalin-Vilkovisky algebra. In the second part of this article, Poissonstructures and the Poisson bicomplex for bialgebroids are introduced, whichsimultaneously generalise, for example, classical Poisson as well as cyclichomology. In case the bialgebroid U is commutative, a Poisson structure on Uleads to a Batalin-Vilkovisky algebra structure on Tor_U(A,A). As anillustration, we show how this generalises the classical Koszul bracket ondifferential forms, and conclude by indicating how classical Lie-Rinehartbialgebras (or, geometrically, Lie bialgebroids) arise from left bialgebroids.

Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids