Bialgebroids, ×A-Bialgebras and Duality

An equivalence between Luʹs bialgebroids, Xuʹs bialgebroids with an anchor, and Takeuchiʹs ×A-bialgebras is explicitly proven. A new class of examples of bialgebroids is constructed. A (formal) dual of a bialgebroid, termed bicoalgebroid, is defined. A weak Hopf algebra is shown to be an example of such a bicoalgebroid.