2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60593An equivalence between Lu's bialgebroids, Xu's bialgebroids with an anchor and Takeuchi's $\times_{A}$-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.16 pages, LaTeX; more concise presentation; Theorem 4.1 improved; final version to appear in J. AlgebraQuantum AlgebraRings and Algebrastext