2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71550An A_\infty-bialgebra is a DGM H equipped with structurally compatible operations {ω^{j,i} : H^{\otimes i} --> H^{\otimes j}} such that (H,ω^{1,i}) is an A_\infty-algebra and (H,ω^{j,1}) is an A_\infty-coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End(TH).15 pages, 5 figures. This final version (3-19-05) features improved graphics and other minor improvementsAlgebraic Topology55P35; 55P99 ; 52B05text