Pre-anti-flexible bialgebra

In this paper, we derive pre-anti-flexible algebras structures in term of zero weight's Rota-Baxter operators defined on anti-flexible algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce the notion of pre-anti-flexible bialgebras and establish equivalences among matched pair of anti-flexible algebras, matched pair of pre-anti-flexible algebras and pre-anti-flexible bialgebras. Investigation on special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation. Both dual bimodules of pre-anti-flexible algebras and dendriform algebras have the same shape and this induces that both pre-anti-flexible Yang-Baxter equation and $\mathcal{D}$-equation are identical. Symmetric solution of pre-anti-flexible Yang-Baxter equation gives a pre-anti-flexible bialgebra. Finally, we recall and link $\mathcal{O}$-operators of anti-flexible algebras to bimodules of pre-anti-flexible algebras and built symmetric solutions of anti-flexible Yang-Baxter equation.