Pre-anti-flexible bialgebras

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 as a splitting introduce the notion bialgebras and establish equivalences among matched pair bialgebras. Investigation special class leads to establishment Yang-Baxter equation. Both dual bimodules dendriform have same shape induces that both equation $\mathcal{D}$-equation are identical. Symmetric solution gives bialgebra. Finally, recall link $\mathcal{O}$-operators built symmetric solutions