LAUSR

Abstract In this paper, we derive pre-anti-flexible algebras structures in term of zero weight’s Rota-Baxter operators, built underlying left-symmetric algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce pre-anti-flexible bialgebras and establish its equivalences among matched pair of anti-flexible algebras and matched pair of pre-anti-flexible algebras. Special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation which is the same with -equation. Symmetric solution of pre-anti-flexible Yang-Baxter equation gives a pre-anti-flexible bialgebra. Finally, we recall and link -operators of anti-flexible algebras to bimodules of pre-anti-flexible algebras and built symmetric solutions of anti-flexible Yang-Baxter equation.