Relative (pre-)anti-flexible algebrasnand associated algebraic structures

Pre-anti-flexible family algebras are introduced and used to define and describe the notions of Ωc-relative anti-flexible algebras, left and right pre-Lie family algebras and Ωc-relative Lie algebras. The notion of Ωc-relative pre-anti-flexible algebras are introduced and also used to characterize pre-anti-flexible family algebras, left and right pre-Lie family algebras and significant identities associated to these algebraic structures are provided. Finally, a generalization of the Rota-Baxter operators defined on an Ωc-relative anti-flexible algebra is introduced and it is also proved that both Rota-Baxter operators and its generalization provide Ωc-relative pre-antiflexible algebras structures and related consequences are derived.