Relative (pre-)anti-flexible algebras and associated algebraic structures

Pre-anti-flexible family algebras are introduced and linked with the notions of relative anti-flexible algebras, left and right pre-Lie family algebras and relative Lie algebras which are for mostly newly defined. Relative pre-anti-flexible algebras are given and their underlying algebras structures such as pre-anti-flexible family algebras, left and right pre-Lie family algebras, and other are investigated and significant identities linking those introduced structures are derived. In addition, a generalization of the Rota-Baxter operators defined on a relative anti-flexible algebra is introduced and both Rota-Baxter operators and its generalization are used to build relative pre-anti-flexible algebras structures underlying relative anti-flexible algebras and related consequences are derived.