  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.
