Extending Structures for Rota-Baxter family Hom-associative Algebras

TL;DR

This paper develops a comprehensive framework for extending, factorizing, and classifying Rota-Baxter family Hom-associative algebras, introducing new algebraic structures and solving key theoretical problems.

Contribution

It introduces extending datums, unified products, and matched pairs for Rota-Baxter family Hom-associative algebras, and provides solutions to extension, factorization, and classification problems.

Findings

Defined extending datums and unified products.

Solved the extending structure problem theoretically.

Introduced matched pairs and solved the factorization problem.

Abstract

In this paper, we first define extending datums and unified products of Rota-Baxter family Hom-associative algebras, and theoretically solve the extending structure problem. Moreover, we consider flag datums as an application, and give an example of the extending structure problem. Second, we introduce matched pairs of Rota-Baxter family Hom-associative algebras, and theoretically solve the factorization problem. Finally, we define deformation maps on a Rota-Baxter family Hom extending structure, and theoretically solve the classifying complements problem.