Extending structures for perm algebras and perm bialgebras

We investigate the theory of extending structures by the unified product for perm algebras, and the factorization problem as well as the classifying complements problem in the setting of perm algebras. For a special extending structure, non-abelian extension, we study the inducibility of a pair of automorphisms associated to a non-abelian extension of perm algebras, and give the fundamental sequence of Wells in the context of perm algebras. For a special extending structure, bicrossed product, we introduce the concept of perm bialgebras, equivalently characterized by Manin triples of perm algebras and certain matched pairs of perm algebras. We introduce and study coboundary perm bialgebras, and our study leads to the ''$\mathcal{S}$-equation" in perm algebras, which is an analogue of the classical Yang-Baxter equation. A symmetric solution of $\mathcal{S}$-equation gives a perm bialgebra.