Extended $\mathcal{O}$-operators, extended perm Yang-Baxter equations and related structures

In this paper, we introduce the notions of extended $\mathcal{O}$-operators on perm algebras, as generalization of $\mathcal{O}$-operators, and the extended perm Yang-Baxter equation. We demonstrate that an $\mathcal{O}$-operator of weight $\lambda$ on a perm algebra gives rise to a post-perm algebra, and an extended $\mathcal{O}$-operator yields a new perm algebra structure. We also give an equivalent characterization of extended $\mathcal{O}$-operators in terms of ordinary $\mathcal{O}$-operators via the ${\pi }_{\pm }$ decomposition. The notion of the generalized perm Yang-Baxter equations is introduced, and their relationship with extended $\mathcal{O}$-operators is established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended perm Yang-Baxter equation (extended perm-YBE), which generalizes the perm Yang-Baxter equation. We establish that a solution of the extended perm-YBE with $(R,\text{ad})$-invariant skew-symmetric part is characterized by an extended $\mathcal{O}$-operator. Furthermore, relationships among extended $\mathcal{O}$-operators, the perm Yang-Baxter equation and the extended perm-YBE are investigated through the framework of quadratic perm algebras and semi-direct product perm algebras.