On Some Unification Theorems: Yang–Baxter Systems; Johnson–Tzitzeica Theorem

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aiming to unify algebraic and coalgebraic structures. The unification of the algebra structures and the coalgebra structures leads to an extension for the duality between finite dimensional algebras and finite dimensional coalgebras to the category of finite dimensional Yang–Baxter structures. In the same manner, we attempt to unify the Tzitzeica–Johnson theorem and its dual version, obtaining a new theorem about circle configurations.