Triple systems with bilinear forms and the associated graph theory

We establish a decomposition theorem for the category of triple systems with multiplicative bases equipped with bilinear forms. We demonstrate that any object in this category can be expressed as a direct sum of irreducible, orthogonal ideals. This structural decomposition is intrinsically linked to graph theory. By constructing an appropriate directed graph associated with the triple system and its bilinear form, this decomposition can be directly retrieved from the graph's structure, thus highlighting a potent connection between the algebraic structure and its graph representation. Additionally, we provide a necessary condition for the simplicity of this class of triple systems, through the path symmetry of their associated graphs.