On an analogue of Frobenius formalism for 3-algebras and pentagon equations solutions arising from projectors

Ruth J.Lawrence introduced a notion of a 3-algebra to construct invariants of 3-manifolds based on their triangulations in her paper "Algebras and triangular relations". Her primary definition is suitable for certain triangulations only although a hint to handle arbitrary ones has been proposed. Here I introduce an analogue of Frobenius compatibility for one class of 3-algebras hence I obtain a way to construct a full 3-algebra. Additionally, I provide with examples of a 3-algebra and invariants for lens spaces. Moreover, it leads to a new family of pentagon equations solutions: arising from projectors.