Let (A, u) and (B, b) be two pointed sets. Given a family of three maps F = {f1 : A → A; f2 : A × A → A; f3 : A × A → B}, this family provides an adequate decomposition of A \ {u} as the orthogonal disjoint union of well-described F-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak H∗ -algebras.