# Article

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Keywords:
Set; application; graded algebra; involutive algebra; quadratic algebra; weak $H^*$-algebra; structure theory
Summary:
Let $({\mathfrak A} , {\epsilon }_{u})$ and $({\mathfrak B} , {\epsilon }_{b})$ be two pointed sets. Given a family of three maps ${\mathcal F}=\{f_1\colon {{\mathfrak A} } \to {\mathfrak A} ; f_2\colon {{\mathfrak A} } \times {\mathfrak A} \to {\mathfrak A} ; f_3\colon {{\mathfrak A} } \times {\mathfrak A} \to {\mathfrak B} \}$, this family provides an adequate decomposition of ${\mathfrak A} \setminus \{ \epsilon _u \}$ as the orthogonal disjoint union of well-described ${\mathcal F}$-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^*$-algebras.
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