Let $(𝔄,{ϵ}_u)$ and $(𝔅,{ϵ}_b)$ be two pointed sets. Given a family of three maps $ℱ=\{f_1:𝔄\to 𝔄;f_2:𝔄\times 𝔄\to 𝔄;f_3:𝔄\times 𝔄\to 𝔅\}$, this family provides an adequate decomposition of $𝔄\setminus \left\{{ϵ}_u\right\}$ 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.

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra $s{l}_2$, the Verma module over a Kac-Moody algebra, the Verma module...