The concept of hierarchy of algebras and graphs (Q2143837)

\textit{J. P. Tian}'s monograph [Evolution algebras and their applications. Berlin: Springer (2008; Zbl 1136.17001)] defines a hierarchical structure for evolution algebras via a sequence of decompositions. In the paper under review, the authors generalizes the concept of hierarchy to any algebra \(\mathfrak g\) with a fixed basis \(\mathcal{B}= \{e_1,e_2,\ldots, e_n\}\). This generalization involves the concepts of persistent and transient subsets of \(\mathcal{B}\). A subset \(S\) of \(\mathcal{B}\) is persistent if the vector subspace of \(\mathfrak g\) spanned by \(S\) is a subalgebra of \(A\). Otherwise, the set \(S\) is said to be transient. The authors define the persistent graph \(G\) of \(\mathfrak{g}\). Its vertices are the elements of \(\mathcal{B}\). Let \(P(\mathcal{B})\) the set of all subsets \(S\) of \(\mathcal{B}\) such that the vector subspace spanned by \(S\) is a subalgebra of \(\mathfrak{g}\). A pair \((e_i,e_j)\) is an edge of the persistent graph \(G\) of \(\mathfrak{g}\) if \(e_i \in \cap \{ S\in P(\mathcal{B}) \, | \, e_j\in S\}\). The paper also translates the concept of hierarchy in algebras to their associated persistent graphs.