Chains in evolution algebras (Q2029869)

Evolution algebras are non-associative algebras introduced in order to model self fertilization [\textit{I. M. H. Etherington}, Proc. R. Soc. Edinb., Sect. B, Biol. 61, 24--42 (1941; Zbl 0063.01290), p. 34] and non-mendelian genetics [\textit{J. P. Tian}, Evolution algebras and their applications. Berlin: Springer (2008; Zbl 1136.17001)]. An evolution algebra is an algebra over a field \(\mathbb{K}\) which has a basis \(\mathscr{B}=\left\{ e_{i}\right\} _{i\in\Lambda}\) such that \(e_{i}e_{j}=0\) for every \(i,j\in\Lambda\).\medskip{} In this paper the authors give two constructions of an evolution algebra \(\overline{A}\) from a given finite evolution algebra \(A\). The first construction, called of type 1, is obtained by adjunction to \textbf{\(A\)} of an annihilator element. In the second construction, called of type 2, the evolution algebra \(A\) is an ideal of \(A\) such that \(\overline{A}\setminus A\) is one-dimensional. They apply these two constructions to evolution algebras to get 3-dimensional evolution algebras from 2-dimensional evolution algebras, and they use two parameters: the \emph{annihilator stabilizing index} \(\text{asi}(A)\) and the \emph{socle stabilizing index }\(\text{ssi}(A)\)\emph{,} to classify these algebras. For an evolution algebra \(A\), the parameter \(\text{asi}(A)\) is defined by \[ \text{asi}\left(A\right)=\min\left\{ q:\text{ann}^{\left(q\right)}\left(A\right)=\text{ann}^{\left(q+1\right)}\left(A\right)\right\} \] where the increasing sequence \(\left(\text{ann}^{\left(q\right)}\left(A\right)\right)_{q\geq0}\) of annihilators of \(A\), is defined by \[ \text{ann}^{\left(0\right)}\left(A\right)=\left\{ 0\right\} \text{ and }\text{ann}^{\left(q\right)}\left(A\right)/\text{ann}^{\left(q-1\right)}\left(A\right)=\text{ann}\left(A/\text{ann}^{\left(q-1\right)}\left(A\right)\right). \] And the parameter\emph{ }\(\text{ssi}(A)\) is defined by \[ \text{ssi}\left(A\right)=\min\left\{ n:\text{Soc}^{\left(n\right)}\left(A\right)=\text{Soc}^{\left(n+k\right)}\left(A\right),k>0\right\} \] where the increasing chain of socles \(\left(\text{Soc}^{\left(n\right)}\left(A\right)\right)_{n\geq1}\) of \(A\),\( \)is \[ \text{Soc}^{\left(0\right)}\left(A\right)=\text{Soc}\left(A\right)\text{ and }\text{Soc}\left(A/\text{Soc}^{\left(n\right)}\left(A\right)\right)=\text{Soc}^{\left(n+1\right)}\left(A\right)/\text{Soc}^{\left(n\right)}\left(A\right). \]