Left symmetric algebras from DNA insertion
In this paper, we construct some left-symmetric algebras arising from the operation of DNA insertion. We define a new operation of insertion by modifying the simplified insertion as follows: $x\Rightarrow y:=f(\mid x\mid, \mid y \mid)\sum\limits_{i=o}^{q} y_{1}y_{2}\cdots y_{i} x y_{i+1}\cdots y_{q}$ where $x = x_{1}x_{2}\cdots x_{p}$ and $y = y_{1}y_{2}\cdots y_{q}$. Then it is proved that $(x, y, z)_2= (y, x, z)_2$ if and only if $f$ satisfies the following conditions: $$f(m, n) f(m+n, p)=f(n, p) f(m, n+p)= f(m, p) f(n, m+p)$$ where $m, n, p\in \mathbb{N}$. The paper also give an important example: $f(m, n)=\exp\{g(m, n)\}$, where $g(m, n)=k\cdot mn,$ $k$ is a fixed positive number. We also give simple case to make clear that the intermediate version of the insertion operation which is the open problem numbered [5] and the operator of synchronized insertion given in the \cite{Bremner} by Bremner do not satisfy the left-symmetric identities.