Finitely Generated Nil but Not Nilpotent Evolution Algebra

TL;DR

This paper explores the construction of finitely generated evolution algebras over rings with nilpotent elements that are nil but not nilpotent, addressing modeling needs in population dynamics.

Contribution

It develops computational criteria for nilpotency in evolution algebras over rings and demonstrates how to construct finitely generated examples that are nil but not nilpotent.

Findings

Criteria for nilpotency over rings with nilpotent elements

Construction methods for finitely generated nil but not nilpotent evolution algebras

Application to modeling population dynamics with extinction and gamete introduction

Abstract

To use evolution algebras to model population dynamics that both allow extinction and introduction of certain gametes in finite generations, nilpotency must be built into the algebraic structures of these algebras with the entire algebras not to be nilpotent if the populations are assumed to evolve for a long period of time. To adequately address this need, evolution algebras over rings with nilpotent elements must be considered instead of evolution algebras over fields. This paper develops some criteria, which are computational in nature, about the nilpotency of these algebras, and shows how to construct finitely generated evolution algebras which are nil but not nilpotent.