On the subalgebra lattice of solvable evolution algebras
Manuel Ladra, Pilar Páez-Guillán, Andrés Pérez-Rodríguez
TL;DR
This work investigates how the subalgebra lattice of solvable evolution algebras encodes distributive and modular properties, with a focus on nilpotent versus non-nilpotent cases. By introducing and leveraging the notions of maximum index of solvability and supersolvability, the authors connect lattice-theoretic properties to concrete algebraic structures, notably via the derived series $\\mathcal{E}^{(k)}$ and explicit matrix forms. They establish an equivalence between distributivity, modularity, and chain-like subalgebra lattices in the nilpotent case, and they clarify how modularity behaves for solvable non-nilpotent algebras, including a precise characterization in terms of block-lower-triangular structure and basic ideals. The results illuminate when the subalgebra lattice attains favorable modular/distributive behavior, highlight the role of quasi-ideals, and provide a framework for understanding lattice properties through the algebraic architecture of evolution algebras, with implications for the interaction between genetic-algebraic structure and lattice theory.
Abstract
The main objective of this paper is to study the relationship between a solvable evolution algebra and its subalgebra lattice, emphasizing two of its main properties: distributivity and modularity. First, we will focus on the nilpotent case, where distributivity is characterised, and a necessary condition for modularity is deduced. Subsequently, we comment on some results for solvable non-nilpotent evolution algebras, finding that the ones with maximum index of solvability have the best properties. Finally, we characterise modularity in this particular case by introducing supersolvable evolution algebras and computing the terms of the derived series.