On the subalgebra lattice of solvable evolution algebras

TL;DR

This paper investigates the subalgebra lattice structure of solvable evolution algebras, focusing on properties like distributivity and modularity, and characterizes these properties in nilpotent and supersolvable cases.

Contribution

It provides new characterizations of distributivity and modularity in the subalgebra lattice of solvable evolution algebras, especially in nilpotent and supersolvable cases.

Findings

Distributivity is characterized for nilpotent evolution algebras.

A necessary condition for modularity in nilpotent cases is established.

Maximum index of solvability correlates with better algebraic properties.

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.