Order Structures around Evolution Algebras
Alejandro González Nevado
TL;DR
This paper addresses how to structure and compare the substructures of evolution algebras through order-theoretic lenses. It develops two complementary approaches: one via distinguished substructures (evolution subalgebras/ideals) and another via distinguished elements (idempotents, natural elements) to study the socle. It introduces the brset framework and evlattices to generalize order structures, and proves that evolution ideals yield evlattices and that lattices definable inside evlattices can be extracted, including maximal lattices via Zorn's lemma. It also analyzes idempotent generation of minimal and evolution ideals, the composition of the socle (natural vs non-natural parts), and provides conditions under which idempotents exist and generate minimal ideals, contributing a foundation for further decompositions and classifications of evolution algebras.
Abstract
We consider evolution algebras and their related substructures: evolution ideals and evolution subalgebras. After exposing some of the concepts related to them in the literature, we explore the order structures that arise in the sets of substructures of an evolution algebra. This leads to the introduction of the socle of an evolution algebra and to the study of its connection with some distinguished elements within the algebra (such as idempotents and natural elements). Finally, we examine the order structures that emerge among these elements when we consider the substructures that they generate inside the algebra. Thus, we develop two order-theoretic approaches (one using distinguished substructures and other using distinguished elements). These two strategies could be used in many ways. Particularly, we direct them towards the study of the socle of an evolution algebra.