Order Structures around 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.