Connecting ideals in evolution algebras with hereditary subsets of its associated graph

Abstract

In this article, we introduce a relation including ideals of an evo-lution algebra and hereditary subsets of vertices of its associated graph andestablish some properties among them. This relation allows us to determinemaximal ideals and ideals having the absorption property of an evolution alge-bra in terms of its associated graph. We also define a couple of order-preservingmaps, one from the sets of ideals of an evolution algebra to that of hereditarysubsets of the corresponding graph, and the other in the reverse direction.Conveniently restricted to the set of absorption ideals and to the set of heredi-tary saturated subsets, this is a monotone Galois connection. According to thegraph, we characterize arbitrary dimensional finitely-generated (as algebras)evolution algebras under certain restrictions of its graph. Furthermore, thesimplicity of finitely-generated perfect evolution algebras is described on thebasis of the simplicity of the graph.