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

Abstract

In this article, we introduce a relation including ideals of an evolution algebra A and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and ideals having the absorption property of an evolution algebra in terms of its associated graph. In particular, the maximal ideals can be determined through maximal hereditary subsets of vertices except for those containing \(A^2\). We also define a couple of order-preserving maps, one from the sets of ideals of an evolution algebra to that of hereditary subsets of the corresponding graph, and the other in the reverse direction. Conveniently restricted to the set of absorption ideals and to the set of hereditary saturated subsets, this is a monotone Galois connection. According to the graph, we characterize arbitrary dimensional finitely-generated (as algebras) evolution algebras under certain restrictions of its graph. Furthermore, the simplicity of finitely-generated perfect evolution algebras is described on the basis of the simplicity of the graph.