On evoluti̇on algebras

In this thesis, it is aimed to compile some algebraic results obtained in the studies on evolution algebras to date and to bring together the basic information about the subject in case of a study in this field. While preparing this thesis, which is purely theoretical, previous studies on evolution algebras were examined. Books and internet resources have been utilized to provide a basis for this subject. By analyzing evolution algebras of arbitrary dimension, the concepts of evolution subalgebras, evolution ideals and non-degeneracy are investigated and examples are presented. Moreover, some algebraic properties of evolution algebras are studied by defining a directed graph whose vertices are labeled with the base elements of the evolution algebra. A relation involving the ideals of an evolution algebra and the hereditary subsets of the vertices of the associated graph is introduced and some properties are established between them. This relation allows to determine the maximal ideals of an evolution algebra and the ideals with the absorption property based on the associated graph. We also consider a pair of order-preserving maps, one defined from the set of ideals of an evolution algebra to the inherited subsets of the associated graph, and one defined in the opposite direction. Under favorable conditions, this map forms a monotone Galois connection. Also, under certain restrictions, finitely generated evolution algebras of arbitrary dimension (as algebras) are characterized with respect to the associated graph. The simplicity of finitely generated perfect evolution algebras is studied in terms of the simplicity of the graph. The study of evolution algebras will help to better understand the connection of this structure with other known algebraic structures. For this reason, comprehending the related works in the literature and analyzing the structural properties of the evolution algebra according to the associated graph will make important contributions to further studies.