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This framework is used to show that, in nondegenerate evolution algebras, any family of natural and orthogonal vectors can be extended to a natural basis. The work also characterizes orthogonal linearly independent families that admit such an extension. Additional results establish ideal structure in perfect evolution algebras and localize third-order nilpotent elements via principal minors of the structure matrix, and relate these properties to the adjoint evolution algebra.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/natural-families-in-evolution-algebras/157012/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/natural-families-in-evolution-algebras/157012.png","ImageObject",300,407,{"name":92,"@type":93},"Evangeline","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-17","2026-08-29",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the evolution rank and why is it introduced?","Question",{"text":113,"@type":114},"The paper introduces evolution rank to analyze how evolution algebras can be decomposed into structured subspaces. It is used to control extending subspaces of rank one and to support later extension and basis results.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the paper characterize families that can be extended to a natural basis?",{"text":118,"@type":114},"It characterizes orthogonal linearly independent families of vectors that satisfy a condition ensuring the extension property. In nondegenerate evolution algebras, natural and orthogonal families can be completed to a natural basis.",{"name":120,"@type":111,"acceptedAnswer":121},"How are nilpotent elements of order three detected in perfect evolution algebras?",{"text":122,"@type":114},"In a perfect evolution algebra where every element is a square, third-order nilpotent elements can be localized by checking the structure matrix: a vanishing principal minor yields the relevant localization. Conversely, for a perfect evolution algebra over an arbitrary field, existence of such a nilpotent element forces a vanishing principal minor in the structure matrix.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},157012,1787971162,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},13056703019662,"https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188","Publ. Mat. 66 (2022), 159{181 DOI: 10.5565/PUBLMAT6612206  \nNATURAL FAMILIES IN EVOLUTION ALGEBRAS  \nNadia Boudi, Yolanda Cabrera Casado, and Mercedes Siles Molina  \nAbstract: In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator and extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in nondegenerate evolution algebras any family of natural and orthogonal vectors can be extended to a natural basis. The central results are the characterization of those families of orthogonal linearly independent vectors which can be extended to a natural basis.  \nWe also consider ideals in perfect evolution algebras and prove that they coincide with basic ideals.  \nNilpotent elements of order three can be localized (in a perfect evolution algebra over a 􀀌eld in which every element is a square) by merely looking at the structure matrix: any vanishing principal minor provides one. Conversely, if a perfect evolution algebra over an arbitrary 􀀌eld has a nilpotent element of order three, then its structure matrix has a vanishing principal minor.  \nWe 􀀌nish by considering the adjoint evolution algebra and relating its properties to the corresponding ones in the initial evolution algebra.  \n2010 Mathematics Subject Classi􀀌cation: 17D92, 17A60, 17A36, 15A99 .  \nKey words: evolution algebra, natural family, decomposition, property (2LI), natural basis, evolution ideal, evolution subalgebra, extension property.  \n1. Introduction and preliminaries  \nThe study of evolution algebras has its starting point in 2006 in the paper [11], followed by the monograph [9] . A systematic study of (arbitrary-dimensional) evolution algebras was initiated in [3], where a characterization of simple evolution algebras was accomplished. The problem of the classi􀀌cation of 􀀌nite-dimensional evolution algebras is a complex one. As far as we know, there have been classi􀀌ed (up to isomorphisms) the 2-and 3-dimensional evolution algebras and perfect nonsimple 4-dimensional algebras (with some mild restrictions); see [1, 2 , 4 , 6] . A useful tool for classifying 4-dimensional evolution algebras has been the notion, introduced in [2], of basic ideal: an ideal having a natural basis which can be extended to a natural basis of the whole algebra. An  \n160 N. Boudi, Y. Cabrera Casado, M. Siles Molina  \ninteresting matter related to basic ideals is when a vector in an evolution algebra is a natural vector (i.e., it is part of a natural basis of the evolution algebra) . A natural follow-up is when a family of orthogonal and independent vectors can be completed to a natural basis of the algebra. These are the questions which initially motivated this paper.  \nWe have divided this article into 􀀌ve sections. The 􀀌rst one contains an introduction and preliminaries. Speci􀀌cally, we see that the linear span of a subset does not necessarily coincide with a subalgebra or ideal. Homomorphisms are recalled and an example showing that in general automorphisms of evolution algebras do not coincide with automorphismsof vector spaces. In fact, linear isomorphisms do not necessarily preserve natural bases. Evolution homomorphisms, de􀀌ned in [9], cannot coincide with algebra homomorphisms: an example of an algebra homomorphism such that the image does not have the extension property is given.  \nIn Section 2 we consider natural families in evolution algebras. In the study of the classi􀀌cation of evolution algebras, an important aspect is when all the natural bases are essentially the same, i.e., all of them can be obtained from one by simply changing the order of the elements or multiplying each element by a scalar. We say in this paper that an evolution algebra having this property \\has a unique natural basis\". It was proved in [7, Theorem 4.4] that when an evolution algebra A is perfect (meaning A2 = A), then it has a unique natural basis. Corollary 2.7 􀀌nds a condition for an evol","cbCaihHqf42P7FIs","https://ap.wps.com/l/cbCaihHqf42P7FIs","pdf",340617,23,"English","# Introduction and preliminaries\n## Natural families and property (2LI)\n## Decomposition via extending evolution subspaces\n## Extension property and orthogonal natural vectors\n# Main results (e.g., Theorem 3.6)","[{\"question\":\"What is the evolution rank and why is it introduced?\",\"answer\":\"The paper introduces evolution rank to analyze how evolution algebras can be decomposed into structured subspaces. It is used to control extending subspaces of rank one and to support later extension and basis results.\"},{\"question\":\"How does the paper characterize families that can be extended to a natural basis?\",\"answer\":\"It characterizes orthogonal linearly independent families of vectors that satisfy a condition ensuring the extension property. In nondegenerate evolution algebras, natural and orthogonal families can be completed to a natural basis.\"},{\"question\":\"How are nilpotent elements of order three detected in perfect evolution algebras?\",\"answer\":\"In a perfect evolution algebra where every element is a square, third-order nilpotent elements can be localized by checking the structure matrix: a vanishing principal minor yields the relevant localization. Conversely, for a perfect evolution algebra over an arbitrary field, existence of such a nilpotent element forces a vanishing principal minor in the structure matrix.\"}]","NATURAL FAMILIES IN EVOLUTION ALGEBRAS | PDF",58]