[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84084-en":3,"doc-seo-84084-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84084,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Chains and Antichains inside Many-One Degrees and Variants","The work studies relationships between many-one degrees and one-one degrees in recursion theory, including how recursive and nonrecursive many-one degrees constrain the existence of chains and infinite antichains. It resolves an open problem attributed to Odifreddi by generalizing Batyrshin’s construction: every nonrecursive and nonirreducible many-one degree contains infinite antichains of one-one degrees. The paper further analyzes finite-one and bounded finite-one degrees, embedding recursive partial orders in appropriate cases while establishing contrasting bounded examples.","arXiv :2607 .06218v1 [math .LO] 7 Jul 2026  \nChains and Antichains inside Many-One Degrees and Variants  \nLinus Richter 1 , Frank Stephan2,3 and Xiaoyan Zhang2  \n1 Flinders University, College of Science and Engineering, Level 3, Tonsley Building 1, 1284 South Road, Tonsley SA 5042, Australia; [linus.richter@flinders.edu.au](linus.richter@flinders.edu.au. Linus Richter)[. Linus Richter](linus.richter@flinders.edu.au. Linus Richter) did most of his work on this paper while he worked for the Department of Mathematics, National University of Singapore, during his previous employment.  \n2 Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Block S17, Singapore 119076, Republic of Singapore, [fstephan@nus.edu.sg](fstephan@nus.edu.sg) and [zhangxy@u.nus.edu](zhangxy@u.nus.edu).  \n3 School of Computing, National University of Singapore, 13 Computing Drive, Block COM1, Singapore 117417, Republic of Singapore.  \n~~ Abstract ~~  \nThe relations between many-one degrees and one-one degrees have been studied since the beginning of recursion theory; early results from the 1960s include that many-one degrees always have a largest one-one degree and either that one-one degree is the only one-one degree inside the many-one degree or every countable linear order is noneffectively embeddable into the structure of one-one degrees inside the given many-one degree. Furthermore, the greatest recursive many-one degree is a special case, as it allows to embed ascending infinite chains but not descending infinite chains, all other many-one degrees fall into the two cases mentioned above. It remained open whether infinite antichains can always be embedded when the many-one degree is nonrecursive and nonirreducible; Odifreddi stated in his survey from the year 1981 and in his book Classical Recursion Theory in the year 1989 this question explicitly as an open problem. Dëgtev had already in 1976 constructed antichains of one-one degrees inside all nonrecursive and nonirreducible recursively enumerable many-one degrees and Batyrshin generalised the result to all nonrecursive and nonirreducible limit-recursive many-one degrees. Recently, Cintioli [5] showed that there is a measure 1 class of sets whose many-one degrees contain infinite antichains of one-one degrees. This class contains all rigid many-one degrees. The present work generalises Batyrshin’s result to all nonrecursive and nonirreducible many-one degrees and solves therefore Odifreddi’s open problem.  \nThe present work also proposes to deepen the study of reducibilities between one-one and many-one in recursion theory in order to get a more complete and detailed picture for the structures inside many-one degrees. It namely proposes to study in more detail than before the finite-one and bounded finite-one degrees. Odifreddi’s Open Problem is solved by showing that every nonrecursive finite-one degree which does not coincide with the greatest one-one degree in a many-one degree contains an infinite antichain of one-one degrees and furthermore allows to embed any recursive partial order effectively into the structure of one-one degrees inside the finite-one degree. This is done by starting with a representative A of the finite-one degree and then constructing an array B0 , B1 , B2 , . . . of sets given by finite-one reductions to A which are also all one-one above A and which form an antichain or embed a given recursive partial order. In contrast to this, there are nonrecursive bounded finite-one degrees consisting of a linearly ordered set of one-one degrees without any incomparable pair of one-one degrees inside it. Furthermore, some initial results about the structure of finite-one degrees inside many-one degrees are obtained.  \nKeywords and phrases Structures inside degrees; one-one degree; finite-one degree; bounded finite-one degree; many-one degree; infinite antichains.  \n2 Chains and Antichains inside Many-One Degrees and Variants  \nFunding. Partial support","cbCaidfopoh5338X","https://ap.wps.com/l/cbCaidfopoh5338X","pdf",485658,4,1,20,"English","en",105,"# Introduction\n## Many-one reducibility and variants\n## Structure of one-one degrees inside many-one degrees\n# Chains and antichains\n## Infinite antichains and Odifreddi’s open problem\n## Finite-one and bounded finite-one degrees","[{\"question\":\"What central relationship does the paper investigate in recursion theory?\",\"answer\":\"It investigates structural connections between many-one degrees and one-one degrees, focusing on how reducibility variants shape the internal ordering of degrees.\"},{\"question\":\"How is Odifreddi’s open problem addressed?\",\"answer\":\"The paper solves it by showing that every nonrecursive finite-one degree under the stated condition contains an infinite antichain of one-one degrees and can realize any recursive partial order effectively.\"},{\"question\":\"What contrasting behavior appears for bounded finite-one degrees?\",\"answer\":\"Some nonrecursive bounded finite-one degrees form a linear order of one-one degrees with no incomparable one-one degrees, unlike the antichain-rich 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central relationship does the paper investigate in recursion theory?","Question",{"text":75,"@type":76},"It investigates structural connections between many-one degrees and one-one degrees, focusing on how reducibility variants shape the internal ordering of degrees.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is Odifreddi’s open problem addressed?",{"text":80,"@type":76},"The paper solves it by showing that every nonrecursive finite-one degree under the stated condition contains an infinite antichain of one-one degrees and can realize any recursive partial order effectively.",{"name":82,"@type":73,"acceptedAnswer":83},"What contrasting behavior appears for bounded finite-one degrees?",{"text":84,"@type":76},"Some nonrecursive bounded finite-one degrees form a linear order of one-one degrees with no incomparable one-one degrees, unlike the antichain-rich 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