[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-153168-105":59,"doc-detail-153168-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","universal-algebra-tutorial-lecture-ii-ross-willard","Universal Algebra tutorial - Lecture II - Ross Willard","","Lecture II by Ross Willard develops a framework linking universal algebra and finite relational structures through the correspondence between generated varieties, posets of relational structures, and Mal'cev classes. It reviews the CSP dichotomy conjecture: for every finite relational structure, CSP is either in P or NP-complete, and presents key reduction steps via cores and endo-rigid expansions with constants. A central goal is locating the dichotomy dividing line within the endo-rigid class.",{"@graph":69,"@context":122},[70,84,105],{"@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/universal-algebra-tutorial-lecture-ii-ross-willard/153168/",{"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/universal-algebra-tutorial-lecture-ii-ross-willard/153168.png","ImageObject",300,407,{"name":92,"@type":93},"วิน","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-21","2026-08-27",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How does Lecture II connect universal algebra with finite relational structures?","Question",{"text":112,"@type":113},"It relates generated varieties and a poset of finite relational structures via an anti-isomorphism between the structures defined from varieties and the poset of relational structures. Mal'cev classes in the variety lattice induce filters and ideals in the relational poset.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the constraint satisfaction problem CSP(G) in this lecture?",{"text":117,"@type":113},"For a fixed finite relational structure G with finite language L, CSP(G) asks whether, given any finite L-structure I, there exists a homomorphism from I to G (equivalently, a G-homomorphism or G-coloring).",{"name":119,"@type":110,"acceptedAnswer":120},"What reductions are used as initial steps toward proving the CSP dichotomy conjecture?",{"text":121,"@type":113},"The lecture reduces testing dichotomy to cores, using that CSP(H)=CSP(core(H)). It then reduces further to the endo-rigid case by expanding structures with constants so that only the identity endomorphism remains, preserving the CSP difficulty.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},153168,1787871730,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},2336475104736,"https://ap-avatar.wpscdn.com/avatar/22000c4c5e0e5b17e70?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786591360781797222","Tutorial on Universal Algebra, Mal'cev Conditions, and Finite Relational Structures: Lecture II  \nRoss Willard  \nUniversity of Waterloo, Canada  \nBLAST 2010  \nBoulder, June 2010  \n Ross Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 1 / 22   \nRecap  \n[K3]  \n[1]  \n(REL 􀀌n ; 􀀔 pp)  \n􀀌n. rel. structures􀀃  \n[var(1)]  \n[var(2)]  \n(ALG 􀀌n ; 􀀔)  \n􀀌n. gen'd varieties  \n􀀒  \n[Triv]  \n[Set](L; 􀀔) varieties  \n Interpretation relation on varieties gives us L.  \n Sitting inside L is the ^-closed sub-poset ALG 􀀌n .  \n Pp-de􀀌nability relation on 􀀌nite structures gives us REL 􀀌n .  \n REL 􀀌n and ALG 􀀌n are anti-isomorphic via [H] 7! [var(PolAlg(H))] .  \n Mal'cev classes in L induce 􀀌lters on ALG 􀀌n and ideals on REL 􀀌n .  \n Ross Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 2 / 22   \nOne more set to de􀀌ne:  \nREL!􀀌n :=  \n= f [H] 2 REL 􀀌n : language of H is 􀀌niteg  \n􀀒  \nConvention: henceforth, all mentioned relational structures under consideration have 􀀌nite languages.  \n Ross Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 3 / 22   \nTheorem (Hell, Ne􀀔set􀀔ril, 1990)  \nSuppose G is a 􀀌nite undirected graph (without loops) .  If G is bipartite, then CSP (G) is in P.  \nOtherwise, CSP (G) is NP-complete.  \nWhat the heck is \\CSP(G)\"?  \nDe􀀌nition  \nGiven a 􀀌nite relational structure G with 􀀌nite language L, the constraint satisfaction problem with 􀀌xed template G, written CSP (G), is the following decision problem:  \nInput: an arbitrary 􀀌nite L-structure I.  \nQuestion: does there exist a homomorphism I ! G?  \nAlso called the G-homomorphism (or G-coloring) problem.  \n Ross Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 4 / 22   \nSome context  \n [Classical]: CSP (K2 ) 􀀑 checking bipartiteness, which is in P.  \nCSP (Kn) 􀀑 graph n-colorability, which is NP-complete  \nfor n 􀀕 3 (Karp) .  \n Key fact [Essentially due to Bulatov & Jeavons, unpubl.]:  \nIf G ; H are 􀀌nite structures in 􀀌nite languages and G 􀀞 pp H ,  \nthen CSP (G) is no harder than CSP (H) .  \nConsequences:  \n If CSP (G) is in P [resp. NP-complete], then same is true 8 H 2 [G] .  \n f [G] : CSP (G) is in Pg is a down-set in REL!􀀌n .  \n f [G] : CSP (G) is NP-completeg is an up-set in REL!􀀌n .  \n In fact:  \n f [G] : CSP (G) is in Pg is an ideal in (REL!􀀌n ; _) . (Not hard)  \n Ross Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 5 / 22   \nPictorially:  \n[K3]  \n[1]  \nCSP(-) is NP-complete  \nCSP(-) is in P  \nREL!􀀌n :  \nHell-Ne􀀔set􀀔ril theorem: there is dichotomy for undirected graphs.  \nThe CSP dichotomy conjecture (Feder, Vardi (1998)  \nThere is general dichotomy. I.e. , for every 􀀌nite relational structure G in a 􀀌nite language, CSP (G) is either in P or is NP-complete.   \nRoss Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 6 / 22   \nInitial steps towards a proof of the Dichotomy Conjecture 1. Reduction to cores.  \nLet G ; H be 􀀌nite relational structures in the same language.  \n G is core if all of its endomorphisms are automorphisms. G is a core of H if G is core and is a retract of H.  \nFacts:  \n Every 􀀌nite relational structure H has a core, which is unique up to isomorphism; call it core (H) .  \n CSP (H) = CSP (core(H)) .  \nHence when testing dichotomy, we need only consider cores.  \n Ross Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 7 / 22   \n2. Reduction to the endo-rigid case.  \nDe􀀌nition  \nLet H = (H ; frelationsg) be a relational structure.  H is endo-rigid if its only endomorphism is idH .  \n Hc := (H ; frelationsg [ f fag : a 2 Hg) . (\\H with constants\")  \nFacts:  \n Endo-rigid ) core.  \n Hc is endo-rigid.  \nProposition (Bulatov, Jeavons, Krokhin, 2005)  \nIf H is core, then CSP (H) and CSP (Hc ) have the same di􀀎culty.  \nHence when testing general dichotomy, we need only consider structures with constants (equivalently, endo-rigid structures) .  \nRoss Willard (Waterloo)  Universal Algebra tutorial  BLAST 2010 8 / 22   \nThe reductions in pictures:  \nREL!􀀌n :  \nendo-rigid  \n[Hc ]  \n[H] where H = core(G)  \n","cbCaitXA0fwNmcsh","https://ap.wps.com/l/cbCaitXA0fwNmcsh","pdf",309512,22,"English","# Recap\n## Varieties and relational structures\n# CSP dichotomy and graph cases\n## Hell–Nešetřil–Neumann theorem\n## CSP definition and homomorphism problem\n# Reduction steps toward dichotomy conjecture\n## Reduction to cores\n## Reduction to endo-rigid structures with constants\n## Locating the dividing line in endo-rigid class","[{\"question\":\"How does Lecture II connect universal algebra with finite relational structures?\",\"answer\":\"It relates generated varieties and a poset of finite relational structures via an anti-isomorphism between the structures defined from varieties and the poset of relational structures. Mal'cev classes in the variety lattice induce filters and ideals in the relational poset.\"},{\"question\":\"What is the constraint satisfaction problem CSP(G) in this lecture?\",\"answer\":\"For a fixed finite relational structure G with finite language L, CSP(G) asks whether, given any finite L-structure I, there exists a homomorphism from I to G (equivalently, a G-homomorphism or G-coloring).\"},{\"question\":\"What reductions are used as initial steps toward proving the CSP dichotomy conjecture?\",\"answer\":\"The lecture reduces testing dichotomy to cores, using that CSP(H)=CSP(core(H)). It then reduces further to the endo-rigid case by expanding structures with constants so that only the identity endomorphism remains, preserving the CSP difficulty.\"}]","Universal Algebra tutorial - Lecture II - Ross Willard | PDF",55]