{"id":"https://openalex.org/W7213435268","doi":"https://doi.org/10.48550/arxiv.2609.16632","title":"The prime spectrum of the talented monoid of a higher-rank graph and applications","display_name":"The prime spectrum of the talented monoid of a higher-rank graph and applications","publication_year":2026,"publication_date":"2026-09-15","ids":{"openalex":"https://openalex.org/W7213435268","doi":"https://doi.org/10.48550/arxiv.2609.16632"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.16632","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16632","pdf_url":"https://arxiv.org/pdf/2609.16632","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.16632","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088812206","display_name":"Roozbeh Hazrat","orcid":"https://orcid.org/0000-0001-8494-2529","observed_orcids":["https://orcid.org/0000-0001-8494-2529"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hazrat, Roozbeh","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5048401886","display_name":"Promit Mukherjee","orcid":"https://orcid.org/0009-0003-5283-2714","observed_orcids":["https://orcid.org/0009-0003-5283-2714"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mukherjee, Promit","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.7562999725341797,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.7562999725341797,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.12610000371932983,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.04490000009536743,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/higher-rank-graphs","display_name":"higher-rank graphs","score":0.8339999914169312},{"id":"https://openalex.org/keywords/prime-spectrum","display_name":"prime spectrum","score":0.8399999737739563},{"id":"https://openalex.org/keywords/spectral-spaces","display_name":"spectral spaces","score":0.5419999957084656},{"id":"https://openalex.org/keywords/lattice-isomorphism","display_name":"lattice isomorphism","score":0.8569999933242798}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8126999735832214},{"id":"https://openalex.org/C2779467367","wikidata":"https://www.wikidata.org/wiki/Q863912","display_name":"Prime ideal","level":3,"score":0.6521999835968018},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5885999798774719},{"id":"https://openalex.org/C149685015","wikidata":"https://www.wikidata.org/wiki/Q5500266","display_name":"Free monoid","level":3,"score":0.5715000033378601},{"id":"https://openalex.org/C206901836","wikidata":"https://www.wikidata.org/wiki/Q208237","display_name":"Monoid","level":2,"score":0.5533000230789185},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.5249000191688538},{"id":"https://openalex.org/C2778740170","wikidata":"https://www.wikidata.org/wiki/Q7575210","display_name":"Spectral space","level":2,"score":0.5127999782562256},{"id":"https://openalex.org/C175958551","wikidata":"https://www.wikidata.org/wiki/Q872088","display_name":"Boolean prime ideal theorem","level":3,"score":0.5099999904632568},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.5090000033378601},{"id":"https://openalex.org/C119903167","wikidata":"https://www.wikidata.org/wiki/Q3321297","display_name":"Syntactic monoid","level":4,"score":0.4810999929904938},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.4334999918937683},{"id":"https://openalex.org/C116335209","wikidata":"https://www.wikidata.org/wiki/Q240651","display_name":"Prime element","level":3,"score":0.4198000133037567},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41929998993873596},{"id":"https://openalex.org/C45804927","wikidata":"https://www.wikidata.org/wiki/Q17098947","display_name":"Minimal ideal","level":4,"score":0.40700000524520874},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4052000045776367},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.39649999141693115},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.3871999979019165},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3774999976158142},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3698999881744385},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.33980000019073486},{"id":"https://openalex.org/C101908085","wikidata":"https://www.wikidata.org/wiki/Q5051859","display_name":"Category of rings","level":5,"score":0.3379000127315521},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3212999999523163},{"id":"https://openalex.org/C107836120","wikidata":"https://www.wikidata.org/wiki/Q4809264","display_name":"Associated prime","level":3,"score":0.3188000023365021},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.3158000111579895},{"id":"https://openalex.org/C118211362","wikidata":"https://www.wikidata.org/wiki/Q1203540","display_name":"Maximal ideal","level":3,"score":0.31209999322891235},{"id":"https://openalex.org/C43494777","wikidata":"https://www.wikidata.org/wiki/Q1193403","display_name":"Regular space","level":3,"score":0.3100000023841858},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.26440000534057617},{"id":"https://openalex.org/C162860070","wikidata":"https://www.wikidata.org/wiki/Q142140","display_name":"Finitely-generated abelian group","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C162329640","wikidata":"https://www.wikidata.org/wiki/Q332648","display_name":"Covering space","level":2,"score":0.2590999901294708},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.2587999999523163}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.16632","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16632","pdf_url":"https://arxiv.org/pdf/2609.16632","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.16632","doi":"https://doi.org/10.48550/arxiv.2609.16632","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.16632","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.16632","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16632","pdf_url":"https://arxiv.org/pdf/2609.16632","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320330043","display_name":"Western Sydney University","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213435268.pdf","grobid_xml":"https://content.openalex.org/works/W7213435268.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4404405869","https://openalex.org/W7172016185","https://openalex.org/W7203497084"],"related_works":["https://openalex.org/W4404405869","https://openalex.org/W7203497084","https://openalex.org/W2255272153","https://openalex.org/W7161644232","https://openalex.org/W3186666506","https://openalex.org/W1790235503","https://openalex.org/W4411329689","https://openalex.org/W7213586823","https://openalex.org/W2100943313","https://openalex.org/W3012876883"],"abstract_inverted_index":{"In":[0],"this":[1,60,167],"paper,":[2],"we":[3,86,217],"further":[4],"investigate":[5],"the":[6,9,37,65,70,73,92,96,128,136,144,152,159,192,208,223,231,235,243],"role":[7],"of":[8,29,42,52,67,95,99,110,122,130,135,146,154,158,166,174,178,214,225,230,237,242],"graded":[10,132,240],"Grothendieck":[11],"group":[12],"$K_0^{\\gr}$":[13],"and":[14,57,76,105,151,189,234],"its":[15],"positive":[16],"cone":[17],"(the":[18,40],"talented":[19,97,160,232],"monoid)":[20],"as":[21],"an":[22,212],"effective":[23],"tool":[24],"for":[25],"distinguishing":[26],"structural":[27],"types":[28],"algebras":[30],"associated":[31],"to":[32,89,142],"higher-rank":[33],"$k$-graphs.":[34],"We":[35],"study":[36],"prime":[38,44,93,133,148,156,209],"spectrum":[39,94],"space":[41,61,129,145,153],"all":[43,131,147,155,226,238],"$\u0393$-order":[45,78,176,183],"ideals":[46,134,150,177,229,241],"equipped":[47],"with":[48,106],"a":[49,53,84,100,107,170,179,182,219,247],"Zariski-like":[50],"topology)":[51],"general":[54],"commutative":[55],"$\u0393$-monoid":[56],"establish":[58,218],"that":[59,91,127],"is":[62,80,112,140,186,201],"spectral":[63],"in":[64,207],"sense":[66],"Hochster,":[68],"provided":[69],"monoid":[71,98,161,233],"has":[72],"refinement":[74,180],"property":[75],"every":[77],"ideal":[79,184],"finitely":[81],"generated.":[82],"As":[83,211],"result":[85,165],"are":[87],"able":[88],"show":[90],"row-finite":[101,116],"$k$-graph":[102,117],"without":[103,119],"sources":[104],"finite":[108],"set":[109,197],"vertices":[111],"spectral.":[113],"For":[114],"any":[115],"$\u039b$":[118],"sources,":[120],"one":[121],"our":[123],"main":[124,164],"results":[125],"says":[126],"Kumjian--Pask":[137,244],"algebra":[138,245],"$\\KP(\u039b)$":[139],"homeomorphic":[141],"both":[143],"$\\mathbb{Z}^k$-order":[149,228],"$\\mathbb{Z}^k$-filters":[157],"$T_\u039b$.":[162],"Another":[163],"paper":[168],"provides":[169],"complete":[171],"topological":[172],"description":[173],"regular":[175,187,202,205,227,239],"$\u0393$-monoid:":[181],"$J$":[185],"if":[188,191],"only":[190],"corresponding":[193],"closed":[194,203],"(resp.,":[195,199,204],"open)":[196,206],"$V(J)$":[198],"$D(J)$)":[200],"spectrum.":[210],"application":[213],"these":[215],"results,":[216],"lattice":[220,224,236],"isomorphism":[221],"between":[222],"via":[246],"spectrum-theoretic":[248],"approach.":[249]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-17T00:00:00"}
