{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T23:22:15Z","timestamp":1783984935333,"version":"3.55.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":4884,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2000,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.<\/jats:p>","DOI":"10.2307\/421058","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:14:28Z","timestamp":1146986068000},"page":"311-330","source":"Crossref","is-referenced-by-count":27,"title":["Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings"],"prefix":"10.1017","volume":"6","author":[{"given":"Jan","family":"Kraj\u00ed\u010dek","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Scanlon","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600006405_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/PL00011162"},{"key":"S1079898600006405_ref018","first-page":"514","volume-title":"Proceedings of the 30th IEEE Symposium on the Foundations of Computer Science","author":"Toda","year":"1989"},{"key":"S1079898600006405_ref010","first-page":"193","volume-title":"Logic, methodology and philosophy of science","author":"Macintyre","year":"1982"},{"key":"S1079898600006405_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/s002220050284"},{"key":"S1079898600006405_ref013","first-page":"1133","volume-title":"Handbook of Mathematical Logic","author":"Paris","year":"1978"},{"key":"S1079898600006405_ref004","volume-title":"Non standard finite fields over I \u03940 + \u03a91","author":"D'Aquino","year":"1999"},{"key":"S1079898600006405_ref002","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-73.1.1"},{"key":"S1079898600006405_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0084232"},{"key":"S1079898600006405_ref007","unstructured":"Haskell D. , K0(\u211ap) = 0, unpublished notes, 1999."},{"key":"S1079898600006405_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1970573"},{"key":"S1079898600006405_ref011","volume-title":"Algebraic Curves Over Finite Fields","author":"Moreno","year":"1993"},{"key":"S1079898600006405_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240070103"},{"key":"S1079898600006405_ref012","doi-asserted-by":"publisher","DOI":"10.2307\/2269958"},{"key":"S1079898600006405_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BF01200117"},{"key":"S1079898600006405_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771574"},{"key":"S1079898600006405_ref019","volume-title":"London mathematical society lecture note series","volume":"248","author":"van den Dries","year":"1998"},{"key":"S1079898600006405_ref008","volume-title":"Proceedings of the London Mathematical Society","author":"Kraj\u00ed\u010dek","year":"1997"},{"key":"S1079898600006405_ref003","volume-title":"Encyclopedia of Mathematics and Its Applications","volume":"60","author":"Kraj\u00ed\u010dek","year":"1995"},{"key":"S1079898600006405_ref014","doi-asserted-by":"publisher","DOI":"10.1017\/S0022481200028061"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600006405","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T20:44:41Z","timestamp":1557348281000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600006405\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,9]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2000,9]]}},"alternative-id":["S1079898600006405"],"URL":"https:\/\/doi.org\/10.2307\/421058","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,9]]}}}