{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T20:00:10Z","timestamp":1762459210764},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,6,26]],"date-time":"2014-06-26T00:00:00Z","timestamp":1403740800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2014,8]]},"abstract":"Finitary$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories are$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories with finite objects only, where$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-subobjects. In this paper, we show that in finitary$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories have a unique$\\mathcal{E}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>factorisation and initial pushouts, and the existence of an$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-initial object implies we also have finite coproducts and a unique$\\mathcal{E}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>\u2032-$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>pair factorisation. Moreover, we can show that the finitary restriction of each$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive category is a finitary$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive category, and finitarity is preserved under functor and comma category constructions based on$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories. This means that all the classical results are also valid for corresponding finitary$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-$\\mathcal{M}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-adhesive categories.<\/jats:p>","DOI":"10.1017\/s0960129512000321","type":"journal-article","created":{"date-parts":[[2014,6,26]],"date-time":"2014-06-26T14:22:13Z","timestamp":1403792533000},"source":"Crossref","is-referenced-by-count":9,"title":["Finitary -adhesive categories"],"prefix":"10.1017","volume":"24","author":[{"given":"KARSTEN","family":"GABRIEL","sequence":"first","affiliation":[]},{"given":"BENJAMIN","family":"BRAATZ","sequence":"additional","affiliation":[]},{"given":"HARTMUT","family":"EHRIG","sequence":"additional","affiliation":[]},{"given":"ULRIKE","family":"GOLAS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,6,26]]},"reference":[{"key":"S0960129512000321_ref25","doi-asserted-by":"publisher","DOI":"10.1142\/3303"},{"key":"S0960129512000321_ref22","doi-asserted-by":"crossref","unstructured":"MacLane S. (1971) Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag.","DOI":"10.1007\/978-1-4612-9839-7"},{"key":"S0960129512000321_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-15928-2_15"},{"key":"S0960129512000321_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24727-2_20"},{"key":"S0960129512000321_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129508007202"},{"key":"S0960129512000321_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0025714"},{"key":"S0960129512000321_ref17","unstructured":"Klyne G. and Carroll J. J. (2004) Resource Description Framework (RDF): Concepts and Abstract Syntax, World Wide Web Consortium (W3C). Available at http:\/\/www.w3.org\/TR\/2004\/REC-rdf-concepts-20040210\/."},{"key":"S0960129512000321_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-15928-2_17"},{"key":"S0960129512000321_ref1","volume-title":"Abstract and Concrete Categories","author":"Ad\u00e1mek","year":"1990"},{"key":"S0960129512000321_ref15","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008734426504"},{"key":"S0960129512000321_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(00)00382-0"},{"key":"S0960129512000321_ref23","unstructured":"Modica T. et al. (2010) Low- and High-Level Petri Nets with Individual Tokens. Technical Report 2009\/13, Technische Universit\u00e4t Berlin. Available at http:\/\/www.eecs.tu-berlin.de\/menue\/forschung\/forschungsberichte\/2009."},{"key":"S0960129512000321_ref19","doi-asserted-by":"publisher","DOI":"10.1051\/ita:2005028"},{"key":"S0960129512000321_ref5","unstructured":"Braatz B. and Brandt C. (2008) Graph transformations for the Resource Description Framework. In: Ermel C. , Heckel R. and de Lara J. (eds.) Proceedings GT-VMT 2008. Electronic Communications of the EASST 10."},{"key":"S0960129512000321_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/11841883_4"},{"key":"S0960129512000321_ref3","unstructured":"Baldan P. , Bonchi F. , Heindel T. and K\u00f6nig B. (2008) Irreducible Objects and Lattice Homomorphisms in Adhesive Categories. In: Pfalzgraf J. (ed.) Proceedings of ACCAT '08 (Workshop on Applied and Computational Category Theory)."},{"key":"S0960129512000321_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-15928-2_16"},{"key":"S0960129512000321_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2010.09.007"},{"key":"S0960129512000321_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2010.09.006"},{"key":"S0960129512000321_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-53832-1_52"},{"key":"S0960129512000321_ref12","first-page":"111","article-title":"Categorical Frameworks for Graph Transformations and HLR Systems based on the DPO Approach.","volume":"102","author":"Ehrig","year":"2010","journal-title":"Bulletin of the EATCS"},{"key":"S0960129512000321_ref24","doi-asserted-by":"publisher","DOI":"10.1007\/s10485-007-9106-3"},{"key":"S0960129512000321_ref11","volume-title":"Fundamentals of Algebraic Graph Transformation","author":"Ehrig","year":"2006"},{"key":"S0960129512000321_ref13","first-page":"1","article-title":"Adhesive High-Level Replacement Systems: A New Categorical Framework for Graph Transformation.","volume":"74","author":"Ehrig","year":"2006","journal-title":"Fundamenta Informaticae"},{"key":"S0960129512000321_ref4","unstructured":"Braatz B. (2009) Formal Modelling and Application of Graph Transformations in the Resource Description Framework, Dissertation, Technische Universit\u00e4t Berlin."}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129512000321","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,11]],"date-time":"2019-08-11T23:22:26Z","timestamp":1565565746000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129512000321\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,26]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2014,8]]}},"alternative-id":["S0960129512000321"],"URL":"https:\/\/doi.org\/10.1017\/s0960129512000321","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,26]]},"article-number":"240403"}}