{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:27:35Z","timestamp":1760243255881,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2014,5,30]],"date-time":"2014-05-30T00:00:00Z","timestamp":1401408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Allegories are enriched categories generalizing a category of sets and binary relations. In this paper, we extend a new, recently-introduced conceptual data model based on allegories by adding support for modal operators and developing a modal interpretation of the model in any allegory satisfying certain additional (but natural) axioms. The possibility of using different allegories allows us to transparently use alternative logical frameworks, such as fuzzy relations. Mathematically, our work demonstrates how to enrich with modal operators and to give a many world semantics to an abstract algebraic logic framework. We also give some examples of applications of the modal extension.<\/jats:p>","DOI":"10.3390\/axioms3020260","type":"journal-article","created":{"date-parts":[[2014,5,30]],"date-time":"2014-05-30T12:00:34Z","timestamp":1401451234000},"page":"260-279","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Modalities for an Allegorical Conceptual Data Model"],"prefix":"10.3390","volume":"3","author":[{"given":"Bartosz","family":"Zieli\u0144ski","sequence":"first","affiliation":[{"name":"Department of Computer Science, Faculty of Physics and Applied Informatics,University of \u0141\u00f3d\u017a ul. 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North-Holland Mathematical Library."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Tarski, A., and Givant, S. (1987). A Formalization of Set Theory Without Variables, American Mathematical Society. Number t. 41 in A Formalization of Set Theory without Variables.","DOI":"10.1090\/coll\/041"},{"key":"ref_3","unstructured":"Lippe, E., and Hofstede, A. (1994). A Category Theory Approach to Conceptual Data Modeling, University of Nijmegen, Computing Science Institute."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1007\/BFb0000487","article-title":"Selective attribute elimination for categorical data specifications","volume":"Volume 1349","author":"Johnson","year":"1997","journal-title":"Algebraic Methodology and Software Technology"},{"key":"ref_5","first-page":"391","article-title":"Relational databases and indexed categories","volume":"Volume 13","author":"Rosebrugh","year":"1992","journal-title":"Proceedings of the International Category Theory Meeting, CMS Conference Proceedings"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S1571-0661(04)00305-6","article-title":"Sketch data models, relational schema and data specifications","volume":"61","author":"Johnson","year":"2002","journal-title":"Electron. Notes Theor. Comput. 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An Introduction to Algebraic Structures, Holden-Day."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Blackburn, P., van Benthem, J., and Wolter, F. (2006). Handbook of Modal Logic, Elsevier Science. Studies in Logic and Practical Reasoning.","DOI":"10.1002\/9780470996751.ch27"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Blackburn, P., de Rijke, M., and Venema, Y. (2002). Modal Logic, Cambridge University Press. Cambridge Tracts in Theoretical Computer Science.","DOI":"10.1017\/CBO9781107050884"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.tcs.2007.02.068","article-title":"Ultraproducts and possible worlds semantics in institutions","volume":"379","author":"Diaconescu","year":"2007","journal-title":"Theor. Comput. 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