{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:31:40Z","timestamp":1760243500670,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2013,7,23]],"date-time":"2013-07-23T00:00:00Z","timestamp":1374537600000},"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>It is known that several discrete integrals, including the Choquet and Sugeno integrals, as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of the class of comonotonically modular functions, we axiomatically identify more general families of discrete integrals that are comonotonically modular, including signed Choquet integrals and symmetric signed Choquet integrals, as well as natural extensions of Sugeno integrals.<\/jats:p>","DOI":"10.3390\/axioms2030390","type":"journal-article","created":{"date-parts":[[2013,7,23]],"date-time":"2013-07-23T13:28:05Z","timestamp":1374586085000},"page":"390-403","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Discrete Integrals Based on Comonotonic Modularity"],"prefix":"10.3390","volume":"2","author":[{"given":"Miguel","family":"Couceiro","sequence":"first","affiliation":[{"name":"Lamsade, University Paris-Dauphine, Place du Mar\u00e9chal de Lattre de Tassigny, Paris Cedex 16 F-75775, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Luc","family":"Marichal","sequence":"additional","affiliation":[{"name":"Mathematics Research Unit, University of Luxembourg, 6 Rue Coudenhove-Kalergi, Luxembourg L-1359, Luxembourg"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,7,23]]},"reference":[{"key":"ref_1","unstructured":"Beliakov, G., Pradera, A., and Calvo, T. (2007). Aggregation Functions: A Guide for Practitioners, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Grabisch, M., Marichal, J.-L., Mesiar, R., and Pap, E. (2009). Aggregation Functions, Cambridge University Press.","DOI":"10.1017\/CBO9781139644150"},{"key":"ref_3","unstructured":"Murofushi, T., and Sugeno, M. (2000). Studies in Fuzziness and Soft Computing, Physica-Verlag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1142\/S0218488511006964","article-title":"Axiomatizations of signed discrete Choquet integrals","volume":"19","author":"Cardin","year":"2011","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.fss.2011.05.006","article-title":"Axiomatizations of Lov\u00e1sz extensions of pseudo-Boolean functions","volume":"181","author":"Couceiro","year":"2011","journal-title":"Fuzzy Sets Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/0165-0114(92)90037-5","article-title":"Characterization and comparison of Sugeno and Choquet integrals","volume":"52","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","first-page":"1329","article-title":"Monotone Set Functions-Based Integrals","volume":"Volume II","author":"Pap","year":"2002","journal-title":"Handbook of Measure Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s00010-011-0091-0","article-title":"Axiomatizations of quasi-Lov\u00e1sz extensions of pseudo-Boolean functions","volume":"82","author":"Couceiro","year":"2011","journal-title":"Aequ. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s00010-009-2969-7","article-title":"Axiomatizations of quasi-polynomial functions on bounded chains","volume":"78","author":"Couceiro","year":"2009","journal-title":"Aequ. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s00010-010-0039-9","article-title":"Quasi-polynomial functions over bounded distributive lattices","volume":"80","author":"Couceiro","year":"2010","journal-title":"Aequ. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1016\/j.fss.2009.10.008","article-title":"Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices","volume":"161","author":"Couceiro","year":"2010","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","first-page":"65","article-title":"Representations and characterizations of polynomial functions on chains","volume":"16","author":"Couceiro","year":"2010","journal-title":"J. Mult. Valued Log. Soft Comput."},{"key":"ref_13","first-page":"247","article-title":"Polynomial functions over bounded distributive lattices","volume":"18","author":"Couceiro","year":"2012","journal-title":"J. Mult. Valued Log. Soft Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1287\/opre.26.2.305","article-title":"Minimizing a submodular function on a lattice","volume":"26","author":"Topkis","year":"1978","journal-title":"Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1109\/TFUZZ.2010.2077641","article-title":"The ordered modular averages","volume":"19","author":"Mesiar","year":"2011","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1016\/j.disc.2008.01.019","article-title":"Weighted lattice polynomials","volume":"309","author":"Marichal","year":"2009","journal-title":"Discret. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/2\/3\/390\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:48:09Z","timestamp":1760219289000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/2\/3\/390"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,7,23]]},"references-count":16,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2013,9]]}},"alternative-id":["axioms2030390"],"URL":"https:\/\/doi.org\/10.3390\/axioms2030390","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2013,7,23]]}}}