{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:45:44Z","timestamp":1760240744208,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,4]],"date-time":"2019-09-04T00:00:00Z","timestamp":1567555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"undefined  &lt;span style=&quot;color:gray;font-size:10px;&quot;&gt;undefined&lt;\/span&gt;","award":["DZRO K217"],"award-info":[{"award-number":["DZRO K217"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many real-life problems are well represented only by sets which allow repetition(s), such as the multiset. Although not limited to the following, such cases may arise in a database query, chemical structures and computer programming. The set of roots of a polynomial, say     f ( x )    , has been found to correspond to a multiset, say F. If     f ( x )     and     g ( x )     are polynomials whose sets of roots respectively correspond to the multisets     F ( x )     and     G ( x )    , the set of roots of their product,     f ( x ) g ( x )    , corresponds to the multiset     F \u228e G    , which is the sum of multisets F and G. In this paper, some properties of the algebraic sum of multisets \u228e and some results on selection are established. Also, the count function of the image of any function on Dedekind multisets is defined and some of its properties are established. Some applications of these multisets are also given.<\/jats:p>","DOI":"10.3390\/sym11091125","type":"journal-article","created":{"date-parts":[[2019,9,5]],"date-time":"2019-09-05T03:22:36Z","timestamp":1567653756000},"page":"1125","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Results on Functions on Dedekind Multisets"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3305-529X","authenticated-orcid":false,"given":"\u0160\u00e1rka","family":"Ho\u0161kov\u00e1-Mayerov\u00e1","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, University of Defence Brno, Kounicova 65, 66210 Brno, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3737-4044","authenticated-orcid":false,"given":"Babatunde Oluwaseun","family":"Onasanya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ibadan, Ibadan 200284, Oyo State, Nigeria"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,4]]},"reference":[{"key":"ref_1","unstructured":"Dedekind, R. (1963). Essays on the Theory of Numbers, Dover."},{"key":"ref_2","first-page":"36","article-title":"Multiset theory","volume":"30","author":"Blizard","year":"1989","journal-title":"Notre Dame Form. Log."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0304-3975(93)90351-S","article-title":"Dedekind multisets and function shells","volume":"110","author":"Blizard","year":"1993","journal-title":"Theor. Comput. Sci."},{"key":"ref_4","first-page":"347","article-title":"Mathematics of Multisets","volume":"Volume 2235","author":"Syropoulos","year":"2001","journal-title":"WMC 2000: Multiset Processing"},{"key":"ref_5","unstructured":"Wildberger, N.J. (2003). A New Look at Multisets, School of Mathematics, UNSW."},{"key":"ref_6","unstructured":"Knuth, D. (1981). The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison Wesley."},{"key":"ref_7","first-page":"643","article-title":"On multisets and multigroups","volume":"6","author":"Nazmul","year":"2013","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_8","first-page":"211","article-title":"Sets, fuzzy sets, multisets and functions","volume":"12","author":"Lake","year":"1976","journal-title":"J. Lond. Math. Soc."},{"key":"ref_9","first-page":"251","article-title":"Multigroups and multicosets","volume":"41","author":"Onasanya","year":"2019","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_10","first-page":"597","article-title":"Multi-fuzzy group induced by multisets","volume":"41","author":"Onasanya","year":"2019","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_11","first-page":"73","article-title":"An overview of the applications of multisets","volume":"37","author":"Singh","year":"2007","journal-title":"Novi Sad J. Math."},{"key":"ref_12","first-page":"33","article-title":"A systemisation of fundamenals of multisets","volume":"29","author":"Singh","year":"2008","journal-title":"Lect. Mat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"47","DOI":"10.9790\/5728-0754752","article-title":"Symmetric groups under multiset perspective","volume":"7","author":"Yohanna","year":"2013","journal-title":"IOSR J. Math."},{"key":"ref_14","first-page":"525","article-title":"Congruences of Multialgebras","volume":"15","author":"Ameri","year":"2009","journal-title":"Multivalued Log. Soft Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.3233\/JIFS-169940","article-title":"Multiset filters of residuated lattices and its application in medical diagnosis","volume":"36","author":"Vimala","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_16","first-page":"21","article-title":"Computer representation of multisets","volume":"6","author":"Tella","year":"2011","journal-title":"Sci. World J."},{"key":"ref_17","unstructured":"Ameri, R., and Hoskova-Mayerova, S. (2016, January 2\u20134). Fuzzy Continuous Polygroups. Proceedings of the Aplimat\u201415th Conference on Applied Mathematics, Bratislava, Slovakia."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1080\/03081078608934952","article-title":"On the theory of bags","volume":"13","author":"Yager","year":"1986","journal-title":"Int. J. Gen. Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1125\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:16:51Z","timestamp":1760188611000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1125"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,4]]},"references-count":18,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["sym11091125"],"URL":"https:\/\/doi.org\/10.3390\/sym11091125","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,9,4]]}}}