Editing Multiset (section)

==History==
Wayne Blizard traced multisets back to the very origin of numbers, arguing that "in ancient times, the number ''n'' was often represented by a collection of ''n'' strokes, [[tally mark]]s, or units."<ref name="Blizard1989">{{cite journal|last=Blizard|first=Wayne D|year=1989|title=Multiset theory|journal=[[Notre Dame Journal of Formal Logic]]|volume=30|issue=1|pages=36–66|url=http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093634995|doi=10.1305/ndjfl/1093634995|doi-access=free}}</ref> These and similar collections of objects can be regarded as multisets, because strokes, tally marks, or units are considered indistinguishable. This shows that people implicitly used multisets even before mathematics emerged.

Practical needs for this structure have caused multisets to be rediscovered several times, appearing in literature under different names.<ref name="Blizard1991">{{cite journal|last=Blizard|first=Wayne D.|year=1991|title=The Development of Multiset Theory|journal=Modern Logic|volume=1|issue=4|pages=319–352|url=http://projecteuclid.org/download/pdf_1/euclid.rml/1204834739}}</ref>{{rp|p=323}} For instance, they were important in early [[AI]] languages, such as QA4, where they were referred to as ''bags,'' a term attributed to [[L. Peter Deutsch|Peter Deutsch]].<ref name="Rulifson1972">{{cite tech report|last1=Rulifson|first1=J. F.|last2=Derkson|first2=J. A.|last3=Waldinger|first3=R. J.|date=November 1972|institution=SRI International|title=QA4: A Procedural Calculus for Intuitive Reasoning|number=73}}</ref> A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set).<ref name="Blizard1991" />{{rp|p=320}}<ref name="Singh2007">{{cite journal | last1=Singh|first1=D.| last2=Ibrahim|first2=A. M.| last3=Yohanna|first3=T.|last4=Singh|first4=J. N.|year=2007|title=An overview of the applications of multisets | journal=Novi Sad Journal of Mathematics|volume=37|issue=2|pages=73–92}}</ref>

Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician [[Bhāskara II|Bhāskarāchārya]] circa 1150, who described permutations of multisets.<ref name="Knuth1998"/>{{rp|p=694}} The work of [[Marius Nizolius]] (1498–1576) contains another early reference to the concept of multisets.<ref name="Angelelli1965">{{cite journal| last=Angelelli|first=I.| year=1965| title=Leibniz's misunderstanding of Nizolius' notion of 'multitudo'|journal=Notre Dame Journal of Formal Logic| issue=6| pages=319–322}}</ref> [[Athanasius Kircher]] found the number of multiset permutations when one element can be repeated.<ref>{{cite book|last=Kircher|first=Athanasius|author-link=Athanasius Kircher|year=1650|title=Musurgia Universalis| url=https://archive.org/details/bub_gb_97xCAAAAcAAJ|publisher=Corbelletti|location=Rome}}</ref> [[Jean Prestet]] published a general rule for multiset permutations in 1675.<ref>{{cite book|last=Prestet|first=Jean|year=1675|title=Elemens des Mathematiques|publisher=André Pralard|location=Paris}}</ref> [[John Wallis]] explained this rule in more detail in 1685.<ref>{{cite book|last=Wallis|first=John|author-link=John Wallis|year=1685|title=A treatise of algebra|publisher=John Playford|location=London}}</ref>

Multisets appeared explicitly in the work of [[Richard Dedekind]].<ref>{{cite book|last=Dedekind|first=Richard|author-link=Richard Dedekind|year=1888|title=Was sind und was sollen die Zahlen?|publisher=Vieweg|location=Braunschweig | page = 114}}</ref><ref name=syropoulos>{{cite conference
 | last = Syropoulos | first = Apostolos
 | editor1-last = Calude | editor1-first = Cristian
 | editor2-last = Paun | editor2-first = Gheorghe
 | editor3-last = Rozenberg | editor3-first = Grzegorz
 | editor4-last = Salomaa | editor4-first = Arto
 | contribution = Mathematics of multisets
 | doi = 10.1007/3-540-45523-X_17
 | pages = 347–358
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Multiset Processing, Mathematical, Computer Science, and Molecular Computing Points of View [Workshop on Multiset Processing, WMP 2000, Curtea de Arges, Romania, August 21–25, 2000]
 | volume = 2235
 | year = 2000| isbn = 978-3-540-43063-6
 }}</ref>

Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century.  For example, [[Hassler Whitney]] (1933) described ''generalized sets'' ("sets" whose [[characteristic function]]s may take any [[integer]]  value: positive, negative or zero).<ref name="Blizard1991" />{{rp|p=326}}<ref name="Whitney">{{cite journal |last1=Whitney |first1=Hassler |title=Characteristic Functions and the Algebra of Logic |journal=[[Annals of Mathematics]] |date=1933 |volume=34 |issue=3 |pages=405–414 |doi=10.2307/1968168|jstor=1968168 }}</ref>{{rp|p=405}}  Monro (1987) investigated the [[category (mathematics)|category]] '''Mul''' of multisets and their [[morphism]]s, defining a ''multiset'' as a set with an [[equivalence relation]] between elements "of the same ''sort''", and a ''morphism'' between multisets as a [[function (mathematics)|function]] that respects ''sorts''.  He also introduced a ''multinumber'': a function ''f''{{hairsp}}(''x'') from a multiset to the [[natural number]]s, giving the ''multiplicity'' of element ''x'' in the multiset.  Monro argued that the concepts of multiset and multinumber are often mixed indiscriminately, though both are useful.<ref name="Blizard1991" />{{rp|pp=327–328}}<ref>{{cite journal |last1=Monro |first1=G. P. |title=The Concept of Multiset |journal=Zeitschrift für Mathematische Logik und Grundlagen der Mathematik |date=1987 |volume=33 |issue=2 |pages=171–178 |doi=10.1002/malq.19870330212}}</ref>