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Multisets, heaps, bags, families: What is a multiset?

Published online by Cambridge University Press:  27 January 2020

Helmut Jürgensen
Helmut Jürgensen*
Affiliation:
Department of Computer Science, The University of Western Ontario, London, Ontario, N6C 5B7, Canada
*
*Corresponding author. Email: [email protected]
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Abstract

Is the current formulation of multiset theory, which is based on sets and multiplicities of their elements, adequate? We exhibit both mathematical and metamathematical reasons which should cause one to rethink the definition. Some problems with multiset theory in its accepted formulation concern even the basic operations of union, intersection, and complement; others, more deeply rooted, concern Cartesian products, relations, or morphisms. We compare current definitions and conclude that the problems of multiset theory need to be resolved at the fundamental level of sets and mappings (or equivalent constructs) with multiplicities introduced only as a secondary concept. As a consequence, we propose to define multisets as families. A mapping establishes the connection to the familiar theory of multisets. Without losing anything, our proposal is simple and provides for an elegant mathematical theory.

Keywords

Multisetgeneralised multisetcategories
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 2 , February 2020 , pp. 139 - 158
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

‡

A preliminary version of this paper, without proofs and details, was presented at the International Conference on Recent Advances in Pure and Applied Mathematics, ICRAPAM 2016, held on 19–23 May, 2016, in Bodrum, Turkey (Jürgensen 2016).

†

Deceased.

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