On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity
We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. Inspired by the well-known mathematical structures of quantum gravity and lattice gauge theories in physics and by the application of this framework in economy by other authors, we descr...
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| Format: | Article |
| Language: | English |
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Elsevier
2019-06-01
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| Series: | Heliyon |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2405844018354276 |
| _version_ | 1818265313931689984 |
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| author | Jean-Pierre Magnot |
| author_facet | Jean-Pierre Magnot |
| author_sort | Jean-Pierre Magnot |
| collection | DOAJ |
| description | We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. Inspired by the well-known mathematical structures of quantum gravity and lattice gauge theories in physics and by the application of this framework in economy by other authors, we describe how the same structures arise in pairwise comparisons. First, we provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, and give a full description of elementary properties that extend to our generalized setting. Then, we show a geometric picture where inconsistency in pairwise comparisons can be interpreted as the non-trivial Holonomy on a loop. We continue with probabilistic aspects, again adapting ideas from quantum physics where probabilities modelize uncertainty. We finish by examples where the use of a non-abelian group is necessary. |
| first_indexed | 2024-12-12T19:48:50Z |
| format | Article |
| id | doaj.art-effe499c689c4ae3a3d309df24ba434c |
| institution | Directory Open Access Journal |
| issn | 2405-8440 |
| language | English |
| last_indexed | 2024-12-12T19:48:50Z |
| publishDate | 2019-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Heliyon |
| spelling | doaj.art-effe499c689c4ae3a3d309df24ba434c2022-12-22T00:14:02ZengElsevierHeliyon2405-84402019-06-0156e01821On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravityJean-Pierre Magnot0LAREMA, université d'Angers, Boulevard Lavoisier, F-49045 Angers cedex 01, France; Lycée Jeanne d'Arc, 30 avenue de Grande Bretagne, F-63000 Clermont-Ferrand, France; Correspondence to: LAREMA, université d'Angers, Boulevard Lavoisier, F-49045 Angers cedex 01, France.We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. Inspired by the well-known mathematical structures of quantum gravity and lattice gauge theories in physics and by the application of this framework in economy by other authors, we describe how the same structures arise in pairwise comparisons. First, we provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, and give a full description of elementary properties that extend to our generalized setting. Then, we show a geometric picture where inconsistency in pairwise comparisons can be interpreted as the non-trivial Holonomy on a loop. We continue with probabilistic aspects, again adapting ideas from quantum physics where probabilities modelize uncertainty. We finish by examples where the use of a non-abelian group is necessary.http://www.sciencedirect.com/science/article/pii/S2405844018354276Applied mathematicsApproximate reasoningInconsistencyPairwise comparisonsGroupHolonomy |
| spellingShingle | Jean-Pierre Magnot On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity Heliyon Applied mathematics Approximate reasoning Inconsistency Pairwise comparisons Group Holonomy |
| title | On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity |
| title_full | On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity |
| title_fullStr | On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity |
| title_full_unstemmed | On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity |
| title_short | On mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity |
| title_sort | on mathematical structures on pairwise comparisons matrices with coefficients in an abstract group arising from quantum gravity |
| topic | Applied mathematics Approximate reasoning Inconsistency Pairwise comparisons Group Holonomy |
| url | http://www.sciencedirect.com/science/article/pii/S2405844018354276 |
| work_keys_str_mv | AT jeanpierremagnot onmathematicalstructuresonpairwisecomparisonsmatriceswithcoefficientsinanabstractgrouparisingfromquantumgravity |