TY  - JOUR
T1  - On topology of pairwise comparison matrices
AU  - Koyuncu, Selcuk
AU  - Ahamad, Ali Mryr
AU  - Özel, Cenap
PY  - 2025
DA  - June
Y2  - 2025
DO  - 10.31801/cfsuasmas.1542622
JF  - Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
JO  - Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.
PB  - Ankara University
WT  - DergiPark
SN  - 1303-5991
SP  - 346
EP  - 354
VL  - 74
IS  - 2
LA  - en
AB  - The aim of this paper is to study the topology of the additive (Lie algebra) and multiplicative (Lie group) pairwise comparison matrices ($PCM$). We prove that both additive and multiplicative $PCM$ are connected while only multiplicative consistent $PCM$ is compact. We also provide that there is no upper bound for the Perron norm of an inconsistent pairwise comparison matrix.
KW  - Hankel matrix
KW  - Lie group
KW  - Lie algebra
KW  - pairwise comparison matrices
CR  - Arvanitoyeorgos, A., Lie Transformation Groups and Geometry, in: Mladenov, I. M. (Ed), Ninth International
Conference on Geometry, Integrability and Quantization, SOFTEX, 9 (2008), 11-35. https://doi.org/10.7546/
giq-9-2008-11-35.
CR  - Crawford, G., Williams, C., The analysis of subjective judgment matrices, RAND Report, R-2572-AF, 1985.
https://www.rand.org/content/dam/rand/pubs/reports/2005/R2572-1.pdf.
CR  - Holszynski, W., Koczkodaj, W. W., Convergence of inconsistency algorithms for the pairwise comparisons, Information Processing Letters, 59(4) (1996), 197–202. https://doi.org/10.1016/0020-0190(96)00113-5.
CR  - Koczkodaj, W. W., Marek, V. W., Yaylı, Y., Pairwise comparisons matrix decomposition into approximation and
orthogonal component using Lie theory, International Journal of Approximate Reasoning, 139 (2021), 201–210.
https://doi.org/10.1016/j.ijar.2021.09.009.
CR  - Koczkodaj, W. W., Orlowski, M., An orthogonal basis for computing a consistent approximation to a pairwise comparisons matrix. Computers and Mathematics with Applications, 34(10) (1997), 41–47. https://doi.org/10.1016/S0898-1221(97)00205-8.
CR  - Koczkodaj, W. W., Smarzewski, R., Szybowski, J., On orthogonal Projections on the space of Consistent Pairwise
Comparisons Matrices, Fundamenta Informaticae, 172(4) (2020), 379–397. https://doi.org/10.3233/fi-2020-1910.
CR  - Koczkodaj, W. W., Szarek, S., On distance-based inconsistency reduction algorithms for pairwise comparisons, Logic Journal of the IGPL, 18(6) (2010), 859–869. https://doi.org/10.1093/jigpal/jzp062.
CR  - Koczkodaj, W. W., Szybowski, J., The limit of inconsistency reduction in pairwise comparisons, International Journal of Applied Mathematics and Computer Science, 26(3) (2016), 721–729. https://doi.org/10.1515/amcs-2016-0050.
CR  - Li, C. C., Dong, Y. C., Xu, Y. J., Chiclana, F., Herrera, F., An overview on managing additive consistency of reciprocal preference relations for consistency driven decision making and fusion: taxonomy and future directions, Information Fusion, 52 (2019), 143–156. https://doi.org/10.1016/j.inffus.2018.12.004.
CR  - Mryr, A. A., Topology and Analysis of pairwise comparison matrices. MSc Thesis (2023), King Abdulaziz University. https://www.kau.edu.sa/Show_Res.aspx?Site_ID=306&LNG=EN&RN=76336.
CR  - Tu, T. L., An Introduction to Manifolds, 2nd ed., Springer, 2010. https://doi.org/10.1007/978-1-4419-7400-6.
UR  - https://doi.org/10.31801/cfsuasmas.1542622
L1  - https://dergipark.org.tr/en/download/article-file/4186651
ER  -