Group structures on quotient sets in classification problems. (English. Russian original) Zbl 1323.68473
Cybern. Syst. Anal. 50, No. 4, 507-518 (2014); translation from Kibern. Sist. Anal. No. 4, 27-41 (2014).
Summary: The relationship between the properties of difunctionality and ternary relations is analyzed as applied to information factorization problems. Conditions are found under which arbitrary \(n\)-ary relations induce multialgebraic systems with a common carrier in the form of a Cartesian cube. Multigroup axiomatics is considered. Conditions of existence of groups over equivalence classes are formulated and proved. Examples of producing multigroups and cases when a common carrier is absent are given.
MSC:
| 68T30 | Knowledge representation |
| 68T35 | Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence |
Keywords:
granulation calculus; multialgebraic system; ternary relation; difunctionality; multigroup; quotient setReferences:
| [1] | L. Zhang and B. Zhang, “Fuzzy reasoning model under quotient space structure,” Inform. Sci., 173, No. 3, 353-364 (2005). · Zbl 1088.68170 · doi:10.1016/j.ins.2005.03.005 |
| [2] | Zhang, L.; Zhang, B.; Pedrycz, W. (ed.); Skowron, A. (ed.); Kreinovich, V. (ed.), Quotient spaces and granular computing, 411-424 (2008), Chichester · doi:10.1002/9780470724163.ch18 |
| [3] | L. Zhang and B. Zhang, “The theory and application of tolerance relations,” Intern. J. of Granular Computing, Rough Sets and Intelligent Systems, 1, No. 2, 179-189 (2009). · doi:10.1504/IJGCRSIS.2009.028008 |
| [4] | L. A. Zadeh, “Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic,” Fuzzy Sets Systems, 19, 111-127 (1997). · Zbl 0988.03040 · doi:10.1016/S0165-0114(97)00077-8 |
| [5] | Yao, Y.; Kryszkiewicz, M. (ed.); etal., The art of granular computing, 101-112 (2007), Berlin-Heidelberg · doi:10.1007/978-3-540-73451-2_12 |
| [6] | B. Apolloni et al. (eds.), The Puzzle of Granular Computing, Series: Studies in Computational Intelligence, 138, Springer, Berlin-Heidelberg (2008). · Zbl 1097.68620 |
| [7] | T. Y. Lin, “Granular computing I: The concept of granulation and its formal model,” Intern. J. Granular Comput. Rough Sets and Intelligent Systems, 1, No. 1, 21-42 (2009). · doi:10.1504/IJGCRSIS.2009.026723 |
| [8] | Kreinovich, V.; Pedrycz, W. (ed.); Skowron, A. (ed.); Kreinovich, V. (ed.), Interval computation as an important part of granular computing: An introduction, 3-32 (2008), Chichester |
| [9] | R. Bello, R. Falcón, W. Pedrycz, and J. Kacprzyk (eds.), Granular Computing: At the Junction of Rough Sets and Fuzzy Sets, Series: Studies in Fuzziness and Soft Comput., Springer, Berlin-Heidelberg (2008). · Zbl 1088.68170 |
| [10] | J. Stepaniuk, “Rough-granular computing in knowledge discovery and data mining,” Studies in Computational Intelligence, 152, Springer, Berlin-Heidelberg (2008). · Zbl 1173.68646 |
| [11] | L. A. Zadeh, “Some reflections on soft computing, granular computing and their roles in the conception, design and utilization of information/intelligent systems,” Soft Computing, 2, No. 1, 23-25 (1998). · doi:10.1007/s005000050030 |
| [12] | W. Pedrycz, “Granular computing in multi-agent systems,” in: G. Wang et al. (eds.), Rough Sets and Knowledge Technology; Lecture Notes in Artificial Intelligence, 5009, 3-17, Springer, Berlin-Heidelberg (2008). |
| [13] | L. Polkowski, “Granulation of knowledge in decision systems: The approach based on rough inclusions. The method and its applications,” in: M. Kryszkiewicz et al. (eds.), Rough Sets and Intelligent Systems Paradigms; Lecture Notes in Artificial Intelligence, 4585, 69-79, Springer, Berlin-Heidelberg (2007). |
| [14] | W. Pedrycz, Knowledge-Based Clustering: From Data to Information Granules, Wiley, N.Y. (2005). · Zbl 1100.68096 · doi:10.1002/0471708607 |
| [15] | V. P. Mashtalir and V. V. Shlyakhov, “Properties of multialgebraic systems in problems of comparative recognition,” Cybernetics and Systems Analysis, 39, No. 6, 790-804 (2003). · Zbl 1097.68620 · doi:10.1023/B:CASA.0000020221.98415.65 |
| [16] | S. N. Gerasin and V. V. Shlyakhov, “On external and internal compatibility of arbitrary <Emphasis Type=”Italic“>n-ary relations,” Dop. NANU, No. 1, 77-82 (2006). · Zbl 1091.08500 |
| [17] | V. V. Shlyakhov and S. Ya. Yakovlev, “On the isomorphism of multialgebraic systems,” Dop. NANU, No. 9, 67-70 (2002). · Zbl 1012.08006 |
| [18] | A. Kagramanyan, V. Mashtalir, and V. Shlyakhov, “Multialgebraic systems in information granulation,” Intern. J. “Information Theories and Applications,” 15, No. 1, 55-63 (2008). |
| [19] | A. I. Maltsev, Algebraic Systems [in Russian], Nauka, Moscow (1970). · Zbl 0223.08001 |
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