Abstract
In this paper, we introduce the concept of topological MI-groups, where the MI-group structure, which naturally generalizes the group structure, is enriched by a topology and the respective binary operation and inversion are continuous. To demonstrate that the proposed generalization of topological groups is meaningful, we prove that there are the products of topological MI-groups and the topological quotient MI-groups. The concept of topological MI-group is demonstrated on examples.
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Notes
- 1.
It should be noted that Markov called this novel structure as quasimodule, which is, however, terminologically confusing with the standard denotation, since no scalar operation is consider here.
References
Bica, A.M.: Categories and algebraic structures for real fuzzy numbers. Pure Math. Appl. 13(1–2), 63–67 (2003)
Bica, A.M.: Algebraic structures for fuzzy numbers from categorial point of view. Soft. Comput. 11, 1099–1105 (2007)
Diamond, P., Kloeden, P.E.: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, River Edge (1994)
Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9, 613–626 (1978)
Fechete, D., Fechete, I.: Quotient algebraic structure on the set of fuzzy numbers. Kybernetika 51(2), 255–257 (2015)
Holčapek, M.: On generalized quotient mi-groups. Fuzzy Sets Syst. 326, 3–23 (2017)
Holčapek, M., Štěpnička, M.: Arithmetics of extensional fuzzy numbers - part I: introduction. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Brisbane, pp. 1517–1524 (2012)
Holčapek, M., Štěpnička, M.: Arithmetics of extensional fuzzy numbers - part II: algebraic framework. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Brisbane, pp. 1525–1532 (2012)
Holčapek, M., Štěpnička, M.: MI-algebras: a new framework for arithmetics of (extensional) fuzzy numbers. Fuzzy Sets Syst. 257, 102–131 (2014)
Holčapek, M., Wrublová, M., Bacovský, M.: Quotient MI-groups. Fuzzy Sets Syst. 283, 1–25 (2016)
Hungerford, T.W.: Algebra, vol. XIX, 502 p. Holt, Rinehart and Winston, Inc. New York (1974)
Kelly, J.L.: General Topology. Springer, New York (1975)
Klement, E.P., Puri, M.L., Ralescu, D.A.: Limit theorems for fuzzy random variables. In: Proceedings of the Royal Society of London, Series A, vol. 407, pp. 171–182 (1986)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, New Jersey (1995)
Krätschmer, V.: Limit theorems for fuzzy-random variables. Fuzzy Sets Syst. 126, 253–263 (2002)
Kruse, R.: The strong law of large numbers for fuzzy random variables. Inf. Sci. 28(3), 233–241 (1982)
Li, S., Ogura, Y., Kreinovich, V.: Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables. Springer, Netherlands (2002). https://doi.org/10.1007/978-94-015-9932-0
Mareš, M.: Addition of fuzzy quantities: disjunction-conjunction approach. Kybernetika 2, 104–116 (1989)
Mareš, M.: Multiplication of fuzzy quantities. Kybernetika 5, 337–356 (1992)
Mareš, M.: Computation over Fuzzy Quantities. CRC Press, Boca Raton (1994)
Markley, N.G.: Topological Groups: An Introduction. Wiley, Hoboken (2010)
Markov, S.: On the algebra of intervals and convex bodies. J. Univ. Comput. Sci. 4(1), 34–47 (1998)
Markov, S.: On the algebraic properties of convex bodies and some applications. J. Convex Anal. 7(1), 129–166 (2000)
Markov, S.: Quasilinear space of convex bodies and intervals. J. Comput. Appl. Math. 162, 93–112 (2004)
Qiu, D., Lu, C., Zhang, W., Lan, Y.: Algebraic properties and topological properties of the quotient space of fuzzy numbers based on mareś equivalence relation. Fuzzy Sets Syst. 245, 63–82 (2014)
Qiu, D., Zhang, W., Lu, C.: On fuzzy differential equations in the quotient space of fuzzy numbers. Fuzzy Sets Syst. 295, 72–98 (2016)
Qiu, D., Zhang, W.: Symmetric fuzzy numbers and additive equivalence of fuzzy numbers. Soft. Comput. 17, 1471–1477 (2013)
Sadeghi, N., Fayek, A.R., Pedrycz, W.: Fuzzy Monte Carlo simulation and risk assessment in construction. Comput. Aided Civ. Infrastruct. Eng. 25, 238–252 (2010)
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This work was supported by the project LQ1602 IT4Innovations excellence in science.
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Holčapek, M., Škorupová, N. (2018). Topological MI-Groups: Initial Study. In: Medina, J., Ojeda-Aciego, M., Verdegay, J., Perfilieva, I., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications. IPMU 2018. Communications in Computer and Information Science, vol 855. Springer, Cham. https://doi.org/10.1007/978-3-319-91479-4_50
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