Weightable quasi-metrics related to fuzzy sets

Year 2018, Volume: 47 Issue: 5, 1184 - 1195, 16.10.2018

M.j. Campión R. G. Catalán E. Induráin O. Valero

Abstract

We show that the definition of a fuzzy set is directly related to the existence of a weightable quasi-metric on a universe. This relationship

is also explored in terms of functional equations coming either from the membership function of a fuzzy set or from the disymmetry function of

a quasi-metric.

Keywords

Fuzzy sets , weightable quasi-metrics , functional equations , representable total preorders

References

• Barrenechea, E., Bustince, H., Campión, M.J., Induráin, E. and Knoblauch, V.: Topological interpretations of fuzzy subsets. A unified approach for fuzzy thresholding algorithms, Knowledge-Based Systems 54, 163-171, 2013.
• Bosi, G., Candeal, J.C., Induráin, E., Olóriz, E. and Zudaire, M.: Numerical representations of interval orders, Order 18, 171-190, 2001.
• Bridges, D.S. and Mehta, G.B.: Representations of Preference Orderings, Springer-Verlag, Berlin-Heidelberg-New York, 1995.
• Campión, M.J., Candeal, J.C. and Induráin, E.: On Yi's extension property for totally preordered topological spaces, J. Korean Math. Soc. 43 (1), 159-181, 2006.
• Campión, M.J., Catalán, R.G., Garay, J. and Induráin, E.: Functional equations related to fuzzy sets and representable orderings, J. Intell. Fuzzy Syst. 29(3), 1065-1077, 2015.
• Campión, M.J., Catalán, R.G., Induráin, E. and Ochoa, G.: Reinterpreting a fuzzy subset by means of a Sincov's functional equation, J. Intell. Fuzzy Syst. 27(1), 367-375, 2014.
• Campión, M.J., Induráin, E. and Knoblauch, V.: Topologies generated by nested collections, Bull. Malays. Math. Sci. Soc. 39, 545-561, 2016.
• Campión, M.J., Induráin, E., Ochoa, G. and Valero, O.: Functional equations related to weightable quasi-metrics, Hacet. J. Math. Stat. 44 (4), 775-787, 2015.
• Candeal, J.C., Hervés, C. and Induráin, E.: Some results on representation and extension of preferences, J. Math. Econom. 29 (1), 75-81, 1998.
• Deza, M.M. and Deza E.: Encyclopedia of Distances, (Springer-Verlag, Berlin, 2009).
• Dijkstra, E.W.: A note on two problems in connection with graphs, Numer. Math. 1, 269- 271, 1959.
• Induráin, E. and Knoblauch, V.: On topological spaces whose topology is induced by a binary relation, Quaest. Math. 36(1), 47-65, 2013.
• Künzi, H.P.A. and Vajner V.: Weighted quasi-metrics, Ann. NY Acad. Sci. 728, 64-77, 1994.
• Matthews, S.G.: Partial metric topology, Ann. NY Acad. Sci. 728, 183-197, 1994.
• Romaguera, S. and Sanchis, M.: Applications of utility functions defined on quasi-metric spaces, J. Math. Anal. Appl. 283, 219-235, 2003.
• Romaguera, S., Tirado, P. and Valero, O.: New results on mathematical foundations of asymptotic complexity analysis of algorithms via complexity spaces, Int. J. Comput. Math. 89, 1728-1741, 2012.
• Schellekens, M.: The Smyth completion: A common foundation for denotational semantics and complexity analysis, Electronic Notes Theoret. Comput. Sci. 1, 535-556, 1995.
• Seda, A.K.: Quasi-metrics and the Semantics of Logic Programs, Fund. Inform. 29, 97-117, 1997.
• Sincov, D.M.: Über eine Funktionalgleichung, Arch. Math. Phys. (3) 6, 216-217, 1903.
• Wu, H.-C.: Existence and uniqueness for the construction of fuzzy sets from a solidly nested family, Fuzzy Optim. Decis. Mak. 14 (1), 1-41, 2015.
• Yi, G.: Continuous extension of preferences, J. Math. Econom. 22, 547-555, 1993.
• Zadeh, L.A.: Fuzzy Sets, Information and Control 8, 338-353, 1965.