Weightable quasi-metrics related to fuzzy sets

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

1. 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.
2. Bosi, G., Candeal, J.C., Induráin, E., Olóriz, E. and Zudaire, M.: Numerical representations of interval orders, Order 18, 171-190, 2001.
3. Bridges, D.S. and Mehta, G.B.: Representations of Preference Orderings, Springer-Verlag, Berlin-Heidelberg-New York, 1995.
4. 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.
5. 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.
6. 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.
7. Campión, M.J., Induráin, E. and Knoblauch, V.: Topologies generated by nested collections, Bull. Malays. Math. Sci. Soc. 39, 545-561, 2016.
8. 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.

9. Candeal, J.C., Hervés, C. and Induráin, E.: Some results on representation and extension of preferences, J. Math. Econom. 29 (1), 75-81, 1998.
10. Deza, M.M. and Deza E.: Encyclopedia of Distances, (Springer-Verlag, Berlin, 2009).
11. Dijkstra, E.W.: A note on two problems in connection with graphs, Numer. Math. 1, 269- 271, 1959.
12. Induráin, E. and Knoblauch, V.: On topological spaces whose topology is induced by a binary relation, Quaest. Math. 36(1), 47-65, 2013.
13. Künzi, H.P.A. and Vajner V.: Weighted quasi-metrics, Ann. NY Acad. Sci. 728, 64-77, 1994.
14. Matthews, S.G.: Partial metric topology, Ann. NY Acad. Sci. 728, 183-197, 1994.
15. Romaguera, S. and Sanchis, M.: Applications of utility functions defined on quasi-metric spaces, J. Math. Anal. Appl. 283, 219-235, 2003.
16. 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.
17. Schellekens, M.: The Smyth completion: A common foundation for denotational semantics and complexity analysis, Electronic Notes Theoret. Comput. Sci. 1, 535-556, 1995.
18. Seda, A.K.: Quasi-metrics and the Semantics of Logic Programs, Fund. Inform. 29, 97-117, 1997.
19. Sincov, D.M.: Über eine Funktionalgleichung, Arch. Math. Phys. (3) 6, 216-217, 1903.
20. 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.
21. Yi, G.: Continuous extension of preferences, J. Math. Econom. 22, 547-555, 1993.
22. Zadeh, L.A.: Fuzzy Sets, Information and Control 8, 338-353, 1965.