Abstract
Nontrivial examples of objects and morphisms are fundamentally important to establishing the credibility of a new category or discipline such as lattice-dependent or fuzzy topology; and often the justifications of the importance of certain objects and the importance of certain morphisms are intertwined. In [33], we established classes of variable-basis morphisms between different fuzzy real lines and between different dual real lines, but left untouched the issue of the canonicity of these objects. In this chapter, we attempt to demonstrate the canonicity of these spaces stemming from the interplay between arithmetic operations and underlying topological structures. We shall summarize the definitions of fuzzy addition and fuzzy multiplication on the fuzzy real lines and indicate their joint-continuity—along with that of the addition and multiplication on the usual real line—with respect to the underlying poslat topologies, as well as the quasi-uniform and uniform continuity (in the case of fuzzy addition and addition) with respect to the underlying quasi-uniform, uniform, and metric structures. These results not only help establish fuzzy topology w.r.t. objects, but enrich our understanding of traditional arithmetic operations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9 (1978), 613–626.
D. Dubois and H. Prade, Fuzzy real algebra: some results, Fuzzy Sets and Systems 2 (1979), 327–348.
M. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979), 205–230.
T. E. Gantner, R. C. Steinlage, and R. H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978), 547–562.
R. Goetschel, Jr. and W. Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems 10 (1983), 87–99.
J. A. Goguen, The fuzzy Tychonoff Theorem, J. Math. Anal. Appl. 43 (1973), 734–742.
U. Höhle, Probabilitische Metriken auf der Menge der nicht negativen Verteilungsfunktionen, Aequationes Mathematicae 18 (1978), 345–356.
U. Höhle and A. Šostak, Axiomatic foundations of fixed-basis fuzzy topology (Chapter 3 in this Volume).
B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975), 74–79.
B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. 58 (1977), 559–571.
A. J. Klein, α-Closure in fuzzy topology, Rocky Mount. J. Math. 11 (1981) 553–560.
A. J. Klein, Generating fuzzy topologies with semi-closure operators, Fuzzy Sets and Systems 9 (1983), 267–274.
A. J. Klein, Generalizing the L-fuzzy unit interval, Fuzzy Sets and Systems 12 (1984), 271–279.
W. Kotzé, Uniform spaces (Chapter 8 in this Volume).
W. Kotzé and T. Kubiak, Fuzzy topologies of Scott continuous functions and their relationship to the hypergraph functor, Quaestiones Mathemati-cae 15(1992), 175–187.
W. Kotzé and T. Kubiak, Inserting L-fuzzy-valued functions, Mathematische Nachrichten 164(1993), 5–11.
T. Kubiak, Extending continuous L-real valued functions, Math. Japon 31(1986), 875–887.
T. Kubiak, L-fuzzy normal spaces and Tietze Extension Theorem, J. Math. Anal. Appl. 125 (1987), 141–153.
T. Kubiak, The fuzzy unit interval and the Helly space, Math. Japonica 33 (1988), 253–259.
T. Kubiak, The topological modification of the L-fuzzy unit interval, in: “Applications of Category Theory to Fuzzy Subsets”, S. E. Rodabaugh et al., editors, (Kluwer Academic Publishers, Dordrecht 1992).
T. Kubiak, On L-Tychonoff spaces, Fuzzy Sets and Systems 73(1995), 25–53.
T. Kubiak, Separation axioms: extension of mappings and embeddings of spaces (Chapter 6 in this Volume).
R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978), 446–454.
R. Lowen, On (ℝ(L),⊕), Fuzzy Sets and Systems 10 (1983), 203–209.
S. E. Rodabaugh, The Hausdorff separation axiom for fuzzy topological spaces, Topology and its Applications 11 (1980), 319–334.
S. E. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems 8 (1982), 39–51.
S. E. Rodabaugh, Separation axioms and the L-fuzzy real lines, Fuzzy Sets and Systems 11 (1983), 163–183.
S. E. Rodabaugh, Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines, Fuzzy Sets and Systems 15 (1985), 285–310.
S. E. Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl. 129 (1988), 37–70.
S. E. Rodabaugh, Dynamic topologies and their applications to crisp topologies, fuzzification of crisp topologies, and fuzzy topologies on the crisp real line, J. Math. Anal. Appl. 131 (1988), 25–66.
S. E. Rodabaugh, Lowen, para-Lowen, and α-level functors and fuzzy topologies on the crisp real line, J. Math. Anal. Appl. 131 (1988), 157–169.
S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40 (1991), 297–345.
S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology(Chapter 4 in this Volume).
S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies (Chapter 2 in this Volume).
S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and compactifications (Chapter 7 in this Volume).
E. S. Santos, Topology versus fuzzy topology, Preprint, Youngstown State University (1977).
C. K. Wong, Fuzzy topology: product and quotient theorems, J. Math. Anal. Appl. 45 (1974), 512–521.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media New York
About this chapter
Cite this chapter
Rodabaugh, S.E. (1999). Fuzzy Real Lines And Dual Real Lines As Poslat Topological, Uniform, And Metric Ordered Semirings With Unity. In: Höhle, U., Rodabaugh, S.E. (eds) Mathematics of Fuzzy Sets. The Handbooks of Fuzzy Sets Series, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5079-2_12
Download citation
DOI: https://doi.org/10.1007/978-1-4615-5079-2_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7310-0
Online ISBN: 978-1-4615-5079-2
eBook Packages: Springer Book Archive