Fuzzy Real Lines And Dual Real Lines As Poslat Topological, Uniform, And Metric Ordered Semirings With Unity

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.