Fuzzy real lines and dual real lines as poslat topological, uniform, and metric ordered semirings with unity (Q2702350)

The chapter under review is a survey of the author's results related to the fuzzy real line \(\mathbb{R}(L)\) and \(\mathbb{R}\) with their arithmetic operations, including the interplay between arithmetic operations and underlying (\(L\)-)topological and (\(L\)-)uniform structures. Section 1 reviews basic material on the algebraic, topological, and uniform structure of \(\mathbb{R}(L)\). Section 2 summarizes tools necessary to study the construction and continuity of arithmetic operations on \(\mathbb{R}(L)\). With \(L\) a complete chain with an order-reversing involution, \(\mathbb{R}(L)\) becomes an \(L\)-topological and \(L\)-uniform additive monoid (Section 3), while in Section 4, which involves the multiplicative aspect of \(\mathbb{R}(L)\), the latter is exhibited as an \(L\)-topological complete fuzzy hyperfield as defined by [the author, Fuzzy Sets Syst. 15, 285-310 (1985; Zbl 0572.54006)]. Finally, in Section 5, \(\mathbb{R}\) is exhibited as an \(L\)-topological field and an \(L\)-uniform additive group.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].