Fuzzy addition in the Lowen fuzzy real line
For L a complete chain with order-reversing involution \(\alpha\) \(\to \alpha '\), \textit{S. E. Rodabaugh} [Fuzzy Sets Syst. 8, 39-52 (1982; Zbl 0508.54002)] defined a fuzzy addition \(\oplus\) in the Hutton fuzzy real line \({\mathbb{R}}(L)\) and studied some of its properties. Subsequently \textit{R. Lowen} [Fuzzy Sets Syst. 10, 203-209 (1983; Zbl 0527.54006)] made a considerably simpler description of \(\oplus\) and gave a direct proof of the continuity of \(\oplus\). In this paper, we introduce the fuzzy addition \(\oplus\) in the Lowen fuzzy real line \(\mu_ <({\mathbb{R}})\) [\textit{R. Lowen}, Manuscr. Math. 39, 293-309 (1985; Zbl 0562.54010)]. We give several equivalent descriptions of \(\oplus\) which may be used for investigating some properties of \(\oplus\). Moreover, we prove the closedness of \(\oplus\) in the four important subspaces \({\mathcal A}_ <({\mathbb{R}})\), \({\mathcal F}_ <({\mathbb{R}})\), \(\mu_ <^{{\mathbb{Z}}}({\mathbb{R}})\), \(\mu_ <^{{\mathbb{Q}}}({\mathbb{R}})\) of \(\mu_{<}({\mathbb{R}})\). This implies, in particular, that the sum of two fuzzy rational numbers is fuzzy rational.
scientific article; zbMATH DE number 4071876