On the mutual definability of classes of generalized fuzzy implications and of classes of generalized negations and S-norms

Given the real functions v: <0,1>/spl rarr/<0,1> and /spl sigma/,/spl pi/: <0,1>/sup 2//spl rarr/<0,1>. First we define a functional operator SIMP where SIMP(/spl sigma/,/spl nu/):<0,1>/sup 2//spl rarr/<0,1> and SIMP(/spl sigma/,/spl nu/) is interpreted as the "S-implication" generated by /spl nu/ and /spl sigma/. Secondly, we define functional operators NEG(/spl pi/):<0,1>/spl rarr/<0,1> and SNOR(/spl pi/):<0,1>/sup 2//spl rarr/<0,1> where NEG(/spl pi/) is interpreted as the "negation" generated by /spl pi/ and SNOR(/spl pi/) is interpreted as the "S-norm" (T-conorm) generated by /spl pi/. We investigate under which assumptions these operators are injective (bijective) and which properties of the "argument functions" are translated into the "value functions". Numerous well-known results on negations, S-norms, and implications can be derived within the framework of this general approach. Further results concern the mutual definability or R-implications and T-norms.