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⟩→⟨0,1⟩ and σ,π: ⟨0,1⟩²→⟨0,1⟩. First we define a functional operator SIMP where SIMP(σ,ν):⟨0,1⟩²→⟨0,1⟩ and SIMP(σ,ν) is interpreted as the “S-implication” generated by ν and σ. Secondly, we define functional operators NEG(π):⟨0,1⟩→⟨0,1⟩ and SNOR(π):⟨0,1⟩²→⟨0,1⟩ where NEG(π) is interpreted as the “negation” generated by π and SNOR(π) is interpreted as the “S-norm” (T-conorm) generated by π. 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