Further results on fuzzy negations and implications induced by fuzzy conjunctions and disjunctions

In this article, we deeply investigate some properties of fuzzy negations induced from fuzzy conjunctions (resp. disjunctions), which are then applied to characterizing the fuzzy negations. We further use the obtained characterization of fuzzy negations to explore some properties of $(D,N)$-implications generated from fuzzy disjunctions and negations. We finally describe $(D,N)$-implications (resp. continuous $(D,N)$-implications) generated from fuzzy disjunctions and negations.

💡 Research Summary

The paper investigates the deep interrelations among fuzzy conjunctions, fuzzy disjunctions, fuzzy negations, and fuzzy implications, focusing on the class of (D,N)-implications generated from fuzzy disjunctions (D) and fuzzy negations (N). The authors begin by recalling the standard definitions of t‑norms, t‑conorms, fuzzy negations, and fuzzy implications, and then introduce the notion of a “natural negation” induced by a fuzzy conjunction C or a fuzzy disjunction D. For a conjunction C, the natural negation N_C is defined as the supremum of all t such that C(x,t)=0; for a disjunction D, the natural negation N_D is the infimum of all t such that D(x,t)=1.

The first major contribution is a thorough analysis of the properties of these induced negations. The authors prove that if C is left‑continuous, then C(x,N_C(x))=0 and N_C can be expressed as the maximal t satisfying C(x,t)=0; similarly, if D is right‑continuous, then D(x,N_D(x))=1 and N_D is the minimal t with that property. They also show that left‑continuity of C (or right‑continuity of D) guarantees the continuity of the induced negation.

A central result is the equivalence among four important characteristics of the induced negations when the underlying conjunction or disjunction is commutative: (i) strict monotonicity, (ii) continuity, (iii) strictness (i.e., non‑increasing and strictly decreasing on (0,1)), and (iv) being a strong negation (an involution). The paper demonstrates that any one of these properties implies the others, thereby unifying several previously separate conditions found in the literature. Counter‑examples are provided to illustrate that commutativity is essential for these equivalences.

Building on this foundation, the authors turn to (D,N)-implications, which are defined in direct analogy with the classical equivalence p→q ≡ ¬p ∨ q: I_{D,N}(x,y)=D(N(x),y). They verify that these implications satisfy the standard axioms of fuzzy implications (monotonicity in the first argument, monotonicity in the second, boundary conditions I(0,0)=I(1,1)=1, I(1,0)=0). Moreover, when N is a strong (hence continuous) negation and D is a commutative, continuous disjunction, the resulting (D,N)-implication enjoys additional desirable properties: the neutral property I(1,y)=y, the exchange principle I(x,I(y,z))=I(y,I(x,z)), and the ordering property I(x,y)=1 iff x≤y.

The paper further refines the description of continuous (D,N)-implications by imposing a mild strengthening on the disjunction: D must be left‑ and right‑continuous and satisfy D(x,0)=x and D(0,y)=y. Under these conditions, the implication can be expressed in a closed form as I_{D,N}(x,y)=N^{-1}(D(N(x),y)). This representation shows that (D,N)-implications encompass, as special cases, the well‑known (S,N)-implications (where S is a t‑conorm) and (U,N)-implications (where U is a uninorm), but also generate a broader family that is not captured by residual implications.

In the concluding section, the authors summarize their contributions: (1) a unified characterization of natural negations induced by fuzzy conjunctions and disjunctions, (2) the establishment of continuity, strictness, and strongness as equivalent for such negations under commutativity, and (3) a comprehensive description of (D,N)-implications, including conditions for continuity and the derivation of an explicit formula. They suggest that these results lay a solid theoretical groundwork for future work in fuzzy decision‑making, fuzzy control, and data‑driven fuzzy modeling, where flexible yet well‑behaved implication operators are essential.