Sugeno-Weber T-Norms in Fuzzy Logic

• Sugeno–Weber t-norms are a family of one-parameter binary operations on [0,1] that generalize fuzzy set intersections.
• They interpolate between the minimum and algebraic product norms, offering tunable strictness for fuzzy logic and optimization.
• Their rich algebraic structure, dominance relations, and copula properties underpin robust applications in fuzzy systems, matrix integration, and logical inference.

The Sugeno–Weber family of t-norms comprises a one-parameter class of binary operations on the unit interval [0,1] that are central to fuzzy logic, aggregation theory, and fuzzy optimization. Each Sugeno–Weber t-norm is denoted by $T^\lambda$, parametrized by $\lambda \in [0, \infty]$, and defined as a generalized intersection operator for fuzzy sets, offering tunable behavior between the drastic, minimum, and algebraic product norms. These t-norms are widely studied for their algebraic structure, dominance relations, copula properties, and their role in fuzzy optimization and fuzzy system modeling.

1. Definition and Algebraic Structure

Members of the Sugeno–Weber family are defined by the formula: $$T^\lambda(x, y) = \max\left(0,\, \frac{x+y-1+\lambda xy}{1+\lambda}\right),\quad x, y \in [0,1],\quad \lambda \geq 0.$$ The operation is commutative, associative, monotonic, and has $1$ as the neutral element. For $\lambda = 0$ it yields the minimum t-norm: $T^{0}(x, y)=\min(x,y)$, while for large $\lambda$ it approaches the algebraic product $x y$. The operation interpolates between several classical t-norms, providing adjustable "strictness" and nilpotency depending on $\lambda$.

The Sugeno–Weber family includes the following limiting cases:

• $T^0$ : Minimum t-norm $T_M$.
• $T^\infty$ : Corresponds to the drastic product.

The family itself is pointwise ordered: $T^\lambda \geq T^\mu$ if and only if $\lambda \leq \mu$.

2. Dominance Relation and Order Structure

Dominance in the Sugeno–Weber family is characterized by the inequality: $$T^\lambda(T^\mu(x,y), T^\mu(u,v)) \geq T^\mu(T^\lambda(x,u), T^\lambda(y,v)), \quad \forall x, y, u, v \in [0,1].$$ This defines a partial order, denoted $T^\lambda \gg T^\mu$, with the following complete characterization for pairs $(\lambda, \mu)$ where dominance holds (Kauers et al., 2010):

• $\lambda = 0$
• $\mu = \infty$
• $\lambda = \mu$
• $0 < \lambda < \mu \leq 17 + 12\sqrt{2}$
• For $\mu > 17 + 12\sqrt{2}$, $$0 < \lambda \leq \left(\frac{1 - 3\sqrt{\mu}}{3 - \sqrt{\mu}}\right)^2$$

Equivalently,

• $0 < \lambda < \min(\mu,1)$, or
• $0 < \lambda < \mu$ and $$1 + \sqrt{\lambda\mu} \leq 3( \sqrt{\lambda} + \sqrt{\mu} )$$

The dominance relation is transitive in this family: If $T^a \gg T^b$ and $T^b \gg T^c$, then $T^a \gg T^c$. Therefore, this dominance is a proper order relation over $\{T^\lambda: \lambda \in [0, \infty]\}$.

This property is nontrivial; in general, dominance is not transitive for all continuous t-norms. The characterization of dominance utilizes quantifier elimination via Cylindrical Algebraic Decomposition (CAD), exploiting the polynomial structure of the max-operator within Sugeno–Weber norms.

3. Copula, Additive Generator, and Functional Properties

Sugeno–Weber t-norms belong to the class of Archimedean copulas, due to the existence of a convex additive generator (Wu, 2022). This brings the copula-theoretic property of 2-increasingness and makes these norms suitable for probabilistic and statistical aggregation. The convexity of the generator function ensures the validity of rearrangement inequalities and related optimization properties.

Additive generator (one form): $$f(x) = \ln \left(\frac{1 - \lambda x}{1 - x}\right), \quad \lambda \geq 0$$ Such a representation allows constructing Sugeno–Weber t-norms as copulas for combining fuzzy membership distributions.

A key property is the ability to "lift" the structure to type-2 fuzzy set convolutions: If one defines operations of the form $$(f \curlywedge g)(x) = \sup\{f(y) \star g(z): y \vartriangle z = x\}$$, and demands that $\curlywedge$ be a t-norm, then $\star$ must itself be a t-norm—thus convolutional constructions using Sugeno–Weber behavior must originate from valid t-norm seeds (Wu et al., 2019).

4. Role in Fuzzy Optimization and Relational Systems

Sugeno–Weber t-norms are employed as fuzzy composition operators in relational inequalities and fuzzy optimization (Ghodousian et al., 16 Sep 2025). For instance, in lattice-based nonlinear optimization problems, feasible regions are defined by systems: $$\max_j \{T^\lambda(a_{ij}, x_j)\} \leq b_{1i},\qquad \max_j \{T^\lambda(d_{ij}, x_j)\} \geq b_{2i},\qquad x \in [0,1]^n$$ Feasibility and optimality can be checked and computed precisely, with the feasible set decomposing into unions of convex cells, each described by finitely many minimal solutions dictated by the t-norm structure. For selected $\lambda$, the solution space and optimal values respond sensitively to the parameter's influence on the aggregation behavior.

These methods allow for the exact solution of the minimum-maximum component problem under Sugeno–Weber fuzzy constraints.

5. Extended Constructions and Logical Frameworks

In logical inference, Sugeno–Weber t-norms serve as the core conjunction mechanism in $(S,N,T)$-implication systems (Neres et al., 2021). By specifying: $I_{SW}(x,y) = S_{SW}(N( T_{SW}(x, N(y) ) ), N(x) )$ with $N(x)=1-x$ and $S_{SW}$ as the dual t-conorm, the resulting fuzzy implication reflects the parametric behavior and exchange principles of Sugeno–Weber norms. Methods of reversing implication construction allow one to recover t-norms and t-conorms from these systems, under robust monotonicity and boundary conditions.

Further, in the context of extended ordinal sum constructions (Mesiarova-Zemankova, 2015), Sugeno–Weber t-norms appear as distinguished summands in the decomposition of uninorms into minimal components, each with specific generators and continuity properties.

6. Matrix Integration, Majorization, and Lattice Analogues

Sugeno-type traces on matrix algebras (Nagisa et al., 2021) employ max–min operations analogous to Sugeno–Weber t-norm logic: $$v_\alpha(a) = \bigvee_{i=1}^{n} ( x_i(a) \wedge \alpha(A_i) ),\qquad x_1(a) \geq x_2(a) \geq \dots \geq x_n(a)$$ This aggregation mirrors fuzzy conjunction and provides unitary invariance and homogeneity, meaning that the matrix analogues of Sugeno–Weber structures extend to operator theory, majorization, and positivity properties.

On finite atomistic lattices (He et al., 6 Dec 2024), t-norms can be constructed by atomic specification and extended via combinatorial joins. While the Sugeno–Weber family is formulated on $[0,1]$ with smooth parametric structure and continuity, atomistic methods yield families of discrete t-norms, only matching continuous and left-continuous properties in the Boolean case (i.e., minimum t-norm).

7. Rearrangement Inequalities and Aggregation Behavior

The rearrangement inequality holds for Sugeno–Weber t-norms by virtue of their Archimedean copula status (Wu, 2022). For non-decreasing sequences, expressions such as: $$(x_n \otimes y_1) \oplus (x_{n-1} \otimes y_2) \oplus \dots \oplus (x_1 \otimes y_n)$$ are minimized or maximized according to the ordering of the arguments, holding for all pairs and all permutations. These inequalities govern optimal aggregation in fuzzy systems, ensuring consistent bounds and order-respecting behavior under Sugeno–Weber composition.

Usage Context	Sugeno–Weber Role	Key Structural Point
Fuzzy aggregation	Parameterized conjunction	Tunable strictness/nilpotency
Fuzzy optimization	Relational composition, feasibility	Decomposition into convex cells
Matrix integration	Nonlinear trace (max–min logic)	Homogeneity, majorization
Logical inference systems	$(S,N,T)$-implication core	Exchange, monotonicity, reversibility
Rearrangement inequalities	Archimedean copula	Bounds on aggregated output

The Sugeno–Weber family stands out for its flexibility, order-theoretic richness, and structural properties both in continuous and discrete environments. It provides a foundation for a wide range of fuzzy modeling, optimization, and inference mechanisms, with the mathematical specifics of its dominance, copula properties, and aggregation behavior thoroughly classified.