T-norms¶
A triangular norm (T-norm) is a binary operation \(T:[0,1]^2\rightarrow[0,1]\) used to model fuzzy conjunction. A T-norm is commutative, associative, monotone in both arguments, and satisfies \(T(a,1)=a\).
frsutils uses T-norms both to aggregate feature-level similarities and in
fuzzy-rough upper approximations. All implementations accept scalar or
array-valued inputs through NumPy-compatible backends.
Implemented T-norms¶
| Name | Formula | Parameters | Registered aliases |
|---|---|---|---|
| Minimum | \(\min(a,b)\) | None | minimum, min, goedel, standardintersection |
| Product | \(ab\) | None | product, prod, algebraic |
| Łukasiewicz | \(\max(0,a+b-1)\) | None | lukasiewicz, luk, bounded, boundeddifference |
| Drastic product | \(a\) if \(b=1\); \(b\) if \(a=1\); otherwise \(0\) | None | drastic, drasticproduct |
| Einstein product | \(ab/(2-a-b+ab)\) | None | einstein, einsteinproduct |
| Hamacher product | \(0\) when \(a=b=0\); otherwise \(ab/(a+b-ab)\) | None | hamacher, hamacherproduct |
| Nilpotent minimum | \(\min(a,b)\) if \(a+b>1\); otherwise \(0\) | None | nilpotent, nilpotentminimum |
| Yager | \(1-\min\{1,[(1-a)^p+(1-b)^p]^{1/p}\}\) | \(p>0\) | yager, yg |
The formulas above are the exact conventions implemented by the library. For array reduction, associative T-norms are applied along the aggregation axis; the Yager implementation uses the equivalent multi-argument expression.
References¶
- Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular Norms. Springer. https://doi.org/10.1007/978-94-015-9540-7
- Radzikowska, A. M., & Kerre, E. E. (2002). A comparative study of fuzzy rough sets. Fuzzy Sets and Systems, 126(2), 137–155. https://doi.org/10.1016/S0165-0114(01)00032-X