Triangular norm

In mathematics, triangular norm (or just t-norm) is binary operation T(x,y) on the interval [0,1] which is:

1. associative,
2. commutative,
3. monotonic function in both operands,
4. $T(x,1)=x$ and $T(x,0)=0$ for all $x\in [0,1]$.

Important triangular norms

• Gödel t-norm: $T_{G}(x,y)=min(x,y)$
• Product t-norm: $T_{\Pi }(x,y)=xy$
• Łukasziewicz t-norm: $T_{L}(x,y)=max(x+y-1,0)$
• Degenerated t-norm: $T_{D}(x,y)=0$ for $x,y\in (0,1)$

Note that always $T_{G}(x,y)\geq T_{\Pi }(x,y)\geq T_{L}(x,y)\geq T_{D}(x,y)$.

Residuum of the triangular norm

Residuum is partial inversion of the operation. It is defined as $R_{T}(x,y)=max\{z|T(x,z)\leq y\}$. The residuum has the following properties:

• $R_{T}(x,y)=1$ if and only if $x\leq y$,
• $R_{T}(x,y)\geq y$,
• $R_{T}(x,1)=1$,
• $R_{T}(1,x)=1$,
• if the T is continuous function, then $x=T(y,R_{T}(y,x))$.

See also

• Copula (statistics)
• Fuzzy logic
• Possibility theory

References

E.P. Klement, R. Mesiar, E. Pap: Triangular norms. Kluwer 2000.