Idempotents of T-addition of fuzzy numbers

Rok, strany: 1997, 65 - 72

The addition of fuzzy numbers based on a t-norm $T$ is introduced. Special attention is paid to the $T$-addition of $L$–$R$ fuzzy numbers when $T$ is a continuous Archimedean t-norm. The exact output of

$$ S=(a, α, β)_LR\operatornamewithlimits\oplus_T (b, α,β)_LR $$

in some special cases is given, completing a series of results of Fullér and Keresztfalvi, Hong and Hwang, and Mesiar, by a necessary and sufficient condition on $T$, $L$, $R$, ensuring

$$ S(z)= \cases f^[-1](2f(L(\frac{s-z} {2α}))), &for $s-2α \leqq z \leqq s$, f^[-1](2f(R(((z-s) / (2β))) )), &for $s\leqq z\leqq s+2 β$ , 0 , &else , \endcases $$

where $s= a+b$ and $f$ is an additive generator of $T$. The only idempotent in $(\Bbb R,+)$ is the zero. The role of zero for fuzzy numbers with respect to $\operatornamewithlimits\oplus\limits_T$ can be played by corresponding $\operatornamewithlimits\oplus\limits_T$-idempotents. So, e.g., the only $\operatornamewithlimits\oplus\limits_TM$-idempotent is the crisp zero, i.e., $δ_{\{0\}}$ (Dirac function). On the other hand, in the case of $\operatornamewithlimits\oplus\limits_TW$, each fuzzy number $A$ with $A(z)=1$ iff $z=0$ is the $\operatornamewithlimits\oplus\limits_TW$-idempotent. Note that if $T₁\leqq T₂$, then each $\operatornamewithlimits\oplus\limits_T2$-idempotent is a $\operatornamewithlimits\oplus\limits_T1$-idempotent, too.

Among $L$–$R$ fuzzy numbers, the only $\operatornamewithlimits\oplus\limits_T$-idempotents are of the form $(0, α, β)_LR$, where $α >0$, $β>0$ can be chosen arbitrarily, and the shape functions $L$, $R$ are dependent on $T$. The classes of shape functions leading to $\operatornamewithlimits\oplus\limits_T$-idempotents are studied for continuous Archimedean t-norms.

ISO 690:

Marková-Stupňanová, A. 1997. Idempotents of T-addition of fuzzy numbers. In Tatra Mountains Mathematical Publications, vol. 12, no.3, pp. 65-72. 1210-3195.

APA:

Marková-Stupňanová, A. (1997). Idempotents of T-addition of fuzzy numbers. Tatra Mountains Mathematical Publications, 12(3), 65-72. 1210-3195.