Difference between revisions of "Triangular norm"
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The following are the four basic triangular norms, together with their dual triangular co-norms: | The following are the four basic triangular norms, together with their dual triangular co-norms: | ||
− | i) the minimum | + | i) the minimum $T_{\mathrm{M}}$ and maximum $S_{\mathrm{M}}$, given by |
− | + | $$ | |
− | + | T_{\mathrm{M}}(x,y) = \min(x,y) \ ; | |
− | + | $$ | |
− | + | $$ | |
− | + | S_{\mathrm{M}}(x,y) = \max(x,y) \ . | |
− | ii) the product | + | $$ |
− | + | ii) the product $T_{\mathrm{P}}$ and probabilistic sum $S_{\mathrm{P}}$, given by | |
− | + | $$ | |
− | + | T_{\mathrm{P}}(x,y) = x \cdot y \ ; | |
− | + | $$ | |
− | + | $$ | |
− | iii) the Lukasiewicz triangular norm | + | S_{\mathrm{P}}(x,y) = x+y - x\cdot y \ . |
− | + | $$ | |
− | + | iii) the Lukasiewicz triangular norm $T_{\mathrm{L}}$ and Lukasiewicz triangular co-norm $S_{\mathrm{L}}$, given by | |
− | + | $$ | |
− | + | T_{\mathrm{L}}(x,y) = \max(x+y-1,0) \ ; | |
− | + | $$ | |
− | iv) the weakest triangular norm (or drastic product) | + | $$ |
− | + | S_{\mathrm{L}}(x,y) = \min(x+y,1) \ . | |
− | + | $$ | |
− | + | iv) the weakest triangular norm (or drastic product) $T_{\mathrm{D}}$ and strongest triangular co-norm $S_{\mathrm{D}}$, given by | |
− | + | $$ | |
− | + | T_{\mathrm{D}}(x,y) = \begin{cases} y & \text{if}\ x = 1 \\ x & \text{if}\, y = 1 \\ 0 & \text{otherwise} \end{cases} \ ; | |
+ | $$ | ||
+ | $$ | ||
+ | S_{\mathrm{D}}(x,y) = \begin{cases} y & \text{if}\ x = 0 \\ x & \text{if}\, y = 0 \\ 1 & \text{otherwise} \end{cases} \ . | ||
+ | $$ | ||
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017048.png" /> be a family of triangular norms and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017049.png" /> be a family of pairwise disjoint open subintervals of the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017050.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017051.png" /> is an at most countable index set). Consider the linear transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017052.png" /> given by | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017048.png" /> be a family of triangular norms and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017049.png" /> be a family of pairwise disjoint open subintervals of the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017050.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017051.png" /> is an at most countable index set). Consider the linear transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017052.png" /> given by | ||
Revision as of 19:13, 27 December 2015
t-norm
A binary operation on the unit interval $[0,1]$, i.e., a function $T : [0,1]^2 \rightarrow [0,1]$ such that for all $x,y,z \in [0,1]$ the following four axioms are satisfied:
T1) (commutativity) $T(x,y) = T(y,x)$;
T2) (associativity) $T(x,T(y,z)) = T(T(x,y),z)$;
T3) (monotonicity) $T(x,y) \le T(x,z)$ whenever $y \le z$;
T4) (boundary condition) $T(x,1) = x$.
If $T$ is a triangular norm, then its dual triangular co-norm $S$ is given by $$ S(x,y) = 1 - T(1-x,1-y) \ . $$
A function $T : [0,1]^2 \rightarrow [0,1]$ is a triangular norm if and only if $([0,1], T, {\le})$ is a fully ordered commutative semi-group (cf. [a3] and $o$-group) with neutral element $1$ and annihilator $0$, where ${\le}$ is the usual order on $[0,1]$.
For each $I$-semi-group $([a,b],{\star})$, i.e. a semi-group in which the binary associative operation $\star$ on the closed subinterval $[a,b]$ of the extended real line is continuous and one of the boundary points of $[a,b]$ acts as a neutral element and the other one as an annihilator ([a6], [a7]), there exists a continuous triangular norm $T$ or a continuous triangular co-norm $S$ such that the linear transformation $\phi : [a,b] \rightarrow [0,1]$ given by $$ \phi : x \mapsto \frac{x-a}{b-a} $$ is an isomorphism between $([a,b],{\star})$ and either $([0,1],T)$ or $([0,1],S)$.
The following are the four basic triangular norms, together with their dual triangular co-norms:
i) the minimum $T_{\mathrm{M}}$ and maximum $S_{\mathrm{M}}$, given by
$$
T_{\mathrm{M}}(x,y) = \min(x,y) \ ;
$$
$$
S_{\mathrm{M}}(x,y) = \max(x,y) \ .
$$
ii) the product $T_{\mathrm{P}}$ and probabilistic sum $S_{\mathrm{P}}$, given by
$$
T_{\mathrm{P}}(x,y) = x \cdot y \ ;
$$
$$
S_{\mathrm{P}}(x,y) = x+y - x\cdot y \ .
$$
iii) the Lukasiewicz triangular norm $T_{\mathrm{L}}$ and Lukasiewicz triangular co-norm $S_{\mathrm{L}}$, given by
$$
T_{\mathrm{L}}(x,y) = \max(x+y-1,0) \ ;
$$
$$
S_{\mathrm{L}}(x,y) = \min(x+y,1) \ .
$$
iv) the weakest triangular norm (or drastic product) $T_{\mathrm{D}}$ and strongest triangular co-norm $S_{\mathrm{D}}$, given by
$$
T_{\mathrm{D}}(x,y) = \begin{cases} y & \text{if}\ x = 1 \\ x & \text{if}\, y = 1 \\ 0 & \text{otherwise} \end{cases} \ ;
$$
$$
S_{\mathrm{D}}(x,y) = \begin{cases} y & \text{if}\ x = 0 \\ x & \text{if}\, y = 0 \\ 1 & \text{otherwise} \end{cases} \ .
$$
Let be a family of triangular norms and let
be a family of pairwise disjoint open subintervals of the unit interval
(i.e.,
is an at most countable index set). Consider the linear transformations
given by
![]() |
Then the function defined by
![]() |
is a triangular norm, which is called the ordinal sum of the summands ,
.
The following representations hold ([a1], [a5], [a6]):
A function is a continuous Archimedean triangular norm, i.e., for all
one has
, if and only if there exists a continuous, strictly decreasing function
with
such that for all
,
![]() |
The function is then called an additive generator of
; it is uniquely determined by
up to a positive multiplicative constant.
is a continuous triangular norm if and only if
is an ordinal sum whose summands are continuous Archimedean triangular norms.
Triangular norms are applied in many fields, such as probabilistic metric spaces [a9], [a4], fuzzy sets, fuzzy logics and their applications [a4], the theory of generalized measures [a2], [a8], functional equations [a1] and in non-linear differential and difference equations (see [a4], [a8]).
References
[a1] | J. Aczél, "Lectures on functional equations and their applications" , Acad. Press (1969) |
[a2] | D. Butnariu, E.P. Klement, "Triangular norm-based measures and games with fuzzy coalitions" , Kluwer Acad. Publ. (1993) |
[a3] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) Zbl 0137.02001 |
[a4] | E.P. Klement, R. Mesiar, E. Pap, "Triangular norms" Trends in Logic--Studia Logica Library 8 Kluwer Academic ISBN 0-7923-6416-3 Zbl 0972.03002 |
[a5] | C.M. Ling, "Representation of associative functions" Publ. Math. Debrecen , 12 (1965) pp. 189–212 |
[a6] | P.S. Mostert, A.L. Shields, "On the structure of semigroups on a compact manifold with boundary" Ann. of Math. , 65 (1957) pp. 117–143 |
[a7] | A.B. Paalman-de Miranda, "Topological semigroups" , Tracts , 11 , Math. Centre Amsterdam (1970) |
[a8] | E. Pap, "Null-additive set functions" , Kluwer Acad. Publ. &Ister Sci. (1995) |
[a9] | B. Schweizer, A. Sklar, "Probabilistic metric spaces" , North-Holland (1983) |
Triangular norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangular_norm&oldid=37117