Triangular norm

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 $T_k\,(k \in K)$ be a family of triangular norms and let $\{ (a_k,b_k) : k \in K \}$ be a family of pairwise disjoint open subintervals of the unit interval $[0,1]$ (i.e., $K$ is an at most countable index set). Consider the linear transformations $\phi_k : [a_k,b_k] \rightarrow [0,1]$ given by $$ \phi_k : u \mapsto \frac{u-a_k}{b_k-a_k} \ . $$

Then the function $T : [0,1]^2 \rightarrow [0,1]$ defined by $$ T : (x,y) \mapsto \begin{cases} \phi_k^{-1}(T_k(\phi_k(x),\phi_k(y))) & \text{if}\, (x,y) \in (a_k,b_k)^2 \\ \min(x,y) & \text{otherwise} \end{cases} $$ is a triangular norm, which is called the ordinal sum of the summands $T_k\,(k \in K)$.

The following representations hold ([a1], [a5], [a6]):

A function $T : [0,1]^2 \rightarrow [0,1]$ is a continuous Archimedean triangular norm, i.e., for all $x \in (0,1)$ one has $T(x,x) < x$, if and only if there exists a continuous, strictly decreasing function $f : [0,1] \rightarrow [0,\infty]$ with $f(1) = 0$ such that for all $x,y \in [0,1]$, $$ T(x,y) = f^{-1}(\min(f(x)+f(y),0)) \ . $$

The function $f$ is then called an additive generator of $T$; it is uniquely determined by $T$ up to a positive multiplicative constant.

$T$ is a continuous triangular norm if and only if $T$ 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)