View source for 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. [[#References|[a3]]] and [[O-group|$o$-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 ([[#References|[a6]]], [[#References|[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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017032.png" /> and maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017033.png" />, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017035.png" /></td> </tr></table>

ii) the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017036.png" /> and probabilistic sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017037.png" />, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017038.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017039.png" /></td> </tr></table>

iii) the Lukasiewicz triangular norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017040.png" /> and Lukasiewicz triangular co-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017041.png" />, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017042.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017043.png" /></td> </tr></table>

iv) the weakest triangular norm (or drastic product) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017044.png" /> and strongest triangular co-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017045.png" />, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017046.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017047.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017053.png" /></td> </tr></table>

Then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017054.png" /> defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017055.png" /></td> </tr></table>

is a triangular norm, which is called the ordinal sum of the summands <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017057.png" />.

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

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017058.png" /> is a continuous Archimedean triangular norm, i.e., for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017059.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017060.png" />, if and only if there exists a continuous, strictly decreasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017061.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017062.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017063.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017064.png" /></td> </tr></table>

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017065.png" /> is then called an additive generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017066.png" />; it is uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017067.png" /> up to a positive multiplicative constant.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017068.png" /> is a continuous triangular norm if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120170/t12017069.png" /> is an ordinal sum whose summands are continuous Archimedean triangular norms.

Triangular norms are applied in many fields, such as probabilistic metric spaces [[#References|[a9]]], [[#References|[a4]]], fuzzy sets, fuzzy logics and their applications [[#References|[a4]]], the theory of generalized measures [[#References|[a2]]], [[#References|[a8]]], functional equations [[#References|[a1]]] and in non-linear differential and difference equations (see [[#References|[a4]]], [[#References|[a8]]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Aczél,   "Lectures on functional equations and their applications" , Acad. Press  (1969)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Butnariu,   E.P. Klement,   "Triangular norm-based measures and games with fuzzy coalitions" , Kluwer Acad. Publ.  (1993)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963) {{ZBL|0137.02001}}</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  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}}
</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  C.M. Ling,   "Representation of associative functions"  ''Publ. Math. Debrecen'' , '''12'''  (1965)  pp. 189–212</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  A.B. Paalman-de Miranda,   "Topological semigroups" , ''Tracts'' , '''11''' , Math. Centre Amsterdam  (1970)</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Pap,   "Null-additive set functions" , Kluwer Acad. Publ. &amp;Ister Sci.  (1995)</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  B. Schweizer,   A. Sklar,   "Probabilistic metric spaces" , North-Holland  (1983)</TD></TR>
</table>

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