# T standard

A T-norm , often also a small t-norm , is a mathematical function that has become important in the field of multi-valued logics , especially in fuzzy logic . The term is derived from the English triangular norm , in German Dreiecksnorm , and comes from the fact that a T-norm describes a triangular-like surface in . ${\ displaystyle \ mathbb {R} ^ {3}}$

## properties

A T-norm is defined on the unit interval [0,1]

${\ displaystyle T: [0.1] \ times [0.1] \ rightarrow [0.1]}$

and must have the following properties (for an exact definition of these properties, see the table on T-Norm and T-Conorm at the end of this article):

The T standard is used for valued logics a generalized Konjunktions - operator to provide. The properties mentioned above are, as it were, the most general properties of such an operator: Associativity and commutativity are self-evident. The monotony guarantees a certain regularity in the structure of the definition and target set. The "1" as a neutral element enables conjunctions, the result of which only depends on one operand.

These properties are used in conjunction with fuzzy sets to simulate the intersection operation.

## T-Conormen

Complementary to T-norms, T-Conormen (also called S-Norms ) are used, as an identifier is usually ⊥ or S:

${\ displaystyle \ bot (a, b) = 1- \ top (1-a, 1-b).}$

With the help of De Morgan's laws , the disjunction or union operation can be derived on the basis of a T-norm, which supplies conjunction or intersection, and a negation .

Generalization: It may be a different than the standard negator

${\ displaystyle \ operatorname {n} (x) = 1-x}$

be used. This generalizes the above relationship to

${\ displaystyle \ bot (a, b) = \ operatorname {n} (\ top (\ operatorname {n} (a), \ operatorname {n} (b))).}$

The minimum requirements for an negator are generally: monotony (falling), n (0) = 1, n (1) = 0.
In this context, however, strict monotony and involutivity n (n ( x )) = x, ie n = n −1 , are required:
The triple is then called the De Morgan triplet. ${\ displaystyle (\ top, \ bot, n)}$

## Common T-Norms and T-Conormen

${\ displaystyle {\ begin {matrix} \ mathrm {\ top _ {min}} (a, b) & = & \ min \ {a, b \} & \ mathrm {\ bot _ {max}} (a, b) & = & \ max \ {a, b \} \\\\\ mathrm {\ top _ {Luka}} (a, b) & = & \ max \ {0, a + b-1 \} & \ mathrm {\ bot _ {Luka}} (a, b) & = & \ min \ {a + b, 1 \} \\\\\ mathrm {\ top _ {prod}} (a, b) & = & a \ cdot b & \ mathrm {\ bot _ {sum}} (a, b) & = & a + ba \ cdot b \\\\\ mathrm {\ top _ {- 1}} (a, b) & = & \ left \ {{\ begin {matrix} a, & {\ mbox {if}} b = 1 \\ b, & {\ mbox {if}} a = 1 \\ 0, & {\ mbox {otherwise}} \ end {matrix}} \ right. & \ mathrm {\ bot _ {- 1}} (a, b) & = & \ left \ {{{\ begin {matrix} a, & {\ mbox {if}} b = 0 \\ b, & {\ mbox {if}} a = 0 \\ 1, & {\ mbox {otherwise}} \ end {matrix}} \ right. \ End {matrix}}}$

The specified T-norms are each related to the standard negation N (x) = 1-x to the corresponding T-norm dual, i.e. linked via De Morgan's laws. With other involutive negations, other T-conorms generally also result.

The former is the most widely used because of its simplicity and its properties mentioned below. The 3rd T-Norm and its T-Conorm come from the calculation of probability . The following relationships also apply:

${\ displaystyle {\ begin {matrix} \ mathrm {\ top _ {- 1}} (a, b) & \ leq & \ top (a, b) & \ leq & \ mathrm {\ top _ {min}} (a, b) \\\ mathrm {\ bot _ {max}} (a, b) & \ leq & \ bot (a, b) & \ leq & \ mathrm {\ bot _ {- 1}} (a , b) \ end {matrix}}}$
This means that the drastic T-norm (T -1 ) is the smallest and the minimum T-norm the largest. The opposite applies to the T-Conorm. T (a, b) or ⊥ (a, b) stands for any T-Norm or T-Conorm.

## Relationship between T-Norm and T-Conorm

Due to the De Morgan laws already mentioned, the following complementary relationships arise:

1-⊥ (a, b) = T (1-a, 1-b) and 1-T (a, b) = ⊥ (1-a, 1-b)

The above axioms for T-norms correspond to the following conditions for a T-conorm:

T standard T-Conorm
Zero element: T (0, a) = T (a, 0) = 0 ⊥ (a, 1) = ⊥ (1, a) = 1
Neutral element: T (a, 1) = T (1, a) = a ⊥ (0, a) = ⊥ (a, 0) = a
Associativity: T (a, T (b, c)) = T (T (a, b), c) ⊥ (a, ⊥ (b, c)) = ⊥ (⊥ (a, b), c)
Commutativity: T (a, b) = T (b, a) ⊥ (a, b) = ⊥ (b, a)
Monotony: a ≤ b ⇒ T (a, c) ≤ T (b, c) a ≤ b ⇒ ⊥ (a, c) ≤ ⊥ (b, c)

These relationships apply not only to the standard negator, but to any De Morgan triplet.

## Relationship between T-norm and copula

A T-norm has the positive rectangle property if : ${\ displaystyle a_ {1} \ leq a_ {2}, b_ {1} \ leq b_ {2}}$

${\ displaystyle \ mathrm {\ top} (a_ {1}, b_ {1}) + \ mathrm {\ top} (a_ {2}, b_ {2}) - \ mathrm {\ top} (a_ {1} , b_ {2}) - \ mathrm {\ top} (a_ {2}, b_ {1}) \ geq 0}$

Every T-norm with a positive rectangular property is a bivariate copula (see Grabisch et al. 2009). Copulae of the above examples are at the same time, but not. ${\ displaystyle \ mathrm {\ top _ {min}}, \ mathrm {\ top _ {Luka}}, \ mathrm {\ top _ {prod}}}$${\ displaystyle \ mathrm {\ top _ {- 1}}}$