Classification of t-norms

A t-norm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]². (Similarly for left- and right-continuity.)

A t-norm is called strict if it is continuous and strictly monotone.

A t-norm is called nilpotent if it is continuous and each x in the open interval (0, 1) is nilpotent, that is, there is a natural number n such that x ${\displaystyle *}$ ... ${\displaystyle *}$ x (n times) equals 0.

A t-norm ${\displaystyle *}$ is called Archimedean if it has the Archimedean property, that is, if for each x, y in the open interval (0, 1) there is a natural number n such that x ${\displaystyle *}$ ... ${\displaystyle *}$ x (n times) is less than or equal to y.

The usual partial ordering of t-norms is pointwise, that is,

T₁ ≤ T₂   if   T₁(a, b) ≤ T₂(a, b) for all a, b in [0, 1].

As functions, pointwise larger t-norms are sometimes called stronger than those pointwise smaller. In the semantics of fuzzy logic, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents.

Properties of continuous t-norms

Although real functions of two variables can be continuous in each variable without being continuous on [0, 1]², this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions f_y(x) = T(x, y) are continuous for each y in [0, 1]. Analogous theorems hold for left- and right-continuity of a t-norm.

A continuous t-norm is Archimedean if and only if 0 and 1 are its only idempotents.

A continuous Archimedean t-norm is strict if 0 is its only nilpotent element; otherwise it is nilpotent. By definition, moreover, a continuous Archimedean t-norm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that

${\displaystyle \top (x,y)=f^{-1}(\top _{\mathrm {Luk} }(f(x),f(y))).}$

If on the other hand it is the case that there are no nilpotent elements of T, the t-norm is isomorphic to the product t-norm. In other words, all nilpotent t-norms are isomorphic, the Łukasiewicz t-norm being their prototypical representative; and all strict t-norms are isomorphic, with the product t-norm as their prototypical example. The Łukasiewicz t-norm is itself isomorphic to the product t-norm undercut at 0.25, i.e., to the function p(x, y) = max(0.25, x · y) on [0.25, 1]².

For each continuous t-norm, the set of its idempotents is a closed subset of [0, 1]. Its complement—the set of all elements that are not idempotent—is therefore a union of countably many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such x, y that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm. The theorem can also be formulated as follows:

A t-norm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, Łukasiewicz, and product t-norm.

A similar characterization theorem for non-continuous t-norms is not known (not even for left-continuous ones), only some non-exhaustive methods for the construction of t-norms have been found.

Basic properties of residua

If ${\displaystyle \Rightarrow }$ is the residuum of a left-continuous t-norm ${\displaystyle \top }$, then

${\displaystyle (x\Rightarrow y)=\sup\{z\mid \top (z,x)\leq y\}.}$

Consequently, for all x, y in the unit interval,

${\displaystyle (x\Rightarrow y)=1}$ if and only if ${\displaystyle x\leq y}$

and

${\displaystyle (1\Rightarrow y)=y.}$

If ${\displaystyle *}$ is a left-continuous t-norm and ${\displaystyle \Rightarrow }$ its residuum, then

${\displaystyle {\begin{array}{rcl}\min(x,y)&\geq &x*(x\Rightarrow y)\\\max(x,y)&=&\min((x\Rightarrow y)\Rightarrow y,(y\Rightarrow x)\Rightarrow x).\end{array}}}$

If ${\displaystyle *}$ is continuous, then equality holds in the former.

Residua of prominent left-continuous t-norms

If x ≤ y, then R(x, y) = 1 for any residuum R. The following table therefore gives the values of prominent residua only for x > y.

Residuum of the	Name	Value for x > y	Graph
Minimum t-norm	Standard Gödel implication	y	Standard Gödel implication. The function is discontinuous at the line y = x < 1.
Product t-norm	Goguen implication	y / x	Goguen implication. The function is discontinuous at the point x = y = 0.
Łukasiewicz t-norm	Standard Łukasiewicz implication	1 – x + y	Standard Łukasiewicz implication.
Nilpotent minimum	Fodor implication	max(1 – x, y)	Residuum of the nilpotent minimum. The function is discontinuous at the line 0 < y = x < 1.

Examples of t-conorms

Important t-conorms are those dual to prominent t-norms:

Graph of the maximum t-conorm (3D and contours)

• Maximum t-conorm ${\displaystyle \bot _{\mathrm {max} }(a,b)=\max\{a,b\}}$, dual to the minimum t-norm, is the smallest t-conorm (see the properties of t-conorms below). It is the standard semantics for disjunction in Gödel fuzzy logic and for weak disjunction in all t-norm based fuzzy logics.

Graph of the probabilistic sum

• Probabilistic sum ${\displaystyle \bot _{\mathrm {sum} }(a,b)=a+b-a\cdot b=1-(1-a)\cdot (1-b)}$ is dual to the product t-norm. In probability theory it expresses the probability of the union of independent events. It is also the standard semantics for strong disjunction in such extensions of product fuzzy logic in which it is definable (e.g., those containing involutive negation).

Graph of the bounded sum t-conorm

• Bounded sum ${\displaystyle \bot _{\mathrm {Luk} }(a,b)=\min\{a+b,1\}}$ is dual to the Łukasiewicz t-norm. It is the standard semantics for strong disjunction in Łukasiewicz fuzzy logic.

Graph of the drastic t-conorm. The function is discontinuous at the lines 1 > x = 0 and 1 > y = 0.

• Drastic t-conorm

${\displaystyle \bot _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=0\\a&{\mbox{if }}b=0\\1&{\mbox{otherwise,}}\end{cases}}}$

dual to the drastic t-norm, is the largest t-conorm (see the properties of t-conorms below).

Graph of the nilpotent maximum. The function is discontinuous at the line 0 < x = 1 – y < 1.

• Nilpotent maximum, dual to the nilpotent minimum:

${\displaystyle \bot _{\mathrm {nM} }(a,b)={\begin{cases}\max(a,b)&{\mbox{if }}a+b<1\\1&{\mbox{otherwise.}}\end{cases}}}$

Graph of the Einstein sum

• Einstein sum (compare the velocity-addition formula under special relativity)

${\displaystyle \bot _{\mathrm {H} _{2}}(a,b)={\frac {a+b}{1+ab}}}$

is a dual to one of the Hamacher t-norms.

Properties of t-conorms

Many properties of t-conorms can be obtained by dualizing the properties of t-norms, for example:

• For any t-conorm ⊥, the number 1 is an annihilating element: ⊥(a, 1) = 1, for any a in [0, 1].
• Dually to t-norms, all t-conorms are bounded by the maximum and the drastic t-conorm:

${\displaystyle \mathrm {\bot _{max}} (a,b)\leq \bot (a,b)\leq \bot _{\mathrm {D} }(a,b)}$, for any t-conorm ${\displaystyle \bot }$ and all a, b in [0, 1].

Further properties result from the relationships between t-norms and t-conorms or their interplay with other operators, e.g.:

• A t-norm T distributes over a t-conorm ⊥, i.e.,

T(x, ⊥(y, z)) = ⊥(T(x, y), T(x, z)) for all x, y, z in [0, 1],

if and only if ⊥ is the maximum t-conorm. Dually, any t-conorm distributes over the minimum, but not over any other t-norm.

Non-standard negators

A negator ${\displaystyle n\colon [0,1]\to [0,1]}$ is a monotonically decreasing mapping such that ${\displaystyle n(0)=1}$ and ${\displaystyle n(1)=0}$. A negator n is called

• strict in case of strict monotonocity, and
• strong if it is strict and involutive, that is, ${\displaystyle n(n(x))=x}$ for all ${\displaystyle x}$ in [0, 1].

The standard (canonical) negator is ${\displaystyle n(x)=1-x,\ x\in [0,1]}$, which is both strict and strong. As the standard negator is used in the above definition of a t-norm/t-conorm pair, this can be generalized as follows:

A De Morgan triplet is a triple (T,⊥,n) such that^[1]

1. T is a t-norm
2. ⊥ is a t-conorm according to the axiomatic definition of t-conorms as mentioned above
3. n is a strong negator
4. ${\displaystyle \forall a,b\in [0,1]\colon \ n({\perp }(a,b))=\top (n(a),n(b))}$.