In
mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of
binary operation used in the framework of
probabilistic metric space
In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers , but in distribution functions.
Let ''D+'' be the set of all probability distributio ...
s and in
multi-valued logic
Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...
, specifically in
fuzzy logic. A t-norm generalizes
intersection in a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
and
conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
in
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
. The name ''triangular norm'' refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
of ordinary
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s.
Definition
A t-norm is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
T:
, 1×
, 1→
, 1that satisfies the following properties:
*
Commutativity: T(''a'', ''b'') = T(''b'', ''a'')
*
Monotonicity
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
: T(''a'', ''b'') ≤ T(''c'', ''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d''
*
Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
: T(''a'', T(''b'', ''c'')) = T(T(''a'', ''b''), ''c'')
* The number 1 acts as
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
: T(''a'', 1) = ''a''
Since a t-norm is a
binary algebraic operation on the interval
, 1 infix algebraic notation is also common, with the t-norm usually denoted by
.
The defining conditions of the t-norm are exactly those of a
partially ordered
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
abelian monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ar ...
on the real unit interval
, 1 '' (Cf.
ordered group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' ...
.)'' The monoidal operation of any partially ordered abelian monoid ''L'' is therefore by some authors called a ''triangular norm on L''.
Classification of t-norms
A t-norm is called ''continuous'' if it is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
as a function, in the usual interval topology on
, 1sup>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
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
, that is, there is a natural number ''n'' such that ''x''
...
''x'' (''n'' times) equals 0.
A t-norm
is called ''Archimedean'' if it has the
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typica ...
, that is, if for each ''x'', ''y'' in the open interval (0, 1) there is a natural number ''n'' such that ''x''
...
''x'' (''n'' times) is less than or equal to ''y''.
The usual partial ordering of t-norms is
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
, that is,
: T
1 ≤ T
2 if T
1(''a'', ''b'') ≤ T
2(''a'', ''b'') for all ''a'', ''b'' in
, 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.
Prominent examples
*Minimum t-norm
also called the Gödel t-norm, as it is the standard semantics for conjunction in
Gödel fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest t-norm (see the
properties of t-norms
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy and ...
below).
*Product t-norm
(the ordinary product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in
product fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
. It is a strict Archimedean t-norm.
*Łukasiewicz t-norm
The name comes from the fact that the t-norm is the standard semantics for strong conjunction in
Łukasiewicz fuzzy logic
Łukasiewicz is a Polish surname. It comes from the given name Łukasz (Lucas). It is found across Poland, particularly in central regions. It is related to the surnames Łukaszewicz and Lukashevich.
People
* Antoni Łukasiewicz (born 1983 ...
. It is a nilpotent Archimedean t-norm, pointwise smaller than the product t-norm.
*Drastic t-norm
::
:The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the
properties of t-norms
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy and ...
below). It is a right-continuous Archimedean t-norm.
*Nilpotent minimum
::
:is a standard example of a t-norm that is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm.
*Hamacher product
::
:is a strict Archimedean t-norm, and an important representative of the parametric classes of
Hamacher t-norms and
Schweizer–Sklar t-norms.
Properties of t-norms
The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm:
:
for any t-norm
and all ''a'', ''b'' in
, 1
For every t-norm T, the number 0 acts as null element: T(''a'', 0) = 0 for all ''a'' in
, 1
A t-norm T has
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
s if and only if it has
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval
, ''a''or [0, ''a''), for some ''a'' in
, 1
Properties of continuous t-norms
Although real functions of two variables can be continuous in each variable without being continuous on
, 1sup>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
, 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
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
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
:
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
.25, 1sup>2.
For each continuous t-norm, the set of its idempotents is a closed subset of
, 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
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
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.
Residuum
For any left-continuous t-norm
, there is a unique binary operation
on
, 1such that
:
if and only if
for all ''x'', ''y'', ''z'' in
, 1 This operation is called the ''residuum'' of the t-norm. In prefix notation, the residuum of a t-norm
is often denoted by
or by the letter R.
The interval
, 1equipped with a t-norm and its residuum forms a
residuated lattice In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice ''x'' ≤ ''y'' and a monoid ''x''•''y'' which admits operations ''x''\''z'' and ''z''/''y'', loosely analogous to division or implication, when ' ...
. The relation between a t-norm T and its residuum R is an instance of
adjunction (specifically, a
Galois connection): the residuum forms a right adjoint R(''x'', –) to the functor T(–, ''x'') for each ''x'' in the lattice
, 1taken as a
poset category.
In the standard semantics of t-norm based fuzzy logics, where conjunction is interpreted by a t-norm, the residuum plays the role of implication (often called ''R-implication'').
Basic properties of residua
If
is the residuum of a left-continuous t-norm
, then
:
Consequently, for all ''x'', ''y'' in the unit interval,
:
if and only if
and
:
If
is a left-continuous t-norm and
its residuum, then
:
If
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''.
T-conorms
T-conorms (also called S-norms) are dual to t-norms under the order-reversing operation that assigns 1 – ''x'' to ''x'' on
, 1 Given a t-norm
, the complementary conorm is defined by
:
This generalizes
De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
.
It follows that a t-conorm satisfies the following conditions, which can be used for an equivalent axiomatic definition of t-conorms independently of t-norms:
* Commutativity: ⊥(''a'', ''b'') = ⊥(''b'', ''a'')
* Monotonicity: ⊥(''a'', ''b'') ≤ ⊥(''c'', ''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d''
* Associativity: ⊥(''a'', ⊥(''b'', ''c'')) = ⊥(⊥(''a'', ''b''), ''c'')
* Identity element: ⊥(''a'', 0) = ''a''
T-conorms are used to represent
logical disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...
in
fuzzy logic and
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
in
fuzzy set theory
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined ...
.
Examples of t-conorms
Important t-conorms are those dual to prominent t-norms:
*Maximum t-conorm
, 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
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
and for weak disjunction in all t-norm based fuzzy logics.
*Probabilistic sum
is dual to the product t-norm. In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
it expresses the probability of the union of independent
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of e ...
s. It is also the standard semantics for strong disjunction in such extensions of
product fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
in which it is definable (e.g., those containing involutive negation).
*Bounded sum
is dual to the Łukasiewicz t-norm. It is the standard semantics for strong disjunction in
Łukasiewicz fuzzy logic
Łukasiewicz is a Polish surname. It comes from the given name Łukasz (Lucas). It is found across Poland, particularly in central regions. It is related to the surnames Łukaszewicz and Lukashevich.
People
* Antoni Łukasiewicz (born 1983 ...
.
*Drastic t-conorm
::
:dual to the drastic t-norm, is the largest t-conorm (see the
properties of t-conorms below).
*Nilpotent maximum, dual to the nilpotent minimum:
::
*Einstein sum (compare the
velocity-addition formula
In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different f ...
under special relativity)
::
: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
, 1
* Dually to t-norms, all t-conorms are bounded by the maximum and the drastic t-conorm:
::
, for any t-conorm
and all ''a'', ''b'' in
, 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
, 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