We use relational algebra to define a refinement fuzzy order called

Fuzzy set theory provides a major newer paradigm in modeling and reasoning with uncertainty. Zadeh, a professor at University of California at Berkeley was the first to propose a theory of fuzzy sets and an associated logic, namely, fuzzy logic in [

There are countless applications for fuzzy logic. In fact, some claim that fuzzy logic is the encompassing theory over all types of logic. The items in this list are more common applications that one may encounter in everyday life.

Fuzzy set theory appeared in 1965 [

The calculus of relations has been an important component of the development of logic and algebra since the middle of the nineteenth century [

The demonic calculus of relations [

In Section

Using Tarski [

In this paper, we need both levels of abstraction. The first one will give us our examples and the second one is useful for our proofs and formulas. So, in this section we will present both levels. A relation

The graphs and relations are closely linked.

Every finite relation can be interpreted as a representation graph and vice versa. Graph (1) which corresponds to the relation

The graph associated to relation

In matrix notation, an entry

As relations are sets, they are ordered by inclusion. The least relation between sets

As relations are particular sets, we can apply the usual sets operations, which are union (

Inverse of a relation

For

We remark that

We can now define the abstract algebraic structures having many properties of the relations. They are based on boolean algebras and other operators which are the composition

The origin of relational calculus goes back to last century with the work of de Morgan [

An

Every structure

Every relation

Schröder rule

Tarski rule is valid:

The precedence of the relational operators, from highest to lowest, is the following:

(a) The algebra of binary relations on different sets is an important relation algebra, because it is very useful. Let

(b) The set of all homogeneous binary relations on a set

(c) The algebra of boolean matrices is another important relation algebra.

We recall by the next examples how some of the operators are applied to boolean matrices. To respect the usual convention, we will use the boolean values

(d) We will give another example. The set of matrices whose entries are relations constitutes a relation algebra [

The constant relations are defined as follows:

From Definition

Let

Sometimes, instead of refering to laws 6, 7, 8, and 9, we refer to the operation (

In the following, we will give the definitions of certain properties.

A relation

an

a

a

a

A

In an algebra of boolean matrices, a vector is a matrix in which the rows are constant and a point is a vector with a nonzero row.

Let

Let

Prerestriction of

Postrestriction of

Domain of

The vector represents the subset

The relation

Fuzzy set theory has been studied extensively over the past 30 years. Most of the early interest in fuzzy set theory pertained to representing uncertainty in human cognitive processes [

The concept of a fuzzy relation on a set was defined by Zadeh [

Let

As fuzzy relations are sets, they are ordered by inclusion. The least fuzzy relation between sets

As fuzzy relations are particular fuzzy sets, we can apply the usual fuzzy sets operations, which are union (

Let

Inverse of a fuzzy relation

such that

The max-min composition

From now on, the composition operator symbol (

The semiscalars multiplication

(i) Let

(ii) Let

We have

These operations are illustrated, respectively, by Figures

Fuzzy relation

Fuzzy relations

The max-min composition of fuzzy relations

The max-product composition of fuzzy relations

The max-av composition of fuzzy relations

Different versions of “composition” have been suggested which differ in their results and also with respect to their mathematical properties. The max-min composition has become the best known and the most frequently used. However, often the so-called

The max-prod composition (

The max-av composition (

Let

We will first compute the max-min composition

In analogy to the above computation we now determine the grades of membership for all pairs

For the max-prod we obtain

then

After performing the remaining computations we obtain

The max-av composition finally yields:

i
1
1
2
0.4
3
0.8
4
1.4
5
0.7;

then

The fuzzy relation

The complement of fuzzy relation

The fuzzy relation

The fuzzy relation

The demonic fuzzy relation

A demonic fuzzy relation

Demonic union of fuzzy relations

Angelic union of fuzzy relations

Demonic intersection of fuzzy relations

Angelic intersection of fuzzy relations

The demonic fuzzy relation

The demonic fuzzy relation

Demonic union of fuzzy relations

Angelic union of fuzzy relations

Demonic intersection of fuzzy relations

Angelic intersection of fuzzy relations

The vectors

Just as for relations, the properties of commutativity, associativity, distributivity, involution, and idempotency all hold for fuzzy relations. Moreover, De Morgan’s principles hold for fuzzy relations just as they do for relations, and the empty relation

Let

In the following, we will give some properties of fuzzy relations.

Let

Equality

Inclusion

If

If

Then one has the proper inclusion

Let

associativity of composition:

distributivity over union:

weak distributivity over intersection:

commutativity:

associativity:

distributivity:

idempotency:

identity:

involution:

De Morgans law:

A fuzzy relation

reflexive if and if only

transitive if and if only

symmetric if and if only

antisymmetric if and if only

equivalence if and if only

order if and if only

preorder if and if only

Let

First, we will give the rationale behind the definition of refinement called the

for any input

In the following, we will define the refinement fuzzy ordering (

One says that a fuzzy relation

In other words,

It is easy to show that this definition is equivalent to definition (given [

Let

We have

Then

Let

We have

Then

Let

Let

The fuzzy relation

In this subsection, we will present fuzzy demonic operators and some of their properties.

Let

Their supremum is

and satisfies

The operator (

if and only if

Their infimum, if it exists, is

and it satisfies

The operator (

if and only if

For

This condition is equivalent to

which can be interpreted as follows: the existence condition simply means that, on the intersection of their domains,

We know that

Let

Let

Demonic composition in Example

Demonic composition in Example

Now we need to define the relative fuzzy implication.

In what follows, we will give our definition of the relative fuzzy implication and some examples.

The binary operator (

(a) Let

The

In other words,

if and only if

Consider the following:

Figures

In this paper, we have presented the notion of relational fuzzy calculus specially a fuzzy refinement order (

The author declares that there is no conflict of interests regarding the publication of this paper.

This research project was supported by a grant from the Research Center of the Center for Female Scientific and Medical Colleges, Deanship of Scientific Research, King Saud University.