On the greatest solutions to weakly linear systems of fuzzy relation
inequalities and equations✩
Jelena Ignjatovića , Miroslav Ćirić∗,a , Stojan Bogdanovićb
a
Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, P. O. Box 224, 18000 Niš, Serbia
b Faculty of Economics, University of Niš, Trg Kralja Aleksandra 11, 18000 Niš, Serbia
Abstract
In this paper we study systems of fuzzy relation inequalities and equations of the form U ◦ Vi 6 Vi ◦ U (i ∈ I), where U
is an unknown and Vi (i ∈ I) are given fuzzy relations, the dual systems Vi ◦ U 6 U ◦ Vi (i ∈ I), their conjunctions, the
systems of the form U ◦Vi = Vi ◦U (i ∈ I), and certain special types of these systems. We call them weakly linear systems.
For each weakly linear system, with a complete residuated lattice as the underlying structure of truth values, we
prove the existence of the greatest solution, and we provide an algorithm for computing the greatest solution, which
work whenever the underlying complete residuated lattice is locally finite. Otherwise, we determine some sufficient
conditions under which the algorithm works. The algorithm is iterative, and each its single step can be viewed as
solving a particular linear system.
Weakly linear systems emerged from the fuzzy automata theory, but we show that they also have important applications in other fields, e.g. in the concurrency theory and social network analysis.
Key words: Fuzzy relations; Fuzzy relation inequalities; fuzzy relation equations; residuals of fuzzy relations;
complete residuated lattices; fuzzy quasi-orders; fuzzy equivalences; regular fuzzy relations; post-fixed points
1. Introduction
Systems of fuzzy relation equations and inequalities emerged from the study aimed at medical applications (cf. [76, 78]), and since they have found a much wider field of applications, and have been applied in
fuzzy control, discrete dynamic systems, knowledge engineering, identification of fuzzy systems, prediction
of fuzzy systems, decision-making, fuzzy information retrieval, fuzzy pattern recognition, image compression and reconstruction, and in other areas.
The most studied systems were systems of fuzzy relation equations of the form U ◦ Vi = Wi (i ∈ I), or the
dual systems Vi ◦U = Wi (i ∈ I), where U is an unknown fuzzy relation, Vi and Wi are either given fuzzy relations or given fuzzy sets, and ◦ denotes the composition operation on fuzzy relations, or between fuzzy sets
and fuzzy relations. These systems, along with systems that are obtained from them by replacing equalities
with inequalities, will be called the linear systems. They were first studied by Sanchez [76]–[79], who discussed linear systems over the Gödel structure, but here we consider them in a more general context, over a
complete residuated lattice. It is known that every linear system of inequalities U ◦ Vi 6 Wi (i ∈ I) has a
solution, and moreover, it has the greatest one, but the opposite system Wi 6 U ◦ Vi (i ∈ I) may not have a
solution in general (cf. [69, 70]). Consequently, a linear system of equations U ◦ Vi = Wi (i ∈ I) also need not
be solvable, but if it is solvable, then it has the greatest solution, which is the same as the greatest solution to
U ◦ Vi 6 Wi (i ∈ I) and it was described by Sanchez using fuzzy implication (cf. [77, 79]). In particular, linear
systems of equations with Vi = Wi , for every i, are solvable and they have the greatest solutions. Linear
✩ Research
supported by Ministry of Science and Technological Development, Republic of Serbia, Grant No. 144011
author. Tel.: +38118224492; fax: +38118533014.
Email addresses: [email protected] (Jelena Ignjatović), [email protected] (Miroslav Ćirić), [email protected] (Stojan
Bogdanović)
Preprint submitted to Elsevier
February 24, 2010
∗ Corresponding
systems of fuzzy relation equations and inequalities have been widely considered in the literature. For more
information about them we refer to [22, 47, 69, 70, 71], and the papers cited there. The survey of the main
results can be found in [48, 68].
In this paper we deal with more complex non-linear systems of the form U ◦ Vi 6 Vi ◦ U (i ∈ I), the dual
systems Vi ◦ U 6 U ◦ Vi (i ∈ I), and their conjunctions, the systems of the form U ◦ Vi = Vi ◦ U (i ∈ I),
where U is an unknown fuzzy relation, and Vi (i ∈ I) are given fuzzy relations on the underlying set A.
Solutions to these systems are sought in the set R(A) of all fuzzy relations on A. Since in many applications
we are required to find solutions contained in a given fuzzy relation, these systems are equipped with an
additional inequality U 6 W, where W is a given fuzzy relation. Moreover, in many situations we need
symmetric solutions, and we also introduce three new systems, adding the corresponding equations and
inequalities given in terms of the inverse of U. For the reason that will be explained in the sequel, all these
systems will be called weakly linear.
Weakly linear systems, as well as some related systems, have arisen recently from research in the fuzzy
automata theory. It was shown in [25, 26, 81] that reduction of the number of states of a fuzzy recognizer
A by means of fuzzy equivalences and fuzzy quasi-orders is closely related to solving suitable systems of
fuzzy relation equations and inequalities. Equations and inequalities forming these systems involve fuzzy
transition relations of A , fuzzy sets of initial and terminal states, and an unknown fuzzy relation, and solutions are sought in the set E (A) of all fuzzy equivalences, or the set Q(A) of all fuzzy quasi orders on the set
of states A of the fuzzy recognizer A . Since better reductions are achieved by means of greater solutions
to corresponding systems, the main task of the present study is to prove the existence, and even more
important, to compute the greatest solutions to weakly linear systems.
Our main results are the following. We show that weakly linear systems possess the greatest solutions,
and these greatest solutions are fuzzy quasi-orders, i.e., in the case of ”symmetric” systems, they are fuzzy
equivalences (cf. Theorems 4.1 and 4.5). In the case when solutions are sought in the set of fuzzy quasi-orders
or the set of fuzzy equivalences, we provide some equivalent forms of weakly linear systems (cf. Theorems
4.2-4.4 and 4.6-4.8). While original forms of the systems are convenient for proving the existence of their
greatest solutions, equivalent forms show oneself to be more convenient for computing these greatest solutions. However, there is a problem how to compute the greatest solution to a weakly linear system. We show
that this problem is equivalent to the problem of computing the greatest post-fixed point of a particular isotone function on the lattice of fuzzy quasi-orders or the lattice of fuzzy equivalences. For every weakly linear
system we define a suitable isotone function and a descending chain of fuzzy relations which correspond to
this function and this system. If the underlying structure of truth values is a locally finite residuated lattice,
then this chain must be finite and its smallest element is the greatest solution we are looking for. But, if
this structure is not locally finite, then the chain may not be finite and its infimum may not be equal to the
greatest solution to the considered system. We determine some sufficient conditions for the finiteness of the
descending chains of weakly linear systems (cf. Theorems 5.2 and 5.4), as well as some sufficient conditions
under which the infima of these chains are equal to the greatest solutions to the systems (cf. Theorem 5.6).
Despite certain differences, weakly linear systems are closely related to linear systems, since in the iterative
procedure for computing the greatest solution to a weakly linear system, every single step may be viewed
as the solving a particular linear system. Just for that reason we called these systems weakly linear.
The algorithm for computing the greatest solution to a weakly linear system can be modified so that it
computes the greatest crisp solution to the system, and this algorithm works when the underlying structure
of truth values is an arbitrary complete residuated lattice (cf. Proposition 5.8). Therefore, in cases when the
above mentioned algorithm does not enable to compute the greatest solution to a weakly linear system,
instead of the greatest solution to the system we can use the greatest crisp solution, which can be effectively
computed. However, we show that the greatest crisp solution can be strictly less, and even have a strictly
greater index, than the greatest fuzzy solution to a weakly linear system (cf. Example 5.9).
Weakly linear systems with a given family of fuzzy relations Vi (i ∈ I) and a given fuzzy relation W are
closely related to two types of linear systems of fuzzy relation inequalities. The first type are linear systems
determined by the same family and the fuzzy relation. Their greatest solutions can be effectively computed
when the underlying structure of truth values is an arbitrary complete residuated lattice (cf. Proposition 6.1).
Nevertheless, they have narrower sets of solutions than the corresponding weakly linear systems, and hence,
2
their greatest solutions are less or equal than the greatest solutions to the weakly linear systems, and may
be strictly less. Moreover, their indices are greater or equal than the indices of the greatest crisp solutions to
the corresponding weakly linear systems, and can be strictly greater. This means that, in some sense, the
greatest crisp solution is a ”better approximation” to the greatest solution to a weakly linear system than the
greatest solution to the related linear system. Another important type of linear systems related to weakly
linear systems are those determined by a family of fuzzy relations of the form P ◦ W (P ∈ M ) or the form
W ◦ P (P ∈ M ), where M is the submonoid of the monoid of all fuzzy relations on the underlying set
generated by fuzzy relations Vi (i ∈ I). These systems have larger sets of solutions than the corresponding
weakly linear systems (cf. Proposition 6.4), what means that their greatest solutions are greater than the
greatest solutions to the weakly linear systems. These solutions need not be solutions to the corresponding
weakly linear systems, but in some applications they can be used instead of solutions to weakly linear
systems, and even be more successful. For example, in the state reduction of fuzzy recognizers, solutions to
both types of systems give equivalent reducts, but in general, solutions to weakly linear systems give worse
reductions. However, there is a problem concerning the number of inequalities in the derived linear systems.
This number can be infinite, and even if it is finite, it can be exponential in the number of elements of the
underlying set. In applications in the fuzzy automata theory, computing of the fuzzy relations P ◦ W and
W ◦ P, for all P ∈ M , and formation of the corresponding linear systems, amounts to the determinization
of fuzzy recognizers and their reverse fuzzy recognizers by means of the Nerode automata (cf. [20, 39, 41]).
The structure of the paper is as follows. In Section 2 we give definitions of basic notions and notation. In
Section 3 we discuss the main properties of residuals of fuzzy relations, and introduce several new operations
on fuzzy relations. Section 4 contains definitions of six types of weakly linear systems, proofs of the existence
of their greatest solutions, and provides some equivalent forms of these systems in cases when solutions are
required to be fuzzy quasi-orders or fuzzy equivalences. In Section 5 we provide algorithms for computing
the greatest solutions to weakly linear systems and determine certain sufficient conditions under which
these algorithms work. Section 6 establishes relationships between weakly linear systems and certain types
of linear systems. Finally, in Section 7 we present applications of weakly linear systems in the theory of
fuzzy automata, concurrency theory, and social network analysis.
2. Preliminaries
A residuated lattice is an algebra L = (L, ∧, ∨, ⊗, →, 0, 1) such that
(L1) (L, ∧, ∨, 0, 1) is a lattice with the least element 0 and the greatest element 1,
(L2) (L, ⊗, 1) is a commutative monoid with the unit 1,
(L3) ⊗ and → form an adjoint pair, i.e., they satisfy the adjunction property: for all x, y, z ∈ L,
x ⊗ y 6 z ⇔ x 6 y → z.
(1)
Moreover, L is called a complete residuated lattice if it satisfies (L1), (L3), and
(L2’) (L, ∧, ∨, 0, 1) is a complete lattice with the least element 0 and the greatest element 1.
The operations ⊗ (called multiplication) and → (called residuum) are intended
the conjunction
W for modelingV
and implication of the corresponding logical calculus, and supremum ( ) and infimum ( ) are intended
for modeling of the existential and general quantifier, respectively. An operation ↔ defined by
x ↔ y = (x → y) ∧ (y → x),
(2)
called biresiduum (or biimplication), is used for modeling the equivalence of truth values. Emphasizing
their monoidal structure, in some sources residuated lattices are called integral, commutative, residuated
ℓ-monoids [37].
The most studied and applied structures of truth values, defined on the real unit interval [0, 1] with
x ∧ y = min(x, y) and x ∨ y = max(x, y), are the Łukasiewicz structure (where x ⊗ y = max(x + y − 1, 0),
3
x → y = min(1 − x + y, 1)), the Goguen (product) structure (x ⊗ y = x · y, x → y = 1 if x 6 y, and = y/x
otherwise), and the Gödel structure (x ⊗ y = min(x, y), x → y = 1 if x 6 y, and = y otherwise). More generally,
an algebra ([0, 1], ∧, ∨, ⊗, →, 0, 1) is a complete
W residuated lattice if and only if ⊗ is a left-continuous t-norm
and the residuum is defined by x → y = {u ∈ [0, 1] | u ⊗ x 6 y} (cf. [4]). Another important set of truth
values is the set {a0 , a1 , . . . , an }, 0 = a0 < · · · < an = 1, with ak ⊗ al = amax(k+l−n,0) and ak → al = amin(n−k+l,n) .
A special case of the latter algebras is the two-element Boolean algebra of classical logic with the support
{0, 1}. The only adjoint pair on the two-element Boolean algebra consists of the classical conjunction and
implication operations. This structure of truth values we call the Boolean structure. A residuated lattice L
satisfying x ⊗ y = x ∧ y is called a Heyting algebra, whereas a Heyting algebra satisfying the prelinearity
axiom (x → y) ∨ (y → x) = 1 is called a Gödel algebra. If any finitelly generated subalgebra of a residuated
lattice L is finite, then L is called locally finite. For example, every Gödel algebra, and hence, the Gödel
structure, is locally finite, whereas the product structure is not locally finite.
If L is a complete residuated lattice, then for all x, y, z ∈ L and any {yi }i∈I ⊆ L the following hold:
x 6 y implies x ⊗ z 6 y ⊗ z,
(3)
x 6 y implies z → x 6 z → y and y → z 6 x → z,
x → y 6 x ⊗ z → y ⊗ z,
(4)
(5)
x ⊗ (x → y) 6 y,
(x → y) ⊗ (y → z) 6 (x → z),
_
_
x⊗
yi =
(x ⊗ yi ),
(6)
(7)
x⊗
(9)
i∈I
^
^
i∈I
yi 6
^
(x ⊗ yi ),
i∈I
(xi → yi ) 6
i∈I
^
(xi → yi ) 6
i∈I
(8)
i∈I
^ ^ xi →
yi ,
i∈I
i∈I
i∈I
i∈I
(10)
_ _ xi →
yi .
(11)
For other properties of complete residuated lattices we refer to [3, 4].
Let L be a lattice. An L -fuzzy subset of a set A is any function from A into L. If the structure L of
membership values is known from the context, we will say simply fuzzy subset instead of L -fuzzy subset.
The set of all L -fuzzy subsets of A is denoted by F (A). Let f, 1 ∈ F (A). The equality of f and 1 is defined
as the usual equality of functions, i.e., f = 1 if and only if f (x) = 1(x), for every x ∈ A. The inclusion f 6 1 is
also defined pointwise: f 6 1 if and only if f (x) 6 1(x),Vfor every x ∈ A. EndowedWwith this partial order
F (A) forms a lattice, in which the meet (intersection) i∈I fi and the join (union) i∈I fi of a finite family
{ fi }i∈I of L -fuzzy subsets of A are functions from A into L defined by




^
_
^ 
_ 



fi  (x) =
fi (x),
fi  (x) =
fi (x).
(12)


i∈I
i∈I
i∈I
i∈I
If L is a complete lattice, then in (12) we can allow I to be an infinite set, and in this case F (A) forms a
complete lattice. If L is a residuated lattice, then F (A) also forms a residuated lattice in which the product
f ⊗ 1 is an L -fuzzy subset defined by f ⊗ 1(x) = f (x) ⊗ 1(x), for every x ∈ A.
A crisp subset of a set A is an L -fuzzy relation which takes values only in the set {0, 1}. If f is a crisp
subset of A, then expressions “ f (x) = 1” and “x ∈ f ” will have the same meaning, i.e., f is considered as
an ordinary subset of A. The crisp part of an L -fuzzy subset f of A is a crisp subset f c : A → L defined by
f c (a) = 1, if f (a) = 1, and f c (a) = 0, if f (a) < 1, i.e., f c = {x ∈ A | f (x) = 1}. An L -fuzzy subset f of A is
normalized (or modal, in some sources) if f (x) = 1 for at least one x ∈ A, i.e., if its crisp part is non-empty.
An L -fuzzy relation on A is any function from A × A into L, that is to say, any L -fuzzy subset of A × A,
and equality, inclusion, joins, meets and ordering of L -fuzzy relations are defined as for L -fuzzy sets.
4
The set of all L -fuzzy relations on a set A is denoted by R(A). If the structure L of membership values is
known from the context, then we say simply fuzzy relation instead of L -fuzzy relation.
Remark 2.1. In this paper we will consider fuzzy subsets and fuzzy relations of/on arbitrary sets, not necessarily finite, for what reason the underlying structure L of membership values is required to be a complete
lattice. However, whenever we work with fuzzy subsets and relations of/on a finite set, the assumption that
L is complete become superfluous, and it can be omitted.
In accordance with the previous remark, in the rest of the section, if not noted otherwise, let A be a
non-empty set, not necessarily finite, and let L be a complete lattice.
For L -fuzzy relations R, S ∈ R(A), their composition R ◦ S is an L -fuzzy relation on A defined by
_
(R ◦ S)(a, b) =
R(a, c) ⊗ S(c, b),
(13)
c∈A
for all a, b ∈ A, and for f ∈ F (A) and R ∈ R(A), the compositions f ◦ R and R ◦ f are L -fuzzy subsets of A
defined by
_
_
( f ◦ R)(a) =
f (b) ⊗ R(b, a), (R ◦ f )(a) =
R(a, b) ⊗ f (b),
(14)
b∈A
b∈A
for any a ∈ A. Finally, for f, 1 ∈ F (A) we write
_
f ◦1=
f (a) ⊗ 1(a).
(15)
a∈A
The value f ◦ 1 can be interpreted as the ”degree of overlapping” of f and 1.
For arbitrary R, S, T ∈ R(A) we have that
(R ◦ S) ◦ T = R ◦ (S ◦ T),
(16)
R 6 S implies R ◦ T 6 S ◦ T and T ◦ R 6 T ◦ S,
(17)
and for arbitrary f, 1 ∈ F (A) and R, S ∈ R(A) we can easily verify that
( f ◦ R) ◦ S = f ◦ (R ◦ S),
( f ◦ R) ◦ 1 = f ◦ (R ◦ 1),
(18)
and hence, the parentheses in (18) can be omitted. For n ∈ N, an n-th power of an L -fuzzy relation R ∈ R(A)
is an L -fuzzy relation Rn on A defined inductively by R1 = R and Rn+1 = Rn ◦ R. We also define R0 to be the
equality relation on A.
Moreover, if L is a complete residuated lattice, then for every R, S ∈ R(A) and every {Ri }i∈I , {Si }i∈I ⊆ R(A)
we have that
_
_ _
_ R◦
Si =
(R ◦ Si ),
Ri ◦ S =
(Ri ◦ S),
(19)
i∈I
i∈I
i∈I
i∈I
i∈I
i∈I
i∈I
i∈I
^
^ ^
^ R◦
Si 6
(R ◦ Si ),
Ri ◦ S 6
(Ri ◦ S),
(20)
and then the system (R(A), ∧, ∨, ◦, ∅, ∇A, ∆A ) forms a quantale, where ∇A is the universal relation on A and
∆A is the equality relation on A, i.e., for all a, b ∈ A we have that ∇A (a, b) = 1, and ∆A (a, b) = 1, if a = b, and
∆A (a, b) = 0, if a , b.
We note that if A is a finite set with n elements, then R and S can be treated as n × n fuzzy matrices over
L and R ◦ S is the matrix product, whereas f ◦ R can be treated as the product of a 1 × n matrix f and an
n × n matrix R, and R ◦ f as the product of an n × n matrix R and an n × 1 matrix f t (the transpose of f ).
For an L -fuzzy relation R on A, the L -fuzzy relation R−1 on A defined by R−1 (a, b) = R(b, a), for every
a, b ∈ A, is called the inverse of R. An L -fuzzy relation R on A is said to be
5
(R) reflexive (or fuzzy reflexive) if ∆A 6 R, i.e., if R(a, a) = 1, for every a ∈ A;
(S) symmetric (or fuzzy symmetric) if R−1 6 R, i.e., if R(a, b) = R(b, a), for all a, b ∈ A;
(T) transitive (or fuzzy transitive) if R ◦ R 6 R, i.e., if for all a, b, c ∈ A we have
R(a, b) ⊗ R(b, c) 6 R(a, c).
W
For an L -fuzzy relation R on a set A, an L -fuzzy relation R∞ on A defined by R∞ = n∈N Rn is the least
transitive L -fuzzy relation on A containing R, and it is called the transitive closure of R.
A reflexive and transitive L -fuzzy relation on A is called an L -fuzzy quasi-order, or just a fuzzy quasi-order,
if L is known from the context, and a reflexive and transitive crisp relation on A is called a quasi-order. In
some sources quasi-orders and fuzzy quasi-orders are called preorders and fuzzy preorders. Note that a
reflexive fuzzy relation R is a fuzzy quasi-order if and only if R2 = R. A reflexive, symmetric and transitive
L -fuzzy relation on A is called an L -fuzzy equivalence (or just a fuzzy equivalence), and a reflexive, symmetric
and transitive crisp relation on A is called an equivalence. An L -fuzzy equivalence E on A is called an L -fuzzy
equality (or just a fuzzy equality) if for any a, b ∈ A, E(a, b) = 1 implies a = b. If R is an L -fuzzy quasi-order
on A, then R ∧ R−1 is the greatest L -fuzzy equivalence contained in R, and it is called the natural fuzzy
equivalence of R.
With respect to the ordering of L -fuzzy relations, the set Q(A) of all L -fuzzy quasi-orders on a set A,
and the set E (A) of all L -fuzzy equivalences on A, form complete lattices. The meet both in Q(A) and E (A)
is the ordinary intersection of L -fuzzy relations, but in the general case, the joins in Q(A) and E (A) do not
coincide with the ordinary union of L -fuzzy relations. Namely, if {Ri }i∈I is a family
of L -fuzzy quasi-orders
W
(resp. L -fuzzy equivalences) on A, then its join in Q(A) (resp. in E (A)) is ( i∈I Ri )∞ , the transitive closure
of the union of this family.
Let Q be an L -fuzzy quasi-order on a set A. For each a ∈ A, the Q-afterset of a is the L -fuzzy subset Qa
of A defined by Qa (x) = Q(a, x), for any x ∈ A, and the Q-foreset of a is the L -fuzzy subset Qa of A defined by
Qa (x) = Q(x, a), for any x ∈ A (cf. [1, 7, 51, 81]). The set of all Q-aftersets will be denoted by A/Q, and the set
of all Q-foresets by A\Q. If E is an L -fuzzy equivalence, then for every a ∈ A we have that Ea = Ea , and Ea
is called the equivalence class of a with respect to E (cf. [21]). The set of all equivalence classes of E is denoted
by A/E and called the factor set of A with respect to E. For any L -fuzzy quasi-order Q on a set A and its
natural L -fuzzy equivalence E we have that the set A/Q of all Q-aftersets, the set A\Q of all Q-foresets, and
the factor set A/E have the same cardinality (cf. [81]). This cardinality will be called the index of Q, and it
will be denoted by ind(Q). If A is a finite set with n elements and an L -fuzzy quasi-order Q on A is treated
as an n × n fuzzy matrix over L , then Q-aftersets are row vectors, whereas Q-foresets are column vectors
of this matrix. For any L -fuzzy subset f of A, let L -fuzzy relations Q f , Q f , and E f on A be defined by
Q f (a, b) = f (a) → f (b),
Q f (a, b) = f (b) → f (a),
E f (a, b) = f (a) ↔ f (b),
(21)
for all a, b ∈ A. We have that Q f and Q f are L -fuzzy quasi-orders, and E f is an L -fuzzy equivalence on A.
In particular, if f is a normalized L -fuzzy subset of A, then it is an afterset of Q f , a foreset of Q f , and an
equivalence class of E f .
For more information on lattices and related concepts we refer to [5, 6, 75], and to [3, 4, 48, 68], for information on fuzzy sets and fuzzy relations.
3. Residuals of fuzzy relations
In this section we consider the main properties of residuals of fuzzy relations, and we introduce several
new operations on fuzzy relations that we need in the further work. Some of the results presented here are
actually known, or follow directly from some well-known results given in [69, 70, 71, 77] and other related
articles, but we show them here in the form which we need later. For the sake of completeness, we also
give proofs of some known results.
Let L be a lattice and let V and W be L -fuzzy relations on a non-empty set A. The greatest L -fuzzy
relation U on A such that V ◦ U 6 W, if it exists, is denoted by W/V and called the right residual of W by V.
6
Analogously, the greatest L -fuzzy relation U on A such that U ◦ V 6 W, if it exists, is denoted by W\V
and called the left residual of W by V. In other words, if they exist, W/V is the greatest solution to the linear
inequality V ◦U 6 W, and W\V is the greatest solution to U ◦V 6 W, where U is an unknown fuzzy relation.
Residuals of crisp (Boolean) relations were introduced by Birkhoff [5] as the best possible aproximations
to inverses of relations. For construction and fundamental properties of residuals of crisp relations we
refer to Boyd and Everett [12]. Fan, Liau and Lin [31] studied residuals of fuzzy relations with membership
values in the Gödel structure, but they did not discussed their existence and construction. Here we determine
necessary and sufficient conditions on the structure L of membership values to residuals exist for every
pair of L -fuzzy relations, and we present a method for their construction.
Proposition 3.1. Let L be a lattice and let V and W be L -fuzzy relations on a non-empty set A. Then:
(a) The sets {U ∈ R(A) | V ◦ U 6 W} and {U ∈ R(A) | U ◦ V 6 W} are ideals of the lattice R(A).
(b) W/V exists if and only if {U ∈ R(A) | V ◦ U 6 W} is a principal ideal. If it is true, then this principal ideal is
generated by W/V, i.e., for each U ∈ R(A) we have that
V ◦ U 6 W ⇔ U 6 W/V .
(22)
(c) W\V exists if and only if {U ∈ R(A) | U ◦ V 6 W} is a principal ideal. If it is true, then this principal ideal is
generated by W\V, i.e., for each U ∈ R(A) we have that
U ◦ V 6 W ⇔ U 6 W\V .
(23)
Proof. These assertions follow immediately by (17), (19) and the definition of residuals.
It is worth noting that the conjunction of conditions (22) and (23) is the non-commutative counterpart
of the adjunction property (1).
Theorem 3.2. Let L = (L, ∧, ∨, ⊗, 0, 1) be an algebra satisfying the conditions (L1) and (L2’) of the definition of a
complete residuated lattice and (3). Then the following conditions are equivalent:
(i) There exists → forming an adjoint pair with ⊗;
(ii) For any x, y ∈ L, the set {z ∈ L | x ⊗ z 6 y} has a greatest element;
(iii) Any two L -fuzzy relations on the same set have the right residual;
(iv) Any two L -fuzzy relations on the same set have the left residual.
Proof. (i)⇔(ii). This follows by Theorem 1.22 [4].
(i)⇒(iii). This implication follows by Lemma 2 [69], but for the sake of completeness we give a proof. Let
V and W be arbitrary fuzzy relations on A, and define a fuzzy relation R on A by
^
R(a, b) =
V(c, a) → W(c, b),
(24)
c∈A
for all a, b ∈ A. For arbitrary a, b ∈ A, by (9) and (6) we obtain that
_
_
^
(V ◦ R)(a, b) =
V(a, c) ⊗ R(c, b) =
V(a, c) ⊗
V(d, c) → W(d, b)
c∈A
6
_^
c∈A d∈A
c∈A
d∈A
V(a, c) ⊗ V(d, c) → W(d, b) 6
_
c∈A
V(a, c) ⊗ V(a, c) → W(a, b) 6 W(a, b),
and hence, V ◦ R 6 W. Further, let S be an arbitrary fuzzy relation on A such that V ◦ S 6 W. Then for
arbitrary a, b, c ∈ A we obtain that
_
V(c, a) ⊗ S(a, b) 6
V(c, d) ⊗ S(d, b) = (V ◦ S)(c, b) 6 W(c, b),
d∈A
7
and according to the adjunction property (1) we obtain S(a, b) 6 V(c, a) → W(c, b). Now we have that
^
S(a, b) 6
V(c, a) → W(c, b) = R(a, b),
c∈A
so S 6 R. Thus, R is the greatest solution to the inequality V ◦ U 6 W (with an unknown U), so W/V = R.
Analogously we prove (i)⇒(iv).
(iii)⇒(ii). Consider arbitrary x, y ∈ L and a set A = {a}. Let us define L -fuzzy relations V and W on A by
V(a, a) = x and W(a, a) = y, and set R = W/V and R(a, a) = u. Then
x ⊗ u = V(a, a) ⊗ R(a, a) = (V ◦ R)(a, a) 6 W(a, a) = y.
If v ∈ L such that x ⊗ u 6 y, and if we define an L -fuzzy relation S on A by S(a, a) = v, then V ◦ S 6 W, so
S 6 R, and hence, v 6 u. Therefore, the set {z ∈ L | x ⊗ z 6 y} has the greatest element u.
Similarly we prove (iv)⇒(ii).
In accordance with Remark 2.1, if we discuss only fuzzy relations on finite sets, then in Theorem 3.2 we
can replace (L2’) by (L2).
Corollary 3.3. Let L = ([0, 1], ∧, ∨, ⊗, 0, 1), where ⊗ is a t-norm. Then the following conditions are equivalent:
(i) ⊗ is a left-continuous t-norm;
(ii) Any two L -fuzzy relations on the same set have the right residual;
(iii) Any two L -fuzzy relations on the same set have the left residual.
In the rest of this paper, if not noted otherwise, let A be a non-empty set, not necessarily finite, let L be
a complete residuated lattice, and let L -fuzzy relations be called simply fuzzy relations.
Theorem 3.4. Let V and W be fuzzy relations on A. Then for all a, b ∈ A we have that
^
(W\V)(a, b) =
V(b, c) → W(a, c).
(25)
c∈A
(W/V)(a, b) =
^
V(c, a) → W(c, b),
c∈A
Proof. This follows by the proof of Theorem 3.2.
Theorem 3.5. Let V be a fuzzy relation on A. Then
(a) V/V and V\V are fuzzy quasi-orders;
(b) V/V is the greatest solution to V ◦ U = V, where U is an unknown fuzzy relation on A;
(c) V\V is the greatest solution to U ◦ V = V, where U is an unknown fuzzy relation on A.
Proof. (a) According to (7), for arbitrary a, b, c, d ∈ A we have that
V/V(a, b) ⊗ V/V(b, c) 6 V(d, a) → V(d, b) ⊗ V(d, b) → V(d, c) 6 V(d, a) → V(d, c),
and hence,
V/V(a, b) ⊗ V/V(b, c) 6
^
V(d, a) → V(d, c) = V/V(a, c).
d∈A
Therefore, V/V is transitive. Clearly, V/V is reflexive, so we conclude that V/V is a fuzzy quasi-order.
Analogously we show that V\V is a fuzzy quasi-order.
8
(26)
(b) By the definition of right residual, V/V is the greatest solution to V ◦ U 6 V. On the other hand, by
reflexivity of V/V, for arbitrary a, b ∈ A we have that
V(a, b) = V(a, b) ⊗ V/V(b, b) 6
_
V(a, c) ⊗ V/V(c, b) = (V ◦ (V/V))(a, b),
c∈A
and hence V 6 V ◦ (V/V). Thus, V/V is a solution to V ◦ U = V, and since every solution to V ◦ U = V is a
solution to V ◦ U 6 V, we conclude that V/V is the greatest solution to V ◦ U = V.
In a similar way we prove (c).
Corollary 3.6. Let V be a fuzzy relation on A. Then the following conditions are equivalent:
(i) V is a fuzzy quasi-order;
(ii) V/V = V;
(iii) V\V = V.
Proof. (i)⇔(ii). If V/V = V, then V is a fuzzy quasi-order, according to Theorem 3.5. Conversely, let V be a
fuzzy quasi-order. Since V is reflexive, we have that V/V 6 V ◦ (V/V) = V. On the other hand, both V and
V/V are solutions to V ◦ U = V, and V/V is the greatest one, so V 6 V/V. Therefore, V/V = V.
Analogously we prove (ii)⇔(iii).
Using right and left residuals, as well as composition of fuzzy relations, we define and study several
new operations on fuzzy relations. First, for any fuzzy relation V on A we set V V = (V\V) ∧ (V\V)−1 and
V∥V = (V/V) ∧ (V/V)−1 , i.e., for all a, b ∈ A we have
(V V)(a, b) =
^
V(a, c) ↔ V(b, c),
(V∥V)(a, b) =
c∈A
^
V(c, a) ↔ V(c, b).
(27)
c∈A
By Theorem 3.5, V\V and V/V are fuzzy quasi-orders, so V V and V∥V are respectively the natural fuzzy
equivalences of V\V and V/V.
Next, for fuzzy relations W and V on A we define fuzzy relations W W V and W ⊘ V on A as follows:
W W V = (W ◦ V)\(W ◦ V),
W ⊘ V = (W ◦ V)/(W ◦ V).
(28)
In other words, for arbitrary a, b ∈ A we have
(W W V)(a, b) =
^
(W ◦ V)(b, c) → (W ◦ V)(a, c), (W ⊘ V)(a, b) =
c∈A
^
(W ◦ V)(c, a) → (W ◦ V)(c, b). (29)
c∈A
According to Theorem 3.5, W W V and W ⊘ V are fuzzy quasi-orders, for arbitrary fuzzy relations W and V.
ˆ V on A by
We also define fuzzy relations W Ŵ V and W ⊘
W Ŵ V = (W ◦ V)
(W ◦ V) = (W W V) ∧ (W W V)−1,
ˆ V = (W ◦ V)∥(W ◦ V) = (W ⊘ V) ∧ (W ⊘ V)−1 , (30)
W⊘
ˆ V are respectively the natural fuzzy equivalences of WWV and W⊘V. Hence, for arbitrary
i.e., W Ŵ V and W ⊘
a, b ∈ A we have that
^
^
ˆ V)(a, b) =
(W Ŵ V)(a, b) =
(W ◦ V)(a, c) ↔ (W ◦ V)(b, c), (W ⊘
(W ◦ V)(c, a) ↔ (W ◦ V)(c, b). (31)
c∈A
c∈A
9
4. Existence of the greatest solution to a weakly linear system
In this section, let A be a non-empty set (not necessarily finite), let {Vi }i∈I be a given family of fuzzy
relations on A (where I is also not necessarily finite), and let W be a given fuzzy relation on A.
Let us consider the following systems of fuzzy relation inequalities and equations:
U ◦ Vi 6 Vi ◦ U
(i ∈ I),
U 6 W;
(32)
Vi ◦ U 6 U ◦ Vi
U ◦ Vi = Vi ◦ U
(i ∈ I),
(i ∈ I),
U 6 W;
U 6 W;
(33)
(34)
where U is an unknown fuzzy relation on A. Solutions to (32), (33), and (34) are respectively called right
regular, left regular, and regular fuzzy relations on A with respect to the family of fuzzy relations {Vi }i∈I and
the fuzzy relation W. Clearly, if W = ∇A , then the inequality U 6 W becomes trivial, and it can be omitted. In
this case, solutions to (32), (33), and (34) are called right regular, left regular, and regular fuzzy relations with
respect to the family {Vi }i∈I . We remind that the above three systems, as well as later defined systems (41),
(42), and (43), we have already called weakly linear. The aim of this section is to prove the existence of the
greatest solutions to weakly linear systems.
First we prove the following:
Theorem 4.1. The systems (32), (33), and (34) have the greatest solutions.
If W is a fuzzy quasi-order, then these greatest solutions are also fuzzy quasi-orders.
Proof. We will prove only the assertion concerning the system (32). The assertions concerning the remaining
two systems can be proved analogously.
It is clear that the system (32) has at least one solution, the empty relation on A. Let {R j } j∈J be a family
of all fuzzy relations on A which are solutions to (32), and let Q denote the union (join) of this family. For
every i ∈ I, by (19) we obtain that
Q ◦ Vi =
_
j∈J
_ _
_
R j ◦ Vi =
(R j ◦ Vi ) 6
(Vi ◦ R j ) = Vi ◦
R j = Vi ◦ Q.
j∈J
j∈J
j∈J
Also, it is clear that Q 6 W. Hence, Q is a solution to (32). By its definition, Q is the greatest solution to (32).
Next, according to (17), for every i ∈ I we have
Q ◦ Q ◦ Vi 6 Q ◦ Vi ◦ Q 6 Vi ◦ Q ◦ Q,
and if W is a fuzzy quasi-order, then Q ◦ Q 6 W ◦ W = W, so we obtain that Q ◦ Q is a solution to (32). Since
Q is the greatest solution to (32), we conclude that Q ◦ Q 6 Q. Therefore, Q is transitive. Evidently, ∆A 6 Q,
i.e., Q is reflexive, and hence, Q is a fuzzy quasi-order.
q
q
Next, let us consider the lattice Q(A) of all fuzzy quasi-orders on A, and let us define functions φrr , φlr ,
q
and φr of Q(A) into itself by
^
^
^
q
q
q
q
q
φrr (Q) =
Vi W Q, φlr (Q) =
Q ⊘ Vi , φr (Q) =
(Vi W Q) ∧ (Q ⊘ Vi ) = φrr (Q) ∧ φlr (Q),
(35)
i∈I
i∈I
i∈I
for every Q ∈ Q(A). As we have noted immediately after the definition of the operations W and ⊘, for every
i ∈ I we have that Vi W Q and Q ⊘ Vi are fuzzy quasi-orders (even if Q is not a fuzzy quasi-order). Therefore,
q
q
q
q
q
q
φrr (Q), φlr (Q), and φr (Q) are also fuzzy quasi-orders, so indeed, φrr , φlr , and φr map Q(A) into itself.
The following theorem gives characterizations of those solutions to the system (32) which are fuzzy
quasi-orders, i.e., it characterizes right regular fuzzy quasi-orders with respect to the family {Vi }i∈I and the
fuzzy relation W.
10
Theorem 4.2. Let Q be a fuzzy quasi-order on A. Then the following conditions are equivalent:
(i) Q is a solution to the system (32);
(ii) Q is a solution to the system
U ◦ Vi ◦ U = Vi ◦ U (i ∈ I),
U 6 W,
(36)
where U is an unknown fuzzy relation on A;
(iii) Q is a solution to the system
q
U 6 φrr (U) ,
U 6W,
(37)
where U is an unknown fuzzy relation on A.
Proof. (i)⇔(ii). Let Q be a solution to (32). Consider an arbitrary i ∈ I. By transitivity of Q and the hypothesis
Q ◦ Vi 6 Vi ◦ Q we have that Q ◦ Vi ◦ Q 6 Vi ◦ Q ◦ Q 6 Vi ◦ Q. The opposite inequality follows by reflexivity
of Q, and hence, Q is a solution to (36).
Conversely, let Q be a solution to the system (36). Then for every i ∈ I, by reflexivity of Q we obtain that
Q ◦ Vi 6 Q ◦ Vi ◦ Q = Vi ◦ Q, and hence, Q is a solution to (32).
(ii)⇔(iii). Let Q be a solution to (36). Then for each i ∈ I we have that Q is a solution to U◦(Vi ◦Q) = Vi ◦Q,
where U is an unknown fuzzy relation, and according to Theorem 3.5, Vi W Q = (Vi ◦ Q)\(Vi ◦ Q) is the
greatest solution to U ◦ (Vi ◦ Q) = Vi ◦ Q, whence Q 6 Vi W Q. Therefore,
^
q
Q6
Vi W Q = φrr (Q),
i∈I
and we have proved that Q is a solution to (37).
Conversely, let Q be a solution to (37). Then for every i ∈ I we have that Q 6 Vi W Q = (Vi ◦ Q)\(Vi ◦ Q),
and according to (23), this is equivalent to Q ◦ Vi ◦ Q 6 Vi ◦ Q. Since the opposite inequality follows by
reflexivity of Q, we obtain that Q ◦ Vi ◦ Q = Vi ◦ Q. Therefore, Q is a solution to (36).
Left regular fuzzy quasi-orders with respect to {Vi }i∈I and W are characterized by the following theorem.
Theorem 4.3. Let Q be a fuzzy quasi-order on A. Then the following conditions are equivalent:
(i) Q is a solution to the system (33);
(ii) Q is a solution to the system
U ◦ Vi ◦ U = U ◦ Vi (i ∈ I),
U 6 W,
(38)
where U is an unknown fuzzy relation on A;
(iii) Q is a solution to the system
q
U 6 φlr (U) ,
U 6W,
(39)
where U is an unknown fuzzy relation on A.
Proof. This theorem can be proved in an analogous way as Theorem 4.2.
Combining the above results concerning right and left regular fuzzy quasi-orders we obtain the following
theorem which describes regular fuzzy quasi-orders with respect to {Vi }i∈I and W.
Theorem 4.4. Let Q be a fuzzy quasi-order on A. Then the following conditions are equivalent:
(i) Q is a solution to the system (34);
(ii) Q is a solution to the systems (32) and (33);
11
(iii) Q is a solution to the system
q
U 6 φr (U) ,
U 6 W,
(40)
where U is an unknown fuzzy relation on A.
Proof. This follows immediately by Theorems 4.2 and 4.3.
Next, let us consider the following systems of fuzzy relation inequalities and equations:
U ◦ Vi 6 Vi ◦ U, U−1 ◦ Vi 6 Vi ◦ U−1 ,
Vi ◦ U 6 U ◦ Vi , Vi ◦ U
U ◦ Vi = Vi ◦ U, U
−1
−1
6U
−1
◦ Vi ,
◦ Vi = Vi ◦ U
−1
U 6 W , U−1 6 W ;
(i ∈ I),
(i ∈ I),
6 W;
(42)
−1
6W;
(43)
U 6W, U
(i ∈ I),
U 6 W, U
(41)
−1
where U is an unknown fuzzy relation on A. Clearly, a fuzzy relation R is a solution to (41) (resp. (42), (43))
if and only if both R and R−1 are solutions to (32) (resp. (33), (34)), and moreover, a symmetric fuzzy relation
is a solution to (41) (resp. (42), (43)) if and only if it is solution to (32) (resp. (33), (34)).
Theorem 4.5. The systems (41), (42), and (43) have the greatest solutions.
If W is a fuzzy equivalence, then these greatest solutions are also fuzzy equivalences.
Proof. We will prove only the assertion concerning the system (41). The assertions concerning the remaining
two systems can be proved analogously.
It is clear that the system (41) has at least one solution, the empty relation on A. Let {R j } j∈J be a family
of all fuzzy relations on A which are solutions to (41), and let E denote the union of this family. Then both
E and E−1 are solutions to the system (32), i.e., E is a solution to the system (41). Evidently, E is the greatest
solution to (41).
Let W be a fuzzy equivalence. We have that E is reflexive, since ∆A 6 E. As in the proof of Theorem 4.1 we
show that E is transitive. Since E is the greatest solution to (41), and obviously, E−1 is also a solution to (41),
we obtain that E−1 6 E, so E is symmetric. Therefore, E is a fuzzy equivalence.
Our next task is to give various characterizations of those solutions to (41), (42), and (43) which are fuzzy
equivalences, i.e., of right regular, left regular and regular fuzzy equivalences with respect to the family
{Vi }i∈I and the fuzzy relation W. To do that we consider the lattice E (A) of all fuzzy equivalences on A, and
we define functions φerr , φelr , and φer of E (A) into itself by
^
^
^
ˆ Vi , φer (E) =
ˆ Vi ) = φerr (E) ∧ φelr (E),
E⊘
(Vi Ŵ E) ∧ (E ⊘
(44)
φerr (E) =
Vi Ŵ E, φelr (E) =
i∈I
i∈I
i∈I
ˆ Vi are fuzzy equifor every E ∈ E (A). As we have already noticed, for each i ∈ I we have that Vi Ŵ E and E ⊘
valences (even if E is not a fuzzy equivalence). Therefore, φerr (E), φelr (E), and φer (E) are also fuzzy equivalences,
so indeed, φerr , φelr , and φer are functions of E (A) into itself.
First we consider right regular fuzzy equivalences with respect to {Vi }i∈I and W.
Theorem 4.6. Let E be a fuzzy equivalence on A. Then the following conditions are equivalent:
(i) E is a solution to the system (41);
(ii) E is a solution to the system (32);
(ii) E is a solution to the system (36);
(iv) E is a solution to the system
U 6 φerr (U) ,
U 6W,
(45)
where U is an unknown fuzzy relation on A.
12
Proof. This follows immediately by Theorem 4.2 and the fact that E−1 = E.
The next two theorems characterize right regular, resp. regular, fuzzy equivalences with respect to the
family {Vi }i∈I and the fuzzy relation W.
Theorem 4.7. Let E be a fuzzy equivalence on A. Then the following conditions are equivalent:
(i) E is a solution to the system (42);
(ii) E is a solution to the system (33);
(ii) E is a solution to the system (38);
(iv) E is a solution to the system
U 6 φelr (U) ,
U 6W,
(46)
where U is an unknown fuzzy relation on A.
Proof. This can be proved analogously as Theorem 4.6.
Theorem 4.8. Let E be a fuzzy equivalence on A. Then the following conditions are equivalent:
(i) E is a solution to the system (43);
(ii) E is a solution to the system (34);
(iii) E is a solution to the systems (32) and (33);
(iv) E is a solution to the system (36) and (38);
(v) E is a solution to the system
U 6 φer (U) ,
U 6 W,
(47)
where U is an unknown fuzzy relation on A.
Proof. This follows immediately by Theorems 4.6 and 4.7.
Example 4.9. Let L be the Boolean structure, let A be any three-element set, and let fuzzy relations {Vi }i∈I ,
where I = {1, 2}, be given by

1

V1 = 0

0

0 0
0 0 ,

0 0

1

V2 = 1

1

0 0
1 0 .

0 0
Let Qrr , Qlr , and Qr denote respectively the greatest right regular, left regular and regular fuzzy quasi-order,
and Err , Elr , and Er the greatest right regular, left regular and regular fuzzy equivalence on A with respect
to {Vi }i∈I . Then






1 0 0
1 1 1
1 0 0
0 1 0
1 1 0
0 1 1
r
rr
lr
r
lr
rr
Q = 
 ,
 , Q = 
 , Q = E = E = E = 






0 0 1
0 1 1
1 1 1
and we conclude the following.
err = Qrr ∧ (Qrr )−1 of Qrr is given by
(a) The natural fuzzy equivalence Q

1 0
err = 0 1
Q

0 1

0
1 .

1
13
Hence, the greatest right regular fuzzy equivalence (with respect to a given family of fuzzy relations) is not
necessarily the natural fuzzy equivalence of the greatest right regular fuzzy quasi-order, that is, the natural
fuzzy equivalence of a right regular fuzzy quasi-order is not necessarily a right regular fuzzy equivalence.
(b) We have that


1 0 0
0 1 0
rr
lr
Q ∧ Q = 
 , Qr ,


0 1 1
and therefore, the intersection of the greatest right and left regular quasi-orders (with respect to a given
family of fuzzy relations) do not necessarily coincide with the greatest regular fuzzy quasi-order.
5. Computation of the greatest solution to a weakly linear system
In the preceding section we have proved the existence of the greatest solution to a weakly linear system.
Here we consider the problem how to effectively compute this solution.
Let A be a non-empty set, let K (A) denote any of the complete lattices R(A), Q(A), and E (A), and let
φ : K (A) → K (A) be an isotone function, i.e., let for all R, S ∈ K (A), R 6 S imply φ(R) 6 φ(S). A relation
R ∈ K (A) is called a fixed point of φ if φ(R) = R, a pre-fixed point of φ if φ(R) 6 R, and a post-fixed point of φ if
R 6 φ(R) (cf. [75]). The well-known Knaster-Tarski fixed point theorem (stated and proved in a more general
context, for complete lattices) asserts that the sets of all fixed points, pre-fixed points and post-fixed points
of φ form complete lattices, the least fixed point of φ coincides with the least pre-fixed point, and the greatest
fixed point of φ coincides with the greatest post-fixed point. This theorem establishes existence of the least
and the greatest fixed points, but it does not give an effective procedure for their computing. A majority of
methods for computing the least and greatest fixed points is based on another theorem – the Kleene fixed
point theorem. If φ is a meet-continuous function (a function which preserves all lower-directed meets), the
Kleene fixed point theorem asserts that the greatest fixed point of φ can be ”computed” as the meet of the
descending Kleene chain of φ, which is defined as follows:
S1 = ∇A , Sk+1 = φ(Sk ), for every k ∈ N.
(48)
This sequence is really descending, since φ is isotone and φ(∇A ) 6 ∇A . Without requiring meet-continuity, if
φ is only isotone, we have
^
Ŝ 6
Sk ,
(49)
k∈N
where Ŝ the greatest fixed point of φ. The Kleene fixed point theorem assumes that φ is meet-continuous
to provide equality in (49). This equality enables either to effectively compute (if the sequence stabilizes
at some Sk ) or to approximate the greatest fixed point of φ. However, in many cases the equality in (49) is
provided even without requiring meet-continuity of φ, it may be sufficient if it is isotone. For example, if
the underlying structure L of membership values is finite, and A is a finite set, then K (A) is also a finite
lattice, and the sequence {Sk }k∈N must be finite, and its least element is equal to Ŝ. In particular, this is the
case when L is the Boolean structure, i.e., when we work with crisp relations on a finite set.
On the other hand, in many cases we do not need to find the greatest fixed or post-fixed point, but
to find the greatest post-fixed point of φ which is contained in a given fuzzy relation W ∈ K (A), i.e., the
greatest solution to the system of fuzzy relation inequalities
U 6 φ(U),
U 6 W,
(50)
where U is an unknown fuzzy relation on A. It is not hard to check that the set of all post-fixed points of
φ contained in W also forms a complete lattice, and therefore, there is the greatest post-fixed point of φ
contained in W, but it is not necessarily a fixed-point of φ. For example, the function ψ which maps every
14
element of K (A) into the greatest element ∇A is clearly isotone, every element of K (A) is a post-fixed point
of ψ, and ∇A is the only fixed point of ψ. Thus, if we choose any W ∈ K (A) different than ∇A , then W itself is
the greatest post-fixed point contained in W, and there is no any fixed point of ψ contained in W. However,
the greatest post-fixed point of φ contained in W can be ”computed” in a similar way as the greatest fixed
point of φ. If we just replace ∇A by W in (48), then the sequence defined as in (48) is not necessarily descending, since φ(W) 6 W does not necessarily hold, but we can modify (48), and define the descending sequence
{Rk }k∈N in the following way:
R1 = W, Rk+1 = Rk ∧ φ(Rk ), for every k ∈ N.
(51)
If W = ∇A then (51) yields (48), and moreover, whenever φ(W) 6 W, then we have that φ(Rk ) 6 Rk , for every
k ∈ N, and in this case Rk+1 is given by Rk+1 = φ(Rk ). We can also easily show that
^
R̂ 6
Rk ,
(52)
k∈N
where R̂ the greatest post-fixed point of φ contained in W, and we will search for certain conditions which
provide equality in (52). Especially, we will search for certain conditions under which the sequence {Rk }k∈N
stabilizes at Rk , for some k ∈ N, i.e., there exist k, l ∈ N such that Rk = Rk+l . In this case we can easily verify
that Rk = Rk+1 = R̂, what means that R̂ is computed after a finite number of iterations.
The sequence {Rk }k∈N is called image-finite if the set
[
Im(Rk )
k∈N
is finite. The following lemma can be easily proved.
Lemma 5.1. The sequence {Rk }k∈N defined by (51) is finite if and only if it is image-finite.
The function φ is called image-localized if there exists a finite X ⊆ L such that for every fuzzy relation
R ∈ K (A) we have
Im(φ(R)) ⊆ hX ∪ Im(R)i,
(53)
where hX ∪ Im(R)i denotes the subalgebra of L generated by the set X ∪ Im(R).
Theorem 5.2. Let the function φ be image-localized, let W be a fuzzy relation on A, and let {Rk }k∈N be a sequence of
fuzzy relations on A defined by (51). Then
[
Im(Rk ) ⊆ hX ∪ Im(W)i.
(54)
k∈N
If, moreover, hX ∪ Im(W)i is a finite subalgebra of L , then the sequence {Rk }k∈N is finite.
Proof. By induction we will prove that
Im(Rk ) ⊆ hX ∪ Im(W)i.
(55)
for every k ∈ N. Clearly, (55) holds for k = 1.
Assume that (55) holds for some k ∈ N. Then X ∪ Im(Rk ) ⊆ hX ∪ Im(W)i, so hX ∪ Im(Rk )i ⊆ hX ∪ Im(W)i,
and we have that
Im(φ(Rk )) ⊆ hX ∪ Im(Rk )i ⊆ hX ∪ Im(W)i.
Since hX ∪ Im(W)i is closed under finite meets, and since Im(Rk ) ⊆ hX ∪ Im(W)i, Im(φ(Rk )) ⊆ hX ∪ Im(W)i,
and Rk+1 = Rk ∧ φ(Rk ), we obtain that Im(Rk+1 ) ⊆ hX ∪ Im(W)i. Now, by induction we conclude that (55)
holds for all k ∈ N, and hence, (54) holds.
Finally, if hX ∪ Im(W)i is finite, then the sequence {Rk }k∈N is finite according to (54) and Lemma 5.1.
15
q
q
q
Our main goal is to examine the greatest post-fixed points of the functions φrr , φlr , φr , φerr , φelr , and φer
defined in the previous section. As in the previous section, we will consider a non-empty set A, a family
{Vi }i∈I of fuzzy relations on A, and a fuzzy relation W on A, but here we will assume that the set A is finite,
and that the family {Vi }i∈I is also finite, i.e., the index set I is finite.
q
q
q
Returning to the functions φrr , φlr , φr , φerr , φelr , and φer , we prove the following.
q
q
q
Theorem 5.3. All functions φrr , φlr , φr , φerr , φelr , and φer are isotone and image-localized.
q
Proof. We will prove only the assertion concerning the function φrr . Assertions concerning the remaining
functions can be proved analogously.
q
First we prove that φrr is isotone. Let P, Q ∈ Q(A) such that P 6 Q. Then we have that P ◦ Q = Q. Next,
consider arbitrary i ∈ I and a, b ∈ A. According to (5), for arbitrary c, d ∈ A we have that
(Vi ◦ P)(b, c) → (Vi ◦ P)(a, c) 6 (Vi ◦ P)(b, c) ⊗ Q(c, d) → (Vi ◦ P)(a, c) ⊗ Q(c, d).
Now, by (29) and (11) we obtain that
^
(Vi W P)(a, b) 6
(Vi ◦ P)(b, c) → (Vi ◦ P)(a, c)
c∈A
6
^h
c∈A
6
h_
c∈A
(Vi ◦ P)(b, c) ⊗ Q(c, d) → (Vi ◦ P)(b, c) ⊗ Q(c, d)
i
i
h_
i
(Vi ◦ P)(b, c) ⊗ Q(c, d) →
(Vi ◦ P)(a, c) ⊗ Q(c, d)
c∈A
= (Vi ◦ P ◦ Q)(b, d) → (Vi ◦ P ◦ Q)(a, d) = (Vi ◦ Q)(b, d) → (Vi ◦ Q)(a, d).
Since this holds for every d ∈ A, we conclude that
^
(Vi W P)(a, b) 6
(Vi ◦ Q)(b, d) → (Vi ◦ Q)(a, d) = (Vi W Q)(a, b),
d∈A
and hence, Vi W P 6 Vi W Q. By this it follows that
^
^
q
q
φrr (P) =
Vi W P 6
Vi W Q = φrr (Q),
i∈I
i∈I
q
and we have proved that φrr is isotone.
S
q
Next, for an arbitrary Q ∈ Q(A) we have that Im(φrr (Q)) ⊆ hX ∪ Im(Q)i, where X = i∈I Im(Vi ), and by
q
the hypothesis, I is finite, so X is finite. This confirms that φrr is image-localized.
S
The notation X = i∈I Im(Vi ) will be also used in the next theorem.
q
q
q
Theorem 5.4. Let φ be any of the functions φrr , φlr , and φr (resp. any of φerr , φelr , and φer ), let W be a fuzzy quasi-order
(resp. a fuzzy equivalence) on A, and let {Rk }k∈N be the sequence of fuzzy relations on A defined by (51).
If hX ∪ Im(W)i is a finite subalgebra of L , then the sequence {Rk }k∈N is finite, and if k is the least natural number
such that Rk = Rk+1 , then Rk is the greatest solution to the system U 6 φ(U), U 6 W.
Proof. This follows immediately by Theorems 5.2 and 5.3.
In particular, if L is a locally finite algebra, then hX ∪ Im(W)i is a finite subalgebra of L , since X ∪ Im(W)
is a finite set. In this case, for every finite set A, every finite family {Vi }i∈I of fuzzy relations on A, and every
fuzzy quasi-order (fuzzy equivalence) W on A, based on the above theorems we can effectively compute
the greatest right regular, left regular and regular fuzzy quasi-orders and fuzzy equivalences with respect
to {Vi }i∈I and W. But, if L is not a locally finite algebra, then the sequence {Rk }k∈N of fuzzy relations defined
by (51) is not necessarily finite, as the following example shows.
16
Example 5.5. Let L be the Goguen (product) structure. Assume that A is any two-element set, V is a fuzzy
relation on A given by
"
#
1 0
V=
,
0 21
W = ∇A , φ is a function of Q(A) into itself defined by φ(Q) = V W Q, for each Q ∈ Q(A), and {Rk }k∈N is a
sequence of fuzzy quasi-orders on A defined by (51). Then for every k ∈ N we have that
"
#
1 1
Rk = 1
, k ∈ N,
1
2k−1
and hence, the sequence {Rk }k∈N is infinite. We also have that the greatest solution to the fuzzy relation
inequality U 6 φ(U), i.e., the greatest right regular fuzzy quasi-order on A with respect to V, is given by
"
#
1 1
rr
Q =
.
0 1
Evidently, Qrr is the meet of the sequence {Rk }k∈N .
If L is not a locally finite algebra, and if the sequence {Rk }k∈N is infinite, then the greatest solution to the
system U 6 φ(U), U 6 W, is different than Rk , for any k ∈ N, but if it is the meet of this sequence, then it can
be approximated by Rk , for some k ∈ N. For that reason, in the sequel we will examine certain conditions
under which this greatest solution is the meet of the sequence {Rk }k∈N .
We will consider the case when L = (L, ∧, ∨, ⊗, →, 0, 1) is a complete residuated lattice satisfying the
following conditions:
^ ^
x∨
yi =
(x ∨ yi ),
(56)
i∈I
i∈I
i∈I
i∈I
^ ^
x⊗
yi =
(x ⊗ yi ),
(57)
for all x ∈ L and {yi }i∈I ⊆ L. Let us note that if L = ([0, 1], ∧, ∨, ⊗, →, 0, 1), where [0, 1] is the real unit interval
and ⊗ is a left-continuous t-norm on [0, 1], then (56) follows immediately by linearity of L , and L satisfies
(57) if and only if ⊗ is a continuous t-norm, i.e., if and only if L is a BL-algebra (cf. [3, 4]). Therefore,
conditions (56) and (57) hold for every BL-algebra on the real unit interval. In particular, the Łukasiewicz,
Goguen (product) and Gödel structures fulfill (56) and (57).
We have that the following is true:
q
q
q
Theorem 5.6. Let φ be any of the functions φrr , φlr , and φr (resp. any of φerr , φelr , and φer ), let W be a fuzzy quasi-order
(resp. a fuzzy equivalence) on A, and let {Rk }k∈N be the sequence of fuzzy relations on A defined by (51).
If L is a complete residuated lattice satisfying (56) and (57), then
^
R̂ =
Rk .
(58)
k∈N
where R̂ is the greatest solution to the system of fuzzy relation inequalities U 6 φ(U), U 6 W.
q
Proof. We will prove only the assertion concerning the case when φ = φrr and W is a fuzzy quasi-order. The
remaining cases can be considered similarly.
It was proved in [26] that if (56) holds, then for all descending sequences {xk }k∈N , {yk }k∈N ⊆ L we have
^
^ ^ (xk ∨ yk ) =
xk ∨
yk .
(59)
k∈N
k∈N
k∈N
17
For the sake of simplicity set
^
R=
Rk .
k∈N
Clearly, R is a fuzzy quasi-order, and R̂ 6 R. To prove the opposite inequality it is enough to prove that R is
a solution to U 6 φ(U). First, for arbitrary a, b, c ∈ A, i ∈ I and k ∈ N we have that
R(a, b) 6 Rk+1 (a, b) 6 φ(Rk )(a, b) 6 (Vi ◦ Rk )(b, c) → (Vi ◦ Rk )(a, c),
(60)
and by (60) and (10) we obtain that
R(a, b) 6
^
^
^
(Vi ◦ Rk )(b, c) → (Vi ◦ Rk )(a, c) 6
(Vi ◦ Rk )(b, c) →
(Vi ◦ Rk )(a, c) .
k∈N
k∈N
(61)
k∈N
Next,
^
(Vi ◦ Rk )(b, c) =
k∈N
^_
Vi (b, d) ⊗ Rk (d, c)
k∈N d∈A
=
_ ^ Vi (b, d) ⊗ Rk (d, c)
(by (59))
d∈A k∈N
(62)
^
_
=
Vi (b, d) ⊗
Rk (d, c)
d∈A
=
(by (57))
k∈N
_
Vi (b, d) ⊗ R(d, c) = (Vi ◦ R)(b, c).
d∈A
Use of condition (59) is justified by the facts that A is finite, and that {Rk (d, c)}k∈N is a descending sequence,
so {Vi (b, d) ⊗ Rk (d, c)}k∈N is also a descending sequence. In the same way we prove that
^
(Vi ◦ Rk )(a, c) = (Vi ◦ R)(a, c).
(63)
k∈N
Therefore, by (61), (62) and (63) we obtain that
R(a, b) 6 (Vi ◦ R)(b, c) → (Vi ◦ R)(a, c).
Since this inequality holds for all i ∈ I and c ∈ A, we have that
^
^ ^
R(a, b) 6
(Vi ◦ R)(b, c) → (Vi ◦ R)(a, c) =
(Vi W R)(a, b),
i∈I c∈A
i∈I
and hence,
^
Vi W R = φ(R).
R6
i∈I
Thus, we conclude that R is a solution to U 6 φ(U), what completes the proof of the theorem.
Example 5.7. Let L be the Goguen (product) structure. Assume that A is any three-element set, V is a fuzzy
relation on A given by

 0

V =  0
1
2
1
1
0

1 
1  ,

0
18
q
q
and W = ∇A . Moreover, let φrr be a function of Q(A) into itself defined by φrr (Q) = V W Q, for any Q ∈ Q(A),
let φerr be a function of E (A) into itself defined by φerr (E) = V Ŵ E, for any E ∈ E (A), and let the sequence
{Qk }k∈N of fuzzy quasi-orders and {Ek }k∈N of fuzzy equivalences on A be defined by
q
Q1 = W, Qk+1 = φrr (Qk ), for every k ∈ N,
E1 = W, Ek+1 = φerr (Ek ), for every k ∈ N,
The sequence {Ek }k∈N of fuzzy equivalences is infinite, since for each k ∈ N we have that

1 
1
 1

2k−1 

1 

1
Ek =  1
k−1  .
2

 1
1
1
2k−1
2k−1
On the other hand, the sequence {Qk }k∈N of fuzzy quasi-orders is finite, since

1 1

Q1 = 1 1

1 1


1
 1
1 , Q2 =  1

1
1
2
1
1
1
2

1
1 , Qk = Q2 , for every k ∈ N, k > 3.

1
Note that {Qk }k∈N is a finite sequence although hX ∪ Im(W)i = h0, 12 , 1i is an infinite subalgebra of L .
In some situations we do not need fuzzy solutions to a weakly linear system, but crisp solutions to
this system. Moreover, in cases where our algorithm for computing the greatest solution to a weakly linear
system do not terminate in a finite number of steps, we can search for the greatest crisp solution to this
system. It can be understood as a kind of “approximation” of the greatest fuzzy solution. Here we show that
the above given algorithm for computing the greatest solution to a weakly linear system can be modified to
compute the greatest crisp solution to the system. The new algorithm terminates in a finite number of steps,
independently of the local finiteness of the underlying structure of truth values. We also show that the
greatest crisp solution can not be obtained simply by taking the crisp part of the greatest fuzzy solution.
Let A be a non-empty finite set, let K (A) denote any of the complete lattices R(A), Q(A), and E (A), and
let K c (A) denote the set of all crisp relations from K (A). It is not hard to verify that K c (A) is a complete
sublattice of K (A), i.e., the meet and the join in K (A) of an arbitrary family of crisp relations from K (A)
are also crisp relations. We also have that Rc ∈ K c (A), for every R ∈ K (A).
For each function φ : K (A) → K (A) we define a function φc : K c (A) → K c (A) by
φc (R) = (φ(R))c , for any R ∈ K c (A).
If φ is isotone, then it can be easily shown that φc is also an isotone function.
Proposition 5.8. Let φ : K (A) → K (A) be an isotone function, and let W ∈ K (A) be a given fuzzy relation. A crisp
relation R ∈ K c (A) is the greatest crisp solution to the system
U 6 φ(U),
U 6 W,
(64)
in the complete lattice K (A), if and only if it is the greatest solution to the system
U 6 φc (U),
U 6 Wc ,
(65)
in the complete lattice K c (A), where U is an unknown fuzzy, resp. crisp relation.
Furthermore, a sequence {Rk }k∈N ⊆ K (A) defined by
R1 = W c , Rk+1 = Rk ∧ φc (Rk ), for every k ∈ N,
(66)
is a finite descending sequence of crisp relations, and the least member of this sequence is the greatest solution to the
system (65) in K c (A).
19
Proof. It is well-known that a crisp set is contained in a fuzzy set if and only if it is contained in its crisp
part. By this it follows that R is a solution to (64) if and only if it is a solution to (65), and therefore, R is the
greatest crisp solution to (64) if and only if it is the greatest solution to (65).
By the isotonicity of φ it follows that φc is also isotone, so {Rk }k∈N ⊆ K (A) is a descending sequence.
Clearly, this sequence consists of crisp relations, and since A is a finite set, any sequence of crisp relations
on A must be finite. Finally, as we have mentioned earlier, the least member of this finite sequence is the
greatest post-fixed point of φc contained in W c , i.e., it is the greatest solution to (65).
Let A be a finite non-empty set, let {Vi }i∈I , where I is a finite set, be a family of fuzzy relations on A, and
q
q
q
let W be a fuzzy quasi-order, resp. fuzzy equivalence on A. Taking φ to be any of the functions φrr , φlr , φr ,
φerr , φelr , and φer defined by (35) and (44), Proposition 5.8 gives algorithms for computing the greatest crisp
solutions to the systems (37), (39), (40), (45), (46), and (47).
It is worth noting that in these cases the function φc can be characterized as follows:
q c
(a, b) ∈ φrr (Q) ⇔ (∀i ∈ I)(∀c ∈ A) (Vi ◦ Q)(b, c) 6 (Vi ◦ Q)(a, c),
q c
(a, b) ∈ φlr (Q) ⇔ (∀i ∈ I)(∀c ∈ A) (Q ◦ Vi )(c, a) 6 (Q ◦ Vi )(c, b),
c
(a, b) ∈ φerr (E) ⇔ (∀i ∈ I)(∀c ∈ A) (Vi ◦ E)(b, c) = (Vi ◦ E)(a, c),
c
(a, b) ∈ φelr (E) ⇔ (∀i ∈ I)(∀c ∈ A) (E ◦ Vi )(c, a) = (E ◦ Vi )(c, b),
q c q c q c c c c
φr = φrr ∧ φlr , φer = φerr ∧ φelr ,
for all Q ∈ Q(A), E ∈ E (A), and a, b ∈ A.
Example 5.9. Let L be the Gödel structure, let A be any three-element set, and let fuzzy relations {Vi }i∈I ,
where I = {1, 2}, be given by




0.5 0.6 0.2
 1 0.3 0.4
0.6 0.3 0.4
0.5 1 0.3
V1 = 
 .
 , V2 = 




0.7 0.7 1
0.4 0.6 0.7
Let Qrr , Qlr , and Qr denote respectively the greatest right regular, left regular and regular fuzzy quasi-order,
and Err , Elr , and Er the greatest right regular, left regular and regular fuzzy equivalence on A with respect
to {Vi }i∈I . Then






1 0.6
1 0.7
1 0.6
 1
 1
 1

1 0.6 .
1 0.7 , Err = Er =  1
1 0.6 , Qlr = Elr =  1
Qrr = Qr =  1






0.6 0.6 1
0.7 0.7 1
0.7 0.7 1
On the other hand, there is no any non-trivial crisp quasi-order or equivalence which is right regular, left
regular or regular with respect to the family {Vi }i∈I , i.e., weakly linear systems corresponding to the family
{Vi }i∈I and W = ∇A , have no non-trivial solutions in Q c (A), resp. E c (A).
Furthermore, this example shows that for any fuzzy relation which is a solution to a weakly linear system,
its crisp part need not be a solution to the same system, and also, the greatest fuzzy solution to this system
can have strictly smaller index than the greatest crisp solution to the same system.
6. Some related linear systems
In this section we show that weakly linear systems are closely related to two types of linear systems. First
we consider linear systems having narrower sets of solutions than weakly linear systems, and then we
discuss certain linear systems having larger sets of solutions.
Let A be a non-empty set (not necessarily finite), let {Vi }i∈I be a given family of fuzzy relations on A
(where I is also not necessarily finite), and let W be a given fuzzy relation on A.
20
Let us consider the following linear systems of fuzzy relation inequalities:
U ◦ Vi 6 Vi ,
(i ∈ I),
U 6 W;
(67)
Vi ◦ U 6 Vi ,
(i ∈ I),
U 6 W;
(68)
U ◦ Vi 6 Vi , Vi ◦ U 6 Vi ,
(i ∈ I),
U 6 W;
(69)
U ◦ Vi 6 Vi , U−1 ◦ Vi 6 Vi ,
(i ∈ I),
U 6 W, U−1 6 W ;
Vi ◦ U 6 Vi , Vi ◦ U
−1
6 Vi ,
U ◦ Vi 6 Vi , Vi ◦ U 6 Vi , U
(i ∈ I),
−1
◦ Vi 6 Vi , Vi ◦ U
−1
6 Vi ,
(i ∈ I),
(70)
U 6 W, U
−1
6 W;
(71)
U 6 W, U
−1
6 W;
(72)
where U is an unknown fuzzy relation on A. Clearly, if we restrict the unknown U to take values in the set of
all reflexive fuzzy relations on A, then the above systems become equivalent to the corresponding systems
in which inequalities involving Vi ’s are replaced by equations. We also have that each reflexive solution
to the system (67) (resp. (68), (69), (70), (71), (72)) is also a solution to the system (32) (resp. (33), (34), (41),
(42), (43)). Solutions to (67), (68), and (69) in Q(A) will be called respectively strongly right regular, strongly
left regular, and strongly regular fuzzy quasi-orders with respect to the family of fuzzy relations {Vi }i∈I and the
fuzzy relation W, and solutions to these systems in E (A), i.e., solutions to (70), (71), and (72) in E (A), will be
called strongly right regular, strongly left regular, and strongly regular fuzzy equivalences with respect to {Vi }i∈I
and W. If W = ∇A , then we call them strongly right regular, strongly left regular, and strongly regular with
respect to {Vi }i∈I .
The assertions of the following proposition are mainly already given in different forms in [69, 70, 71, 77]
and other related articles, but we show them here in the form which we need.
Proposition 6.1. The following are true.
(a) The sets of all solutions to the systems (67)–(72) form principal ideals of the lattice R(A).
(b) If W is a fuzzy quasi-order, then the greatest solutions to (67), (68), and (69) are also fuzzy quasi-orders, and
they can be respectively expressed as Qsrr ∧ W, Qslr ∧ W, and Qsr ∧ W, where Qsrr , Qslr , and Qsr are fuzzy
quasi-orders on A defined by
Qsrr =
^
Vi \Vi ,
Qslr =
i∈I
^
Vi /Vi ,
Qsr = Qsrr ∧ Qslr .
(73)
i∈I
(c) If W is a fuzzy equivalence, then the greatest solutions to (70), (71), and (72) are also fuzzy equivalences, and
they can be respectively expressed as Esrr ∧ W, Eslr ∧ W, and Esr ∧ W, where Esrr , Eslr , and Esr are fuzzy
equivalences on A defined by
Esrr =
^
i∈I
Vi Vi ,
Eslr =
^
Vi ∥Vi ,
Esr = Esrr ∧ Eslr .
(74)
i∈I
Proof. It is easy to check that (a) is true.
By Theorem 3.5 we have that Qsrr and Qslr are fuzzy quasi-orders, and they are respectively the greatest
solutions to the systems (67) and (68) with W = ∇A (in which the inequality U 6 W is omitted). Consequently,
Qsr is also a fuzzy quasi-order, and by (a), it is the greatest solution to the system (69) with W = ∇A . It is
evident that Esrr , Eslr , and Esr are respectively the natural fuzzy equivalences of Qsrr , Qsrr , and Qsr , and by
this, by (a) and by symmetry we obtain that Esrr , Esrr , and Esr are respectively the greatest solutions to the
systems (70), (71), and (72) with W = ∇A .
Now, again by (a), we have that Qsrr ∧ W, Qslr ∧ W, Qsr ∧ W, Esrr ∧ W, Eslr ∧ W, and Esr ∧ W are respectively
the greatest solutions to (67), (68), (69), (70), (71), and (72).
21
Example 6.2. Let us consider again the family {Vi }i∈I , I = {1, 2}, of fuzzy relations from Example 5.9. Then

 1

srr
Q = 0.3

0.4

 1

srr
E = 0.2

0.2


0.2 0.2
 1

1 0.3 , Qslr = 0.4


0.4 1
0.4


0.2 0.2
 1
1 0.3 , Eslr = 0.3


0.3 1
0.2


0.3 0.2
 1

1 0.2 , Qsr = 0.3


0.3 1
0.4


0.3 0.2
 1
1 0.2 , Esr = 0.2


0.2 1
0.2

0.2 0.2
1 0.2 ,

0.3 1

0.2 0.2
1 0.2 .

0.2 1
Evidently, all these fuzzy relations are strictly less than the corresponding greatest solutions to the systems
(32)–(34), and (41)–(43), computed in Example 5.9, and they even have strictly greater indices than the mentioned fuzzy quasi-orders and fuzzy equivalences from Example 5.9.
Example 6.3. Let us consider a family {V1 } consisting of the single fuzzy relation V1 from Example 5.9. With
respect to this family, the greatest right regular fuzzy quasi-order Qrr , the greatest right regular crisp quasiorder Qcrr , and the greatest strongly right regular fuzzy quasi-order Qsrr on A are given by



1 1
1 1
 1

1 1 , Qcrr = 1 1
Qrr =  1



0 0
0.7 0.7 1



1
 1 0.3 0.3
1 , Qsrr = 0.3 1 0.3 .



0.4 0.4 1
1
Clearly, ind(Qsrr ) = 3 and ind(Qrr ) = ind(Qcrr ) = 2.
The sets of all crisp solutions to the weakly linear systems (32)–(34) and (41)–(43), as well as the sets of all
solutions to the systems (67)–(72), are respectively subsets of the sets of all solutions to (32)–(34), and the
greatest elements of these subsets are respectively less or equal to the greatest solutions to the weakly linear
systems (32)–(34) and (41)–(43). Therefore, the greatest crisp solutions to (32)–(34) and (41)–(43), and the
greatest solutions to (67)–(72), can be conceived as some kind of “lower approximations” to the greatest
solutions to (32)–(34) and (41)–(43). In the further text we study some linear systems of fuzzy relation
inequalities whose greatest solutions can be conceived as “upper approximations” to the greatest solutions
to (32), (33), (41), and (42). Although these “upper approximations” are not necessarily solutions to (32),
(33), (41), and (42), they can share certain important properties with the solutions to these systems. This
allows, in some cases, to use them instead of the greatest solutions to (32), (33), (41), and (42), even more
successfully, as it will be demonstrated in the next section.
In the sequel, let A be also a non-empty set (not necessarily finite), let {Vi }i∈I be a given family of fuzzy
relations on A (where I is also not necessarily finite), and let W be a given fuzzy quasi-order on A. Next, let
M denote the submonoid of the monoid (R(A), ◦) generated by the set {Vi }i∈I , i.e., let M consist of ∆A and
of all products of the form Vi1 ◦ · · · ◦ Vik , for i1 , . . . , ik ∈ I.
Let us consider the following systems of fuzzy relation inequalities:
U ◦ WPr 6 WPr ,
WPl
◦U 6
U◦
WPr
6
WPl ◦ U 6
WPl ,
WPr ,
WPl ,
U
−1
◦
WPr
−1
WPl ◦ U
6
6
WPr ,
WPl ,
(P ∈ M );
(75)
(P ∈ M );
(76)
(P ∈ M );
(77)
(P ∈ M );
(78)
where WPr = P ◦ W, WPl = W ◦ P, and U is an unknown fuzzy relation on A.
It is clear that a fuzzy relation R is a solution to (77) (resp. (78)) if and only if both R and R−1 are solutions
to (75) (resp. (76)). Solutions to (75) and (76) are respectively called weakly right regular and weakly left regular
relations with respect to {Vi }i∈I and W, and if W = ∇A , then we call them weakly right regular and weakly
left regular with respect to {Vi }i∈I .
Now we are ready to state and prove the following proposition.
22
Proposition 6.4. The following are true.
(a) The sets of all solutions to the systems (75)–(78) form principal ideals of the lattice R(A).
(b) The greatest solutions to (75) and (76) are fuzzy quasi-orders, and they can be respectively expressed as
^
^
Qwrr =
P W W, Qwlr =
W ⊘ P.
P∈M
(79)
P∈M
(c) The greatest solutions to (77) and (78) are fuzzy equivalences, and they can be respectively expressed as
^
^
ˆ P.
Ewrr =
P Ŵ W, Ewlr =
W⊘
P∈M
(80)
P∈M
(d) If W is a transitive fuzzy relation, then every solution to the system (32) (resp. (33), (41), (42)) is a solution to
the system (75) (resp. (76), (77), (78)).
Proof. It is clear that the systems (75)–(78) are special cases of the systems (67), (68), (70), and (71), with the
family {WPr }P∈M (resp. {WPr }P∈M ) taken instead of {Vi }i∈I , and ∇A instead of W, and according to Proposition 6.1
we obtain that (a) is true.
Furthermore, by Proposition 6.1 it follows that
^
^
^
^
W ⊘ P,
P W W,
Qwlr =
WPl /WPl =
Qwrr =
WPr \WPr =
P∈M
P∈M
E
wrr
=
^
WPr
WPr
P∈M
=
^
P∈M
P∈M
P Ŵ W,
E
wlr
P∈M
=
^
WPl ∥WPl
P∈M
=
^
ˆ P,
W⊘
P∈M
and hence, (b) and (c) hold.
Next, let W be a transitive fuzzy relation and let R be a solution to the system (32). By the transitivity of
W we obtain that R 6 W implies R ◦ W 6 W ◦ W 6 W, and hence, the first inequality in (75) (for P = ∆A ) is
satisfied. Let P = Vi1 ◦ · · · ◦ Vik , for some i1 , . . . , ik ∈ I. Then R ◦ P 6 P ◦ R, whence
R ◦ WPr = R ◦ P ◦ W 6 P ◦ R ◦ W 6 P ◦ W = WPr .
Therefore, R is a solution to the system (75).
In a similar way we prove the assertions concerning the systems (33), (41), and (42).
In the systems (75)–(78) we can replace the fuzzy relation W by a fuzzy subset τ of A, what gives related
systems with important applications (cf. Section 7.1). However, such systems involving a fuzzy subset τ can
be turned back into equivalent systems of the form (75)–(78) if we replace τ by a fuzzy relation Wτr (for (75)
and (77)) or a fuzzy relation Wτl (for (76) and (78)), where Wτr (a, b) = τ(a) and Wτl (a, b) = τ(b), for all a, b ∈ A.
Example 6.5. Let L
W on A be given by

1 0 0
0 0 0

V1 = 
0 0 0
0 0 0
Then we have that

1 1
0 1

Qrr = 
0 0

0 0
0
0
1
0
be the Boolean structure, let A be any four-element set, and let fuzzy relations V1 and

0
1
,
0

0

1
1
W = 
1
1

1
1
,
1

1

1
0

Err = 
0

0
1
1
1
1
0
1
0
0
0
0
1
0
0
0
1
0

1
1
.
1

1

0
0
,
0

1
Qwrr

1
1

= 
1

1
1
1
1
1
0
0
1
0

1
1
,
1

1
Ewrr

1
1

= 
0

1
1
1
0
1
0
0
1
0

1
1
.
0

1
Hence, Qrr , Qwrr and Err , Ewrr , and moreover, ind(Qrr ) = ind(Err ) = 4 and ind(Qwrr ) = ind(Ewrr ) = 2.
23
Example 6.6. Consider again fuzzy relations from Example 5.5. The set {WPr | P ∈ M } consists of all relations
represented by matrices of the form
"
#
1 1
1 , n ∈ N.
1
2n
2n
Hence, this set is infinite, i.e., here the system (75) consists of infinitely many inequalities. We also have that
"
#
1 1
Qwrr = Qrr =
.
0 1
Example 6.7. Let L be the Boolean structure, let A = {1, 2, . . . , n}, n ∈ N, n > 3, and let relations V0 , V1 , and
W on A be given by


 1 0 0 0 . . . 0 0 
 0 0 0 0 . . . 0 0 




 0 1 0 0 . . . 0 0 


V0 =  0 0 1 0 . . . 0 0  ,
. . . . . . . . . . . . . . . . . . . . .




 0 0 0 0 . . . 0 0 


0 0 0 0 ... 1 0
that is,


 1 0 0 0 . . . 0 0 
 1 0 0 0 . . . 0 0 




 0 1 0 0 . . . 0 0 


V1 =  0 0 1 0 . . . 0 0  ,
. . . . . . . . . . . . . . . . . . . . .




 0 0 0 0 . . . 0 0 


0 0 0 0 ... 1 0



 1, if i = j = 1 or 3 6 i 6 n, j = i − 1,
V0 (i, j) = 

 0, otherwise,



 1, if i = 1,
W(i, j) = 

 0, otherwise.


 1 1 1 1 . . . 1 1 
 0 0 0 0 . . . 0 0 




 0 0 0 0 . . . 0 0 


W =  0 0 0 0 . . . 0 0  ,
. . . . . . . . . . . . . . . . . . . . .




 0 0 0 0 . . . 0 0 


0 0 0 0 ... 0 0



 1, if i = j = 1 or 2 6 i 6 n, j = i − 1,
V1 (i, j) = 

 0, otherwise,
Then for arbitrary k ∈ N and α1 , α2 , . . . , αk ∈ {0, 1} we have that

1
1 ...
 1

 α1 α1 α1 . . .

Vα1 ◦ · · · ◦ Vαk ◦ W =  α2 α2 α2 . . .
 . . . . . . . . . . . .


αn−1 αn−1 αn−1 . . .
1
α1
α2
...
αn−1





 , if k > n − 1,




 1
 α
 1

. . .

Vα1 ◦ · · · ◦ Vαk ◦ W =  αk
 0


. . .

0
1
α1
...
αk
0
...
0
1
α1
...
αk
0
...
0
...
...
...
...
...
...
...
Therefore, the set {WPr | P ∈ M } consists of all relations represented by matrices of the form

1
1 ...
 1

 α1 α1 α1 . . .

 α2 α2 α2 . . .

 . . . . . . . . . . . .

αn−1 αn−1 αn−1 . . .

1

α1 

. . .

αk  , if k < n − 1.
0 

. . .
0

1 

α1 

α2  ,
. . . 

αn−1
for arbitrary α1, α2 , . . . , αn−1 ∈ {0, 1}, so there are 2n−1 different relations WPr , P ∈ M , i.e., in this case the system
(75) consists of 2n−1 inequalities.
Example 6.8. Consider again fuzzy relations from Example 5.7. Then the set {WPr | P ∈ M } consists of two
relations, which are represented by the matrices

1
1


1

1 1
1 1 ,

1 1

 1
 1

1
2
1
1
1
2

1 
1  ,
1
2
24
and we have that
Qwrr

 1

= Qrr =  1
1
2
1
1
1
2

1
1 ,

1
Ewrr

 1

= Err =  1
1
2
1
1
1
2

1

2
1
 .
2

1

Recall that in Example 5.7 we have proved that Err is computed as the meet of an infinite sequence of fuzzy
equivalences, whereas the sequence corresponding to Qrr is finite.
According to Proposition 6.4, the greatest solutions to the systems (32)–(42)) are respectively less or equal
to the greatest solutions to the systems (75)–(78)), and Example 6.5 shows that these inclusions can be strict.
Also, the indices of Qwrr , Qwlr , Ewrr , and Ewlr can be strictly smaller than the indices of Qrr , Qlr , Err , and Elr .
In the next section we will see that it can be very useful in some applications. However, there is a problem
concerning the number of inequalities in the systems (75)–(78)). Example 6.6 shows that this number can
be infinite. On the other hand, if the underlying structure L of truth values has a locally finite semiring
reduct (with respect to the join and multiplication), then this number must be finite, but by Example 6.7 we
show that it can be exponential in the number of elements of the considered set.
7. Applications in the fuzzy automata theory, concurrency theory, and social network analysis
7.1. Fuzzy automata theory – state reduction
A fuzzy automaton over L , or simply a fuzzy automaton, is a triple A = (A, X, δA ), where A and X are the
set of states and the input alphabet, and δA : A × X × A → L is an L -fuzzy subset of A × X × A, called the fuzzy
transition function. We can interpret δA (a, x, b) as the degree to which an input letter x ∈ X causes a transition
from a state a ∈ A into a state b ∈ A. The input alphabet X is always assumed to be finite, but for methodological reasons the set of states A is allowed to be infinite. A fuzzy automaton whose set of states is finite
is called a fuzzy finite automaton. Cardinality of a fuzzy automaton A = (A, X, δA ), denoted as |A |, is defined
as the cardinality of its set of states A.
Let X∗ denote the free monoid over the alphabet X, and let ε ∈ X∗ be the empty word. The mapping δA
∗
can be extended up to a mapping δA
∗ : A × X × A → L as follows: If a, b ∈ A, then



 1 if a = b
A
δ∗ (a, ε, b) = 
,
(81)

 0 otherwise
and if a, b ∈ A, u ∈ X∗ and x ∈ X, then
_
A
δA
δA
∗ (a, ux, b) =
∗ (a, u, c) ⊗ δ (c, x, b).
(82)
c∈A
According to (8) and Theorem 3.1 [52], we have that
_
A
δA
δA
∗ (a, uv, b) =
∗ (a, u, c) ⊗ δ∗ (c, v, b),
(83)
c∈A
for all a, b ∈ A and u, v ∈ X∗ , i.e., if w = x1 · · · xn , for x1 , . . . , xn ∈ X, then
_
δA
δA (a, x1 , c1 ) ⊗ δA (c1 , x2 , c2 ) ⊗ · · · ⊗ δA (cn−1 , xn , b).
∗ (a, w, b) =
(84)
(c1 ,...,cn−1 )∈An−1
Intuitively, the product δA (a, x1 , c1 ) ⊗ δA (c1 , x2 , c2 ) ⊗ · · · ⊗ δA (cn−1 , xn , b) represents the degree to which the
input word w causes a transition from a state a into a state b through the sequence of intermediate states
c1 , . . . , cn−1 ∈ A, and δA
∗ (a, w, b) represents the supremum of degrees of all possible transitions from a into b
caused by w.
25
For any u ∈ X∗ , and any a, b ∈ A define a fuzzy relation δA
u on A by
A
δA
u (a, b) = δ∗ (a, u, b),
(85)
called the fuzzy transition relation determined by u. Then (83) can be written as
A
A
δA
uv = δu ◦ δv ,
(86)
for all u, v ∈ X∗ .
A fuzzy recognizer is defined as a five-tuple A = (A, X, δA , σA , τA ), where (A, X, δA ) is as above, σA ∈ F (A)
is the fuzzy set of initial states, and τA ∈ F (A) is the fuzzy set of terminal states. Any fuzzy subset of the free
monoid X∗ is called a fuzzy language in X∗ over L , or briefly a fuzzy language. A fuzzy language recognized by
a fuzzy recognizer A = (A, X, δA , σA , τA ), denoted as L(A ), is a fuzzy language in X∗ defined by
_
A
L(A )(u) =
σA (a) ⊗ δA
(87)
∗ (a, u, b) ⊗ τ (b),
a,b∈A
or equivalently,
A
A
A
A
A
A
L(A )(u) = σA ◦ δA
u ◦ τ = σ ◦ δx1 ◦ δx2 ◦ · · · ◦ δxn ◦ τ ,
(88)
for every u = x1 x2 . . . xn ∈ X∗ , where x1 , x2 , . . . , xn ∈ X. In other words, the equality (87) means that the membership degree of the word u to the fuzzy language L(A ) is equal to the degree to which A recognizes or
accepts the word u.
If A = (A, X, δA ) is a fuzzy automaton such that δA is a crisp relation, i.e., if A is a fuzzy automaton taking
membership values in the Boolean structure, then A is an ordinary crisp non-deterministic automaton. It is
also called a Boolean automaton. Analogously we define non-deterministic or Boolean recognizers.
For more information on fuzzy automata and recognizers with membership values in complete residuated lattices (and certain related structures) we refer to the recent papers [25, 26, 38, 39, 40, 41, 52, 81].
Let A = (A, X, δA , σA , τA ) be a fuzzy recognizer and let Q be a fuzzy quasi-order on A. We can define the
fuzzy transition function δA/Q : A/Q × X × A/Q → L, the fuzzy set σA/Q : A/Q → L of initial states, and the
fuzzy set τA/Q : A/Q → L of terminal states by
_
δA/Q (Qa , x, Qb ) =
Q(a, a′ ) ⊗ δA (a′ , x, b′ ) ⊗ Q(b′ , b),
(89)
σA/Q (Qa ) =
_
a′ ,b′ ∈A
σA (a′ ) ⊗ Q(a′ , a),
τA/Q (Qa ) =
a′ ∈A
_
Q(a, a′ ) ⊗ τA (a′ ),
(90)
a′ ∈A
or equivalently
A
b
δA/Q (Qa , x, Qb ) = (Q ◦ δA
x ◦ Q)(a, b) = Qa ◦ δx ◦ Q ,
σ
A/Q
A
A
a
(Qa ) = (σ ◦ Q)(a) = σ ◦ Q ,
A/Q
τ
(91)
A
A
(Qa ) = (Q ◦ τ )(a) = Qa ◦ τ ,
(92)
for all a, b ∈ A and x ∈ X. We have that δA/Q is a well-defined function and A /Q = (A/Q, X, δA/Q , σA/Q , τA/Q )
is a fuzzy recognizer, called the afterset fuzzy recognizer of A with respect to Q. Analogously we can define
the foreset fuzzy recognizer of A with respect to Q, but these two fuzzy recognizers are isomorphic (cf. [81]),
and we will consider only the first one. If E is a fuzzy equivalence on A, then A /E is called the factor fuzzy
recognizer of A with respect to E.
For a fuzzy recognizer A = (A, X, δA , σA , τA ) and a fuzzy quasi-order Q on A we have that A and the
afterset fuzzy recognizer A /Q are equivalent, i.e., L(A ) = L(A /E), if and only if Q is a solution to the
system of fuzzy relation equations
σA ◦ τA = σA ◦ U ◦ τA ,
A
A
A
A
A
A
A
A
σA ◦ δA
x1 ◦ δx2 ◦ · · · ◦ δxn ◦ τ = σ ◦ U ◦ δx1 ◦ U ◦ δx2 ◦ U ◦ · · · ◦ U ◦ δxn ◦ U ◦ τ ,
26
(93)
for all n ∈ N and x1 , x2 , . . . , xn ∈ X, where U is an unknown fuzzy relation and solutions are sought in Q(A)
(cf. [25, 26, 81]). The system (93) is called the general system.
It is well-known that the state minimization problem for fuzzy and non-deterministic finite recognizers
is computationally hard, and a more practical problem is the state reduction problem, where a fuzzy recognizer has to be replaced by an equivalent fuzzy recognizer with as smaller number of states as possible, but
not necessarily minimal. When the state reduction is performed by means of fuzzy quasi-orders or fuzzy
equivalences, this means that we have to find as big a solution to the general system as possible. The set of all
solutions to the general system is non-empty and it is an ideal of the lattice Q(A), but it does not necessarily
have the greatest element. Moreover, the general system may consist of infinitely many equations, and
finding its nontrivial solutions may be a very difficult task. For that reason it is more convenient to consider
some instances of the general system, by which we mean systems, built from the same fuzzy relations, whose
sets of solutions are contained in the set of all solutions to the general system. These instances have to be as
general as possible, but they have to consist of finitely many equations and to be “easier” to solve.
The first equation in (93) has two important instances, the equations σA ◦ U = σA and U ◦ τA = τA . Both
equations are easy to solve. Namely, a fuzzy quasi-order Q is a solution to σA ◦ U = σA if and only if Q 6 Qσ
(where σ = σA ), and Q is a solution to U ◦ τA = τA if and only if Q 6 Qτ (where τ = τA ), what means that
the fuzzy quasi-orders Qσ and Qτ are respectively the greatest solutions to σA ◦ U = σA and U ◦ τA = τA
(cf. [22, 47, 69, 71, 77]). Similarly, a fuzzy equivalence E is a solution to σA ◦ U = σA (resp. U ◦ τA = τA ) if and
only if E 6 Eσ (resp. E 6 Eτ ), i.e., fuzzy equivalences Eσ and Eτ are the greatest solutions to σA ◦ U = σA and
U ◦ τA = τA in E (A). For these reasons, we can aim our attention to the system in the second row of (93),
and we can consider those solutions to this system which are contained in Qσ or Qτ .
τ
If we set I = X, {Vi }i∈I = {δA
x }x∈X and W = Q , then every solution to the system (32) (or the system (36)) in
τ
Q(A), i.e., every right regular fuzzy quasi-order with respect to {δA
x }x∈X and Q , is a solution to the general
system. This also holds for the system (33) (or the system (38)) with I = X, {Vi }i∈I = {δA
x }x∈X and W = Qσ ,
i.e., every left regular fuzzy quasi-order with respect to {δA
x }x∈X and Qσ is a solution to the general system.
State reduction of fuzzy recognizers by means of right and left regular fuzzy quasi-orders and fuzzy
equivalences has been recently studied in [25, 26, 81], and in the case of non-deterministic recognizers in
[16, 17, 18, 42, 43, 44, 45]. In these papers the terms “right” and “left invariant” have been used instead of
“right” and “left regular”.
τ
Strongly right regular fuzzy quasi-orders and fuzzy equivalences with respect to {δA
x }x∈X and Q , as well
A
as strongly left regular fuzzy quasi-orders and fuzzy equivalences with respect to {δx }x∈X and Qσ , have been
also studied in [26, 81]. It has been shown that they are much easier to construct, but give worse reductions
than the right and left regular ones (cf. Examples 5.9 and 6.2). According to Example 6.3,they even give worse
reductions than right and left regular crisp quasi-orders and crisp equivalences. There are also many papers
dealing with state reduction of fuzzy recognizers by means of crisp equivalences (cf. [2, 19, 49, 55, 61, 72]), but
according to Example 5.9, crisp right and left regular quasi-orders and equivalences give worse reductions
than their fuzzy counterparts. However, right and left regular crisp quasi-orders and crisp equivalences
could be used in cases when algorithms presented in Section 5, for computing the greatest right and left
regular fuzzy quasi-orders and fuzzy equivalences, do not work. Finally, it is worth noting that the natural
fuzzy equivalence of a right (resp. left) regular fuzzy quasi-order is not necessarily a right (resp. left) regular
fuzzy equivalence (cf. Example 4.9), but it is a solution to the general system.
Let W σ and W τ be transitive fuzzy relations on A defined by W σ (a, b) = σA (b) and W τ (a, b) = τA (a), for all
τ
a, b ∈ A. In this case, weakly right regular fuzzy quasi-orders with respect to {δA
x }x∈X and W (i.e., solutions
σ
to (75)), and weakly left regular fuzzy quasi-orders with respect to {δA
}
and
W
(solutions
to (76)), are
x x∈X
also solutions to the general system (cf. [81]), and therefore, they can be also used in state reduction of fuzzy
recognizers. According to Example 6.5, the greatest weakly right regular and weakly left regular fuzzy
quasi-orders and fuzzy equivalences give better results in state reduction than right and left regular ones,
i.e., they give smaller afterset and factor fuzzy recognizers. Moreover, in some cases they are even easier to
compute. However, as we have already mentioned in the previous section, the main problem concerning the
systems (75)-(78) is the number of inequalities, which may be exponential in the number of fuzzy relations
{Vi }i∈I , or even infinite. In the case of fuzzy recognizers, the formation of the systems (75)-(78), i.e., the
computing of the fuzzy relations WPr and WPl , for all P ∈ M , amounts to the determinization of the fuzzy
27
recognizer A and its reverse fuzzy recognizer by means of the Nerode automata (cf. [20, 39, 41]).
7.2. Concurrency theory – bisimulations and simulations
Bisimulations are generally considered as one of the most important contributions of concurrency theory
to computer science, although they have been discovered not only in computer science, but also (and roughly
at the same time) in modal logic and set theory. They are employed today in a number of areas of computer
science, such as functional languages, object-oriented languages, types, data types, domains, databases,
compiler optimizations, program analysis, verification tools, etc. For more information about bisimulations
we refer to [28, 32, 54, 59, 60, 74, 80].
In concurrency theory, bisimulations were introduced by Milner [58] and Park [65] as a means for testing
behavioural equivalence among processes, but they have been also very successfully exploited to reduce the
state-space of processes. The most common structures on which bisimulations have been studied are labelled
transition systems. They are essentially labelled directed graphs, or equivalently, non-deterministic automata.
However, in the exactly same way bisimulations can be defined for fuzzy automata and fuzzy recognizers,
and here we present definitions given in [24]. In fact, in [24] simulations and bisimulations have been
defined as fuzzy relations between states of two possibly different fuzzy recognizers, but here we consider
them only between states of a single fuzzy recognizer.
Let A = (A, X, δA , σA , τA ) be a fuzzy recognizer. A reflexive fuzzy relation R on A is a forward simulation
τ
−1
if R−1 is a solution to the system (32) with I = X, {Vi }i∈I = {δA
is right regular with
x }x∈X and W = Q , i.e., if R
A
τ
respect to {δx }x∈X and Q . On the other hand, R is a backward simulation if R is a solution to the system (33)
A
with I = X, {Vi }i∈I = {δA
x }x∈X and W = Qσ , i.e., if R is left regular with respect to {δx }x∈X and Qσ . Further, R is a
−1
forward bisimulation if both R and R are forward simulations, i.e., if R is a solution to the system (41) with
τ
−1
I = X, {Vi }i∈I = {δA
are backward simulations,
x }x∈X and W = Q , and R is a backward bisimulation if R and R
A
i.e., if R is a solution to the system (42) with I = X, {Vi }i∈I = {δx }x∈X and W = Qσ . Disregarding inequalities
involving Qσ and Qτ we obtain definitions of the related notions for fuzzy automata, and assuming that L
is the Boolean structure, we obtain related definitions for non-deterministic automata (labelled transition
systems) and non-deterministic recognizers.
In numerous papers dealing with simulations and bisimulations mostly forward simulations and forward bisimulations have been studied. They have been usually called just simulations and bisimulations,
or strong simulations and strong bisimulations (cf. [59, 60, 74]), and the greatest bisimulation equivalence has
been usually called the bisimilarity. Distinction between forward and backward simulations, and forward
and backward bisimulations, has been made, for instance, in [15, 34, 54] (for various kinds of automata),
but less or more these concepts differ from the concepts having the same name which are considered in
[23, 24] and here. More similar to our concepts of forward and backward simulations and bisimulations are
those studied in [14], and in [35, 36] (for tree automata).
According to the results proved in Section 4, any fuzzy recognizer has the greatest forward and backward
simulations, which are fuzzy quasi-orders, and the greatest forward and backward bisimulations, which
are fuzzy equivalences, and the results in Section 5 give methods for their computing. It is worth noting that
many efficient algorithms have been provided for computing the greatest bisimulation equivalence and
the greatest simulation quasi-order on a labeled transition system. The faster algorithms for computing the
greatest bisimulation equivalence are based on the crucial equivalence between this problem and the relational coarsest partition problem (cf. [28, 32, 46, 64, 73]).
7.3. Social network analysis – positional analysis
Network analysis has originated as a branch of sociology and mathematics which provides formal
models and methods for the systematic study of social structures, and it has an especially long tradition
in sociology, social psychology and anthropology. But, concepts of network analysis capture the common
properties of all networks and its methods are applicable to the analysis of networks in general. For that
reason, methods of network analysis are nowadays increasingly applied to many networks which are not
social networks but share a number of commonalities with social networks, such as the hyperlink structure
on the Web, the electric grid, computer networks, information networks or various large-scale networks
appearing in nature.
28
The key difference between network analysis and other approaches is the focus on relationships among
actors rather than the attributes of individual actors. Network analysis takes a global view on network structures, based on the belief that types and patterns of relationships emerge from individual connectivity and
that the presence (or absence) of such types and patterns have substantial effects on the network and its constituents. In particular, the network structure provides opportunities and imposes constraints on the individual actors by determining the transfer or flow of resources (material or immaterial) across the network. Such
an approach requires a set of methods and analytic concepts that are distinct from the methods of traditional
statistics and data analysis. The natural means to model networks mathematically is provided by the notions
of graphs, relations and matrices, and methods of network analysis primarily originate from graph theory,
semigroup theory and linear algebra. This formality served network analysis to reduce the vagueness in formulating and testing its theories, and contributed to more coherence in the field by allowing researchers to
carry out more precise discussions in the literature and to compare results across studies. More information
on various aspects of the network analysis and its applications can be found in [11, 13, 33, 57, 67, 82].
However, the above mentioned vagueness in social and many other networks can not be completely
avoided, since relations between nodes are in essence vague. This vagueness can be overcame only applying
fuzzy approach to the network analysis, but still, just few authors dealt with this topic (cf. [27, 30, 31, 62, 63]).
Here, a fuzzy network is defined as a fuzzy relational structure A = (A, {Vi }i∈I ), where A is a non-empty set of
nodes or actors (usually finite), and {Vi }i∈I is a family of fuzzy relations on A. Alternatively, fuzzy networks
can be treated as directed fuzzy multigraphs, directed labelled fuzzy graphs (with labels taken from the
index set I), or as fuzzy automata (with I as its input alphabet).
In large and complex systems it is impossible to understand the relationship between each pair of individuals, but to a certain extent, it may be possible to understand the system, by classifying individuals
and describing relationships on the class level. In networks, for instance, nodes in the same class can be
considered to occupy the same position, or play the same role in the network. The main aim of the positional
analysis of networks is to find similarities between nodes which have to reflect their position in a network.
These similarities have been formalized first by Lorrain and White [53] by the concept of a structural equivalence. Informally speaking, two nodes are considered to be structurally equivalent if they have identical
neighborhoods. However, in many situations this concept has shown oneself to be too strong, and weakening it sufficiently to make it more appropriate for modeling social positions, White and Reitz [83] have
introduced the concept of a regular equivalence. Here, two nodes are considered to be regularly equivalent if they are equally related to equivalent others. Afterwards, regular equivalences have been studied in
numerous papers, e.g., in [8, 9, 10, 11, 12, 29, 66, 67].
The notion of a regular equivalence has been extended to the fuzzy framework by Fan et al. [30, 31]. They
have defined a regular fuzzy equivalence on a fuzzy network A as any fuzzy equivalence E on A which satisfies
E ◦ Vi = Vi ◦ E, for each i ∈ I. In our terminology, these are just regular fuzzy equivalences with respect
to {Vi }i∈I , i.e., fuzzy equivalences which are solutions to the system (34) with W = ∇A . On the other hand,
structural fuzzy equivalences, defined in the same manner as in the crisp case (cf., e.g., [50]), are just strongly
regular fuzzy equivalences with respect to {Vi }i∈I , i.e., fuzzy equivalences which are solutions to the system
(69) with W = ∇A . Fan et al. [30, 31] have also provided procedures for computing the greatest regular fuzzy
equivalence and the greatest regular crisp equivalence contained in a given fuzzy (resp. crisp) equivalence,
but they have considered only fuzzy relations over the Gödel structure which is locally finite, so the problems
appearing in Section 5 do not appear in this case. It is worth noting that similarity between regular and
bisimulation equivalences has been pointed out in [56], and the results of this paper make this similarity
even more clear.
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