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Fuzzy relation equations and subsystems of fuzzy transition systems✩
Jelena Ignjatovića , Miroslav Ćirića,∗, Vesna Simovićb
a University
b
of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia
High Economic School for Professional Studies Peć in Leposavić, Serbia
Abstract
In this paper we study subsystems, reverse subsystems and double subsystems of a fuzzy transition system. We characterize them
in terms of fuzzy relation inequalities and equations, as eigen fuzzy sets of the fuzzy quasi-order Qδ and the fuzzy equivalence Eδ
generated by fuzzy transition relations, and as linear combinations of aftersets and foresets of Qδ and equivalence classes of Eδ . We
also show that subsystems, reverse subsystems and double subsystems of a fuzzy transition system T form both closure and opening
systems in the lattice of fuzzy subsets of A, where A is the set of of states of T , and we provide efficient procedures for computing
related closures and openings of an arbitrary fuzzy subset of A. These procedures boil down to computing the fuzzy quasi-order
Qδ or the fuzzy equivalence Eδ , which can be efficiently computed using the well-known algorithms for computing the transitive
closure of a fuzzy relation.
Keywords: Fuzzy transition systems; Fuzzy automata; Fuzzy relation equations; Fuzzy quasi-orders; Fuzzy equivalences; Eigen
fuzzy sets;
1. Introduction
From the very beginning of the theory of fuzzy sets,
fuzzy automata and languages are studied as a means for
bridging the gap between the precision of computer languages and vagueness and imprecision, which are frequently encountered in the study of natural languages.
Study of fuzzy automata and languages was initiated
in 1960s by Santos [81–83], Wee [90], Wee and Fu [91],
and Lee and Zadeh [52]. From late 1960s until early 2000s
mainly fuzzy automata and languages with membership
values in the Gödel structure have been considered (cf.,
e.g., [31, 36, 64]). The idea of studying fuzzy automata with
membership values in some structured abstract set comes
back to Wechler [89], and in recent years researcher’s attention has been aimed mostly to fuzzy automata with
membership values in complete residuated lattices, latticeordered moinoids, and other kinds of lattices. Fuzzy automata taking membership values in a complete residuated lattice were first studied by Qiu in [71, 72], where
some basic concepts were discussed, and later, Qiu and
his coworkers have carried out extensive research of these
fuzzy automata (cf. [73, 74, 92–96]). From a different point
of view, fuzzy automata with membership values in a
complete residuated lattice were studied by Ignjatović,
Ćirić and their coworkers in [18–21, 39, 41, 42, 44, 85, 86].
✩
Research supported by Ministry Education and Science, Republic of
Serbia, Grant No. 174013
∗ Corresponding author. Tel.: +38118224492; fax: +38118533014.
Email addresses: [email protected] (Jelena
Ignjatović), [email protected] (Miroslav Ćirić),
[email protected] (Vesna Simović)
Preprint submitted to Knowledge-Based Systems
Fuzzy automata taking membership values in a latticeordered monoid were investigated by Li and others [55,
56, 58, 60], fuzzy automata over other types of lattices were
the subject of [3, 30, 50, 51, 57, 59, 67–70], and automata
which generalize fuzzy automata over any type of lattices, as well as weighted automata over semirings, have
been studied recently in [15, 29, 46]. During the decades,
fuzzy automata and languages have gained wide field of
application, including lexical analysis, description of natural and programming languages, learning systems, control systems, neural networks, knowledge representation,
clinical monitoring, pattern recognition, error correction,
databases, discrete event systems, and many other areas.
On the other hand, fuzzy relation equations and inequalities were first studied by Sanchez, who used them in
medical research (cf. [76–79]). Later they found a much
wider field of application, and nowadays they are used
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 many other areas (cf., e.g., [24, 28,
31, 32, 49, 66, 69]). Recently, fuzzy relation equations and
inequalities have been very successfully applied in the
theory of fuzzy automata. In [20, 21, 86] they were used in
the reduction of the number of states of fuzzy finite automata, and in [18, 19] (see also [16]) they were employed in
the study of of simulation, bisimulation and equivalence
of fuzzy automata. Here, fuzzy relation equations and inequalities are used in the study of subsystems of fuzzy
transition systems (by which we mean fuzzy automata
without fixed fuzzy sets of initial and terminal states).
September 8, 2011
Subsystems of a fuzzy transition system are defined as
fuzzy subsets of its set of states which are closed under
all fuzzy transition relations. In the case of an ordinary
transition system, they come down to sets of states which
with each of its states also contain all the states that are
accessible from it, and in the case of a deterministic transition system, if it is regarded as a unary algebra, they are
precisely subalgebras of this algebra. Subsystems of fuzzy
transition systems were first studied by Malik, Mordeson
and Sen in [61], who described some of their fundamental properties (see also [64]). Later, subsystems were investigated by Das in [22], who showed that they form
a topological closure system and identified a number of
their topological properties. From a topological point of
view, subsystems were also discussed in [84]. All the mentioned papers dealt with fuzzy transition systems over the
Gödel structure. Here we study subsystems in a more general context, for fuzzy transition systems over a complete
residuated lattice. We also introduce and examine two related concepts. Reverse subsystems are fuzzy subsets of
the set of states which are closed under reverse transitions, and double subsystems are fuzzy subsets which are
both subsystems and reverse subsystems. We show that all
three types of subsystems can be considered as solutions to
some particular systems of fuzzy relation inequalities and
equations. Especially important role in our research play
the fuzzy quasi-order Qδ and and the fuzzy equivalence
Eδ generated by fuzzy transition relations, for which we
demonstrate how they can be efficiently computed using
the well-known algorithms for computing the transitive
closure of a fuzzy relation. In particular, we characterize
subsystems, reverse subsystems and double subsystems,
respectively, as eigen fuzzy sets (in the sense of Sanchez
[80]) of Qδ , Q−1
and Eδ , and we also characterize them
δ
as linear combinations of aftersets and foresets of Qδ and
equivalence classes of Eδ (Theorems 4.2, 4.5 and 4.7). We
also show that subsystems, reverse subsystems and double subsystems of a fuzzy transition system T form both
closure and opening systems in the lattice of fuzzy subsets of A, where A is the set of of states of T (Proposition
5.1), and we provide efficient procedures for computing
related closures and openings of an arbitrary fuzzy subset
of A (Theorem 5.2). These procedures simply boil down to
computing the fuzzy quasi-order Qδ or the fuzzy equivalence Eδ . The results obtained here generalize the results
concerning subsystems of fuzzy transition systems over
the Gödels structure obtained in [22, 61, 64, 84], and the results from [11–14] concerning subsystems, reverse subsystems, double subsystems and direct sum decompositions
of crisp transition systems.
The structure of the paper is as follows. In Section 2
we introduce basic notions and notation concerning complete residuated lattices, fuzzy sets, fuzzy relations, fuzzy
transition systems and fuzzy automata. In Section 3 we
present the basic properties and give the construction of
the fuzzy quasi-order and fuzzy equivalence generated
by fuzzy transition relations of a fuzzy transition sys-
tem. Section 4 contains our main results characterizing
subsystems, reverse subsystems and double subsystems
of a fuzzy transition system. Then in Section 5 we show
that subsystems, reverse subsystems and double subsystems form both closure and opening systems and describe
the corresponding closure and opening operators.
2. Preliminaries
The terminology and basic notions in this section are
according to [4, 5, 27, 31, 32, 34, 35, 49]. For more information on lattices and related concepts we refer to books
[6, 7, 75], as well as to books [4, 5, 32, 49], for more information on fuzzy sets and fuzzy relations.
2.1. Complete residuated lattices
We will use complete residuated lattices as the structures of membership (truth) values. Residuated lattices are
a very general algebraic structure and generalize many algebras with very important applications (cf. [4, 5, 37, 38]).
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)
If, in addition, (L, ∧, ∨, 0, 1) is a complete lattice, then L
is called a complete residuated lattice. Emphasizing their
monoidal structure, in some sources residuated lattices are
called integral, commutative, residuated ℓ-monoids [38].
The operations ⊗ (called multiplication) and → (called
residuum) are intended for modeling the conjunction and
implication W
of the corresponding
logical calculus, and
V
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, and a unary operation ¬
defined by ¬x = x → 0 is called the negation.
It can be easily verified that with respect to 6, ⊗ is
isotone in both arguments, → is isotone in the second and
antitone in the first argument, and for any x, y, z ∈ L and
any {xi }i∈I , {yi }i∈I ⊆ L, the following hold:
_
_ (xi ⊗ x),
(3)
xi ⊗ x =
i∈I
(x ⊗ y) → z = x → (y → z),
^
_ (xi → y),
xi → y =
(5)
x 6 y ⇒ ¬y 6 ¬x,
x 6 ¬¬x.
(6)
(7)
i∈I
2
i∈I
(4)
i∈I
subset f of A is a fuzzy subset defined by f ¬ (x) = ¬( f (x)),
for every x ∈ A, and the crisp part of f is the crisp subset
fˆ = {a ∈ A | f (a) = 1} of A. We will also consider fˆ as a
function fˆ : A → L defined by fˆ(a) = 1, if f (a) = 1, and
fˆ(a) = 0, if f (a) < 1.
A subset C of LA is called a closure system in F (A)
if A ∈ C and C is closed under arbitrary meets. If C is
a closure system in F (A) and X = { fi }i∈I is a family of
fuzzy subsets of A, then
^n
o
(8)
1 ∈ C fi 6 1, for every i ∈ I
CX =
In general, the opposite inequality in (7) does not hold,
and if it holds, i.e., if x = ¬¬x, for each x ∈ L, then we
say that the complete residuated lattice L has the double
negation property. Note that if L has the double negation
property, then the opposite implication in (6) also hold. For
other properties of complete residuated lattices we refer
to [4, 5].
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), 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 residuated lattice if and only if ⊗ is a left-continuous t-norm and the residuum is defined by x → y =
W
{u ∈ [0, 1] | u ⊗ x 6 y} (cf. [5]). 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. Note that the Łukasiewicz structure has the double negation property, but the Goguen and Gödel structure
do not have this property.
is the least element of C containing all members of X , and
it is called the C -closure of the family X . In particular, the
C -closure of a family containing only one fuzzy subset
f is denoted by C f , and it is called the C -closure of f . If
X = { fi }i∈I is a family of fuzzy subsets of A,Wthen it can be
easily shown that CX = C f , where and f = i∈I fi . In other
words, the C -closure of a family of fuzzy sets coincides
with the C -closure of the union of this family.
Analogously, a subset C of LA is called an opening (or
interior) system in F (A) if ∅ ∈ C and C is closed under
arbitrary joins. If C is an opening system in F (A) and
X = { fi }i∈I is a family of fuzzy subsets of A, then
_n
o
(9)
1 ∈ C 1 6 fi , for every i ∈ I
CX =
is the least element of C contained all members of X ,
and it is called the C -opening of X . In particular, the C opening of a family containing only one fuzzy subset f
is called the C -opening of f . It can be also shown that the
C -opening of a family of fuzzy sets coincides with the
C -opening of the intersection of this family.
If C is a closure system in F (A), then for any a ∈ A we
have that the family { f ∈ C | f (a) = 1} is non-empty, since
it contains A, so
^n
o
f ∈ C f (a) = 1 ∈ C .
(10)
Ca =
2.2. Fuzzy sets
In this paper we deal with fuzzy sets, fuzzy relations,
fuzzy transition systems and fuzzy automata with membership values in complete residuated lattices. Therefore,
in the rest of the paper, if not noted otherwise, L will
denote a complete residuated lattice.
A fuzzy subset of a set A over L , or simply a fuzzy
subset of A, is any function from A into L. Ordinary crisp
subsets of A are considered as fuzzy subsets of A taking
membership values in the set {0, 1} ⊆ L. The set of all fuzzy
subsets of A over L will be denoted by LA . Let f and 1 be
two fuzzy subsets of 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
V if f (x) 6 1(x),
for everyWx ∈ A. The meet (intersection) i∈I fi and the join
(union) i∈I fi of an arbitrary family { fi }i∈I of fuzzy subsets
of A are functions from A into L defined by




^
_
^ 
_ 




fi  (x) =
fi (x),
fi  (x) =
fi (x).


i∈I
i∈I
i∈I
In other words, Ca is the C -closure of the crisp subset {a}
of A. The fuzzy set Ca will be called the principal element
of C generated by a, and the set P(C ) = {Ca | x ∈ A} will
be called the principal part of C .
Let S = (S, ⊕, ⊗, 0, 1) be a semiring with the zero 0 and
the identity 1. By a left S -semimodule we mean a commutative monoid (M, +, 0) for which an external multiplication
S×M → M, denoted by (λ, x) 7→ λx and called the left scalar
multiplication, is defined and which for all λ, λ1 , λ2 ∈ S and
x, x1 , x2 ∈ M satisfies the following equalities:
i∈I
In addition, the product f ⊗ 1 is a fuzzy subset defined
by f ⊗ 1(x) = f (x) ⊗ 1(x), for every x ∈ A. The quintuple
F (A) = (LA , ∨, ∧, ∅, A) is a complete lattice with the least
element ∅ and the greatest element A, and it is called the
lattice of fuzzy subsets of A. The complement f ¬ of a fuzzy
3
(λ1 ⊗ λ2 )x = λ1 (λ2 x),
λ(x1 + x2 ) = λx1 + λx2 ,
(11)
(12)
(λ1 ⊕ λ2 )x = λ1 x + λ2 x,
(13)
1x = x,
λ0 = 0x = 0.
(14)
(15)
The definition of a right S -semimodule is analogous, where
the external multiplication is a function M× S → M, denoted by (x, λ) 7→ xλ and called the right scalar multiplication,
and conditions dual to (11)–(15) are satisfied.
An S -bisemimodule is a commutative monoid (M, +, 0)
which is both left and right S -semimodule, and for all
λ, µ ∈ S and x ∈ M the following is true:
(λx)µ = λ(xµ).
denote the universal relation on a set A, which is given by
∇A (a, b) = 1, for all a, b ∈ A, and by ∆A we denote the equality relation on A, which is given by ∆A (a, b) = 1, if a = b,
and ∆A (a, b) = 0, if a , b, for all a, b ∈ A.
For fuzzy relations R, S ∈ LA×A , their composition R ◦ S
is a fuzzy relation on A defined by
(R ◦ S)(a, b) =
(16)
_
R(a, c) ⊗ S(c, b),
(18)
c∈A
If, in addition, left and right scalar multiplications coincide, i.e., if λx = xλ, for all λ ∈ S and x ∈ M, then we say
that (M, +, 0) is a strong S -bisemimodule (cf. [27, 34, 35]).
If L = (L, ∧, ∨, ⊗, →, 0, 1) is a complete residuated lattice, the algebra L ∗ = (L, ∨, ⊗, 0, 1) is a semiring and will
be called the semiring reduct of L . Let us consider the complete lattice F (A) = (LA , ∨, ∧, ∅, A) of all fuzzy subsets of
a set A with membership values in L . For any λ ∈ L and
f ∈ LA we define the left scalar multiplication λ f and the
right scalar multiplication f λ as follows: λ f (a) = λ ⊗ f (a)
and f λ(a) = f (a) ⊗ λ, for every a ∈ A. Due to commutativity of the multiplication ⊗, we have that the left and
right scalar multiplications coincide, i.e., λ f = f λ, for all
a ∈ A, and hence, the monoid (LA , ∨, ∅) forms a strong
L ∗ -bisemimodule. Then, by a scalar multiplication we will
mean either the left or the right scalar multiplication, but
we will use just the left notation (λ, f ) 7→ λ f to denote it.
The lattice F (A) equipped with this scalar multiplication
will be denoted by F⊗ (A) and called the L -lattice of fuzzy
subsets of the set A. A fuzzy subset f ∈ LA is said to be
a linear combination of fuzzy subsets fi ∈ LA (i ∈ I) if
there exist scalars λi ∈ L (i ∈ I) such that f can be expressed in the form
_
f =
λi fi .
(17)
for all a, c ∈ A. If R and S are crisp relations, then R ◦ S =
{(a, b) ∈ A × A | (∃c ∈ A) (a, c) ∈ R & (c, b) ∈ S}, i.e., R ◦ S is
an ordinary composition of relations, and if R and S are
functions, then R ◦ S is an ordinary composition of functions, i.e., (R◦S)(a) = S(R(a)), for every a ∈ A. Furthermore,
if f ∈ LA , R ∈ LA×A and 1 ∈ LA , the compositions f ◦ R and
R ◦ 1 are fuzzy subsets of A which are defined by
Any subset of LA which is closed under scalar multiplication and arbitrary meets and joins, and it contains the least
and the greatest element of F (A) will be called a complete
L -sublattice of F⊗ (A).
(i.e., composition of fuzzy relations is associative), and
( f ◦ R)(a) =
_
f (b) ⊗ R(b, a),
_
R(a, b) ⊗ 1(b),
b∈A
(R ◦ 1)(a) =
(19)
b∈A
for every a ∈ A. In particular, for f, 1 ∈ LA we write
f ◦1=
_
f (a) ⊗ 1(a).
(20)
a∈A
The value f ◦1 can be interpreted as the ”degree of overlapping” of f and 1. In particular, if f and 1 are crisp sets and
R is a crisp relation,
then f ◦ R = a ∈ A
| (∃b ∈ f ) (b, a) ∈ R
and R ◦ 1 = a ∈ A | (∃b ∈ 1) (a, b) ∈ R .
For arbitrary R, S, T ∈ LA×A we have
(R ◦ S) ◦ T = R ◦ (S ◦ T),
i∈I
(21)
R 6 S implies R−1 6 S−1 ,
(22)
R 6 S implies T ◦ R 6 T ◦ S and R ◦ T 6 S ◦ T.
2.3. Fuzzy relations
Further, for any R, S ∈ LA×A and f, 1, h ∈ LA we can easily
verify that
Let A be A non-empty set. A fuzzy relation on a set A
is any function from A × A into L, that is to say, any fuzzy
subset of A × A. Therefore, the equality, inclusion (ordering), joins and meets of fuzzy relations are defined as for
fuzzy sets, and the quintuple R(A) = (LA×A , ∨, ∧, ∅, A × A)
is a complete lattice, which is called the lattice of fuzzy relations on A. In fact, this lattice is nothing other than the
lattice F (A × A) of all fuzzy subsets of A × A. The converse
(in some sources called inverse or transpose) of a fuzzy relation R ∈ LA×A is a fuzzy relation R−1 ∈ LA×A defined by
R−1 (a, b) = R(b, a), for all a, b ∈ A. A crisp relation is a fuzzy
relation which takes values only in the set {0, 1}, and if
R is a crisp relation on A, then expressions “R(a, b) = 1”
and “(a, b) ∈ R” will have the same meaning. By ∇A we
( f ◦ R) ◦ S = f ◦ (R ◦ S),
( f ◦ R) ◦ 1 = f ◦ (R ◦ 1),
(23)
(R ◦ S) ◦ h = R ◦ (S ◦ h)
and consequently, the parentheses in (23) can be omitted,
as well as the parentheses in (21).
For a fuzzy relation R ∈ LA×A and n ∈ N0 (where N
denotes the set of natural numbers without zero included
and N0 = N ∪ {0}), we define the n-th power Rn of R
inductively, as follows: R0 = ∆A , Rn+1 = Rn ◦ R, for each
n ∈ N0 . Clearly, R1 = R.
4
Finally, for all R, Ri , S ∈ LA×A (i ∈ I) we have that
(R ◦ S)−1 = S−1 ◦ R−1 ,
_ −1 _
=
R−1
Ri
i ,
(25)
_
(S ◦ Ri ),
Ri =
(26)
_
(Ri ◦ S),
Ri ◦ S =
(27)
_
i∈I
_
i∈I
(24)
i∈I
i∈I
S◦
will be called the fuzzy equivalence closure, or simply the
E -closure. The sets of all reflexive, symmetric and transitive fuzzy relations are also closure systems in R(A),
and correspondingly we define the reflexive closure, symmetric closure and transitive closure of a family of fuzzy
relations or a single fuzzy relation. For a fuzzy relation
R on A, the reflexive closure Rr of R can be expressed as
Rr = R ∨ ∆A , the symmetric closure Rs of R can be expressed as Rs = R ∨ R−1 , andWthe transitive closure Rt of
R can be expressed as Rt = W n∈N Rn . If A is a finite set
with n elements, then Rt = nk=1 Rk , and besides, if R is
reflexive, then Rt = Rn−1 . Now, the Q-closure Rq of R can
be represented as Rq = (Rr )t = (Rt )r , and it is also called
the reflexive-transitive closure of R, whereas the E -closure
Re of R can be represented as Re = ((Rr )s )t = ((Rs )r )t =
((Rs )t )r . For more information on efficient algorithms for
computing the transitive closure of a fuzzy relation we
refer to [23, 25, 26, 53, 87, 88] and the sources cited there.
i∈I
i∈I
We note that if A is a finite set of cardinality |A| = n,
then R, S ∈ LA×A can be treated as n × n fuzzy matrices
over L , and R ◦ S is the matrix product. Analogously, for
f, 1 ∈ LA we can treat f ◦R as the product of an 1 × n matrix
f and an n × n matrix R (vector-matrix product), R ◦ 1 as
the product of an n × n matrix R and an n × 1 matrix 1t ,
the transpose of 1 (matrix-vector product), and f ◦ 1 as the
scalar product of vectors f and 1.
A fuzzy relation R on a set A is said to be
Let R be a fuzzy relation on a set A. For each a ∈ A,
the R-afterset of a is the fuzzy subset aR ∈ LA defined by
(aR)(b) = R(a, b), for any b ∈ A, and the R-foreset of a is the
fuzzy subset Ra ∈ LA defined by (Ra)(b) = R(b, a), for any
b ∈ A (cf. [2, 9, 24, 54, 86]). Clearly, R is a symmetric fuzzy
relation if and only if aR = Ra, for each a ∈ A. If E is a
fuzzy equivalence, then we write Ea instead of aE = Ea,
and we call Ea the equivalence class of a with respect to E
(cf. [17]). The set of all equivalence classes of E is denoted
by A/E and called the factor set of A with respect to E. If A
is a finite set with n elements and a fuzzy relation R on A
is treated as an n × n fuzzy matrix over L , then R-aftersets
are row vectors, whereas R-foresets are column vectors of
this matrix.
– reflexive (or fuzzy reflexive) if ∆A 6 R, i.e., if R(a, a) = 1,
for every a ∈ A;
– symmetric (or fuzzy symmetric) if R−1 6 R, i.e., if
R(a, b) = R(b, a), for all a, b ∈ A;
– 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).
A reflexive and transitive fuzzy relation on A is called a
fuzzy quasi-order, and a reflexive and transitive crisp relation on A is called a quasi-order. In some sources quasiorders and fuzzy quasi-orders are called preorders and
fuzzy preorders, but we use the original name introduced
by Birkhoff in [6]. Note that a reflexive fuzzy relation R
is a fuzzy quasi-order if and only if R2 = R. A reflexive and symmetric fuzzy relation will be called a fuzzy
proximity relation. Note that in some sources they are also
called fuzzy tolerance, fuzzy similarity and fuzzy compatibility relations. A reflexive, symmetric and transitive fuzzy
relation on A is called a fuzzy equivalence relation (or just a
fuzzy equivalence), and a reflexive, symmetric and transitive crisp relation on A is called an equivalence. If Q is a
fuzzy quasi-order on A, then Q ∧ Q−1 is the greatest fuzzy
equivalence contained in Q, and it is called the natural
fuzzy equivalence of Q. For arbitrary fuzzy quasi-orders Q1
and Q2 on A we have that
Q1 6 Q2 ⇔ Q1 ◦ Q2 = Q2 ◦ Q1 = Q2 .
Let R be a fuzzy relation on a set A. A fuzzy subset α
of A is called forward R-closed if α(a) ⊗ R(a, b) 6 α(b), for all
a, b ∈ A, and backward R-closed if R(a, b) ⊗ α(b) 6 α(a), for all
a, b ∈ A. In other words, α is forward R-closed if α ◦ R 6 α,
and backward R-closed if R ◦ α 6 α. In many sources forward R-closed fuzzy subsets were called just R-closed or
R-congruent (cf. [8, 10, 47]), since the opposite closedness
(backward R-closedness) was never discussed. It is easy to
check that if R is a symmetric fuzzy relation on A and α
is a fuzzy subset of A, then α ◦ R = R ◦ α. In this case, a
fuzzy subset α of A is forward R-closed if and only if it is
backward R-closed, and in this case we simply say that α
is R-closed. Note that E-closed fuzzy subsets, where E is a
fuzzy equivalence, are known as fuzzy subsets extensional
with respect to E (cf., e.g., [17, 48]).
(28)
As we have already noted, the lattice R(A) of fuzzy
relations on A is nothing other than the lattice of fuzzy
subsets of A × A, so the concepts of a closure system and
an opening system in R(A), and the C -closure and the C opening for C ⊆ LA×A , are defined as for fuzzy sets. The
set Q(A) of all fuzzy quasi-orders on a set A, and the set
E (A) of all fuzzy equivalences on A, form closure systems
in R(A). The Q(A)-closure of a family of fuzzy relations
or a single fuzzy relation will be called the fuzzy quasiorder closure, or simply the Q-closure, and the E (A)-closure
Let R be a fuzzy relation on a set A and let α be a fuzzy
subset of A. The left residual of α by R is a fuzzy subset α/R
of A defined by
(α/R)(a) =
^
b∈A
R(a, b) → α(b) =
^
(aR)(b) → α(b), (29)
b∈A
for each a ∈ A, and the right residual of α by R is a fuzzy
5
subset R\α of A defined by
(R\α)(a) =
^
R(b, a) → α(b) =
b∈A
2.4. Fuzzy transition systems and fuzzy automata
A fuzzy transition system over L , or simply a fuzzy transition system, is a triple T = (A, X, δ), where A and X are
non-empty sets, called respectively the set of states and the
input alphabet, and δ : A × X × A → L is a fuzzy subset
of A × X × A, called the fuzzy transition function. We can
interpret δ(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. For methodological reasons we sometimes allow
the set of states A and the input alphabet X to be infinite. A fuzzy transition system whose set of states and the
input alphabet are finite is called a fuzzy finite transition
system.
A fuzzy automaton over L , or simply a fuzzy automaton,
is a quintuple A = (A, X, δ, σ, τ), where A, X and δ are
as above, and σ, τ ∈ LA are fuzzy subsets of A, called
respectively the fuzzy set of initial states and the fuzzy set
terminal states. We can interpret σ(a) and τ(a) as the degrees
to which a is respectively an input state and a terminal
state. A fuzzy automaton whose set of states and input
alphabet are finite is called a fuzzy finite automaton.
Let X+ and X∗ denote respectively the free semigroup
and the free monoid over the alphabet X, and let e ∈ X∗ be
the empty word. We ca extend the fuzzy transition function δ : A × X × A → L up to a function δ∗ : A × X∗ × A → L
as follows: If a, b ∈ A, then



 1, if a = b,
∗
(32)
δ (a, e, b) = 

 0, otherwise,
^
(Ra)(b) → α(b), (30)
b∈A
for each a ∈ A. We think of the left residual α/R as what
remains of α on the left after “dividing” α on the right by
R, and of the right residual R\α as what remains of α on
the right after “dividing” α on the left by R. It is easy to
check that
α ◦ R 6 β ⇔ α 6 β/R,
R ◦ α 6 β ⇔ α 6 R\β, (31)
for all fuzzy subsets α and β of A and every fuzzy relation
R on A. Clearly, if R is a symmetric fuzzy relation, then
α/R = R\α, for every fuzzy subset α of A.
The following proposition can be easily proved using
the basic properties of residuated lattices [4, 5]. For some
related results we refer to [8, 10].
Proposition 2.1. Let R be an arbitrary fuzzy relation on a set
A. Then the following is true:
(a) Functions α 7→ α◦R, α 7→ α/R, α 7→ R◦α and α 7→ R\α
are isotone functions of the lattice F (A) into itself.
(b) If R is reflexive, then functions α 7→ α ◦ R and α 7→ R ◦ α
are extensive, and functions α 7→ α/R and α 7→ R\α are
intensive.
(c) If R is a fuzzy quasi-order, then all functions α 7→ α ◦ R,
α 7→ α/R, α 7→ R ◦ α and α 7→ R\α are idempotent.
and if a, b ∈ A, u ∈ X∗ and x ∈ X, then
_
δ∗ (a, ux, b) =
δ∗ (a, u, c) ⊗ δ(c, x, b).
Let (P, 6) be a partially ordered set and let φ : P → P
be a function. If x 6 y implies φ(x) 6 φ(y), for all x, y ∈ P,
then φ is called isotone, if x 6 φ(x), for each x ∈ P, then
it is called extensive, if φ(x) 6 x, for each x ∈ P, then it is
called intensive, and if φ(φ(x)) = φ(x), for each x ∈ P, then
it is called idempotent. An isotone, extensive and idempotent function is called a closure operator on (P, 6), and each
x ∈ P for which φ(x) = x is called a φ-closed element. On the
other hand, an isotone, intensive and idempotent function
is called an opening operator on (P, 6), and each x ∈ P for
which φ(x) = x is called a φ-open element. In particular, if
φ is a closure operator on the lattice F (A), then the set of
all φ-closed elements form a closure system in F (A), and
conversely, for every closure system C in F (A) there is a
unique closure operator φ on F (A) such that C is the set
of all φ-closed elements. Analogously, if φ is an opening
operator on F (A), then the set of all φ-open elements is an
opening system in F (A), and conversely, for every opening system C in F (A) there is a unique opening operator
φ on F (A) such that C is the set of all φ-open elements.
According to Proposition 2.1, for any fuzzy quasiorder Q on A, functions α 7→ α ◦ Q and α 7→ Q ◦ α are
closure operators on F (A), and functions α 7→ α/Q and
α 7→ Q\α are opening operators on F (A).
(33)
c∈A
By (3) and Theorem 3.1 [58] (see also [71, 72, 74]), we
have that
_
δ∗ (a, uv, b) =
δ∗ (a, u, c) ⊗ δ∗ (c, v, b),
(34)
c∈A
for all a, b ∈ A and u, v ∈ X∗ , i.e., if w = x1 · · · xn , for
x1 , . . . , xn ∈ X, then
_
δ(a, x1 , c1 )⊗· · ·⊗δ(cn−1 , xn , b). (35)
δ∗ (a, w, b) =
(c1 ,...,cn−1 )∈An−1
Intuitively, the product δ(a, x1 , c1 ) ⊗ · · · ⊗ δ(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 δ∗ (a, w, b) represents the supremum of degrees of all possible transitions
from a into b caused by w. Also, we can visualize a fuzzy
finite automaton A representing it as a labelled directed
graph whose nodes are states of A , and an edge from
a node a into a node b is labelled by pairs of the form
x/δ(a, x, b), for any x ∈ X.
The reverse fuzzy transition system of a fuzzy transition
system T = (A, X, δ) is defined as the fuzzy transition
6
system T¯ = (A, X, δ̄) whose fuzzy transition function δ̄ :
A × X × A → L is defined by δ̄(a, x, b) = δ(b, x, a), for all
a, b ∈ A and x ∈ X. In other words, δ̄x = δ−1
x , for each
x ∈ X. Similarly, the reverse fuzzy automaton of a fuzzy
automaton A = (A, X, δ, σ, τ) is the fuzzy automaton A¯ =
(A, X, δ̄, σ̄, τ̄), where the fuzzy transition function δ̄ is given
as above, and the fuzzy sets σ̄ and τ̄ of initial and terminal
states are defined by σ̄ = τ and τ̄ = σ.
3. Q-closures and E -closures of fuzzy transition relations
Let δ be the fuzzy transition function of a fuzzy transition system T = (A, X, δ) or a fuzzy automaton A =
(A, X, δ, σ, τ). Define fuzzy relations Vδ and Qδ on A as
follows:
_
_
Vδ =
δx ,
Qδ =
δu .
(38)
If T = (A, X, δ) is a fuzzy transition system such that
δ : A×X×A → {0, 1}, i.e., δ is a crisp subset of A×X×A, then
T is an ordinary labelled transition system (cf. [1, 62, 63]),
or just a transition system, and id δ is a function from A × X
to A, then T is a deterministic transition system. Also, if
A = (A, X, δ, σ, τ) is a fuzzy automaton such that δ is a
crisp subset of A × X × A and σ and τ are crisp subsets of
A, then A is an ordinary nondeterministic automaton, and if
δ is a function from A × X to A, then A is a deterministic automaton. In other words, ordinary transition systems and
nondeterministic automata are respectively fuzzy transition systems and fuzzy automata over the Boolean structure. If A = (A, X, δ, σ, τ) is a fuzzy automaton such that
δ is a function of A × X into A, σ is a one-element crisp
subset of A, that is, σ = {a0 }, for some a0 ∈ A, and τ
is a fuzzy subset of A, then A is called a deterministic
fuzzy automaton, and it is denoted by A = (A, X, δ, a0 , τ). In
[15, 29] the name crisp-deterministic was used. For more
information on deterministic fuzzy automata we refer to
[3, 39, 41, 44, 58]. Evidently, if δ is a crisp subset of A×X×A,
or a function of A×X into A, then δ∗ is also a crisp subset of
A×X∗ ×A, or a function of A×X∗ into A, respectively. A deterministic fuzzy automaton A = (A, X, δ, a0 , τ), where τ is
a crisp subset of A, is an ordinary deterministic automaton.
Then we have the following:
Theorem 3.1. Let δ be the fuzzy transition function of a fuzzy
transition system T = (A, X, δ) or a fuzzy automaton A =
(A, X, δ, σ, τ). Then Qδ is a fuzzy quasi-order, and it is the
Q-closure of the families {δx }x∈X and {δu }u∈X∗ , and the fuzzy
relation Vδ .
Proof. As we have noted in Section 2.2, the family {δx }x∈X
and its union Vδ have the same Q-closure. For the same
reason, the family {δu }u∈X∗ and its union Qδ also have
the same Q-closure, so it remains to show that Qδ is a
fuzzy quasi-order and that it is the Q-closure of the family {δx }x∈X .
For arbitrary a, b, c ∈ A we have that
_
_
δv (b, c)
δu (a, b) ⊗
Qδ (a, b) ⊗ Qδ (b, c) =
u∈X∗ v∈X∗
6
_ _ _
δu (a, d) ⊗ δv (d, c)
u∈X∗ v∈X∗ d∈A
=
__
u∈X∗ v∈X∗
δuv (a, c) 6
_
δw (a, c)
w∈X∗
= Qδ (a, c),
and thus, Qδ is a transitive fuzzy relation. By ∆A = δe 6 Qδ
we obtain that Qδ is a reflexive fuzzy relation. Therefore,
Qδ is a fuzzy quasi-order, which also means that Qδ is the
Q-closure of the family {δu }u∈X∗ , and it is clear that δx 6 Qδ ,
for every x ∈ X.
Let Q be an arbitrary fuzzy quasi-order on A such that
δx 6 Q, for every x ∈ X. Then δe 6 Q, and for each u ∈ X+
we have that u = x1 · · · xn , for some x1 , . . . , xn ∈ X, n ∈ N,
so δu = δx1 ◦ · · · ◦ δxn 6 Qn = Q. Now
_
δu 6 Q.
Qδ =
(36)
for all a, b ∈ A. The fuzzy relation δu is called the fuzzy
transition relation determined by u. Note that now (34) can
be written as
δuv = δu ◦ δv ,
v∈X∗
u∈X∗
_ _
=
δu (a, b) ⊗ δv (b, c)
Let δ be the fuzzy transition function of a fuzzy transition system T = (A, X, δ) or a fuzzy automaton A =
(A, X, δ, σ, τ). For any u ∈ X∗ we define a fuzzy relation δu
on A by
δu (a, b) = δ∗ (a, u, b),
u∈X∗
x∈X
(37)
u∈X∗
Hence, we have proved that Qδ is the least fuzzy quasiorder on A containing fuzzy transition relations δx , for all
x ∈ X, i.e., it is the Q-closure of the family {δx }x∈X .
for all u, v ∈ X∗ . This means that the algebra M(T ) =
({δu }u∈X∗ , ◦, δe ) is a monoid with the identity δe = ∆A , and it
is called the transition monoid of the fuzzy transition system
T . In the case when we deal with a fuzzy automaton A ,
we talk about the transition monoid of a fuzzy automaton
and we denote it by M(A ). For a procedure for computing
the transition monoid of a fuzzy transition monoid or a
fuzzy automaton we refer to [44] (see also [15]).
The previous theorem gives an efficient procedure for
computing the fuzzy quasi-order Qδ . Namely, computation of Qδ by means of the second formula in (38) (the
definition of Qδ ) could lead to problems, because it requires computation of all fuzzy transition relations δu ,
7
u ∈ X∗ . However, the number of these fuzzy transition
relations can be exponential in the number of states of a
considered fuzzy transition system or a fuzzy automaton,
and for some structures of membership values it may even
be infinite. This makes such a procedure theoretically inefficient. Nevertheless, Qδ can be efficiently computed as the
reflexive-transitive closure of the fuzzy relation Vδ , using
some of the algorithms provided in [23, 25, 26, 53, 87, 88].
The next theorem gives an interesting characterization
of the fuzzy quasi-order Qδ , in terms of fuzzy relation
inequalities.
systems (39) and (40), whereas the assertion concerning
inequality (44) follows directly from the assertions concerning systems (42) and (43).
In Section 2.3 we have seen that the E -closure Re of a
fuzzy relation R can be expressed as Re = ((Rr )s )t . Here
we give another characterization of Re which will be more
convenient for our further work. If R is a fuzzy relation
on a set A, then we define fuzzy relations Ra and R f on A
as follows:
Ra = R ◦ R−1 ,
Theorem 3.2. Let δ be the fuzzy transition function of a fuzzy
transition system T = (A, X, δ) or a fuzzy automaton A =
(A, X, δ, σ, τ). Then Qδ is the least reflexive solution to any of
the following systems of fuzzy relation inequalities and fuzzy
relation inequalities:
U ◦ δx 6 U
(x ∈ X);
(39)
δx ◦ U 6 U
U ◦ δx 6 U,
(x ∈ X);
(x ∈ X);
(40)
(41)
δx ◦ U 6 U
U ◦ Vδ 6 U;
Vδ ◦ U 6 U;
Vδ ◦ U 6 U,
c∈A
f
R (a, b) =
=
(46)
i.e., Ra (a, b) can be understood as the degree of intersection
of the R-aftersets of aR and bR, and R f (a, b) as the degree
of intersection of the R-foresets of Ra and Rb. That is the
reason why we use the notation Ra and R f .
Now we prove the following:
(44)
Proof. For an arbitrary x ∈ X we have that
_ _
Qδ ◦ δx =
δu ◦ δx =
δu ◦ δx
u∈X∗
_
(Ra)(c) ⊗ (Rb)(c),
c∈A
where U is an unknown taking values in R(A).
_
(45)
It is clear that Ra and R f are symmetric fuzzy relations,
and if R is reflexive, then Ra and R f are also reflexive,
R 6 Ra and R 6 R f . Moreover, for arbitrary a, b ∈ A we
have that
_
Ra (a, b) =
(aR)(c) ⊗ (bR)(c),
(42)
(43)
U ◦ Vδ 6 U,
R f = R−1 ◦ R.
Proposition 3.3. For an arbitrary fuzzy relation R on a set A
we have that Re = ((Rr )a )t = ((Rr ) f )t .
u∈X∗
δux 6
u∈X∗
_
v∈X∗
Proof. For the sake of simplicity set E = Re .
Since (Rr )a and (Rr ) f are reflexive and symmetric fuzzy
relations, then ((Rr )a )t and ((Rr ) f )t are fuzzy equivalences
which contain R, so E = Re 6 ((Rr )a )t and E = Re 6
((Rr ) f a)t .
On the other hand, by R 6 E it follows that Rr 6 E, so
δv = Qδ .
Thus, Qδ is a solution to (39). Let R be an arbitrary reflexive
fuzzy relation which is a solution to (39). Then by R ◦ δx 6
R, for each x ∈ X, it follows that R ◦ δu 6 R, for each u ∈ X∗ ,
whence
_ _
(R ◦ δu ) 6 R,
δu =
R ◦ Qδ = R ◦
u∈X∗
(Rr )a = Rr ◦ (Rr )−1 6 E ◦ E−1 = E2 = E,
(Rr ) f = (Rr )−1 ◦ Rr 6 E−1 ◦ E = E2 = E,
u∈X∗
and by reflexivity of R we obtain that Qδ 6 R ◦ Qδ 6
R. Therefore, Qδ is the least reflexive solution to (39).
Next, we will prove that a fuzzy relation R on A is
a solution to system (39) if and only if it is a solution to
inequality (42). Let R be a solution to (39). Then
_ _
(R ◦ δx ) 6 R,
R ◦ Vδ = R ◦
δx =
x∈X
and therefore,
((Rr )a )t 6 Et = E
and
((Rr ) f )t 6 Et = E.
This completes the proof of the proposition.
Further, for the fuzzy transition function δ of a fuzzy
transition system T = (A, X, δ) or a fuzzy automaton A =
(A, X, δ, σ, τ), by Eδ we will denote the E -closure of the
family {δx }x∈X .
x∈X
and hence, R is a solution to (42). Conversely, let R be a
solution to (42). Then by δx 6 Vδ , for each x ∈ X, it follows
that R ◦ δx 6 R ◦ Vδ 6 R, and we conclude that R is a
solution to (39). Therefore, Qδ is also the least reflexive
solution to (42).
Analogously we show that Qδ is the least reflexive
solution to (40) and (43). The assertion concerning system (41) follows directly from the assertions concerning
Theorem 3.4. Let δ be the fuzzy transition function of a fuzzy
transition system T = (A, X, δ) or a fuzzy automaton A =
(A, X, δ, σ, τ). Then Eδ is the E -closure of the family {δu }u∈X∗ ,
and the fuzzy relations Vδ and Qδ .
f
Moreover, Eδ is the transitive closure of Qaδ and Qδ .
8
Proof. By the same arguments used in the proof of the
Theorem 3.1 we obtain that Eδ is the E -closure of Vδ , and
that the family {δu }u∈X∗ and the fuzzy quasi-order Qδ have
the same E -closure.
Since Qδ is the Q-closure of Vδ and Eδ is the E -closure
of Vδ , we conclude that Qδ 6 Eδ . If E is an arbitrary fuzzy
equivalence on A which contains Qδ , then E contains δx ,
for every x ∈ X, and consequently, Eδ 6 E. Therefore, Eδ
is the least fuzzy equivalence on A containing Qδ , i.e., it is
the E -closure of Qδ .
Now, by Proposition 3.3 we obtain that
Now, due to the reflexivity of R, we obtain that
_
n
Eδ 6 R ◦ Eδ = R ◦
(Qδ ◦ Q−1
δ )
n∈N
=
n∈N
4. Subsystems, reverse subsystems and double subsystems of fuzzy transition systems
f
(Qδ )t .
Let T = (A, X, δ) be a fuzzy transition system. A fuzzy
subset α of A will be called a subsystem of T if it is forward
δx -closed, for each x ∈ X, i.e., if α(a) ⊗ δx (a, b) 6 α(b), for all
a, b ∈ A and x ∈ X. Equivalently, α is a subsystem of T if
it is a solution to a system of fuzzy relation inequalities
The fuzzy equivalence Eδ can be also characterized in
terms of fuzzy relation inequalities, as follows.
Theorem 3.5. Let δ be the fuzzy transition function of a fuzzy
transition system T = (A, X, δ) or a fuzzy automaton A =
(A, X, δ, σ, τ). Then Eδ is the least fuzzy equivalence which is a
solution to (39), (40), (42) and (43).
Moreover, Eδ is the least fuzzy proximity relation which is
a solution to (41) and (44).
χ ◦ δx 6 χ
(x ∈ X),
(47)
where χ is an unknown taking values in LA .
Proof. According to (28), from Qδ 6 Eδ it follows that
Eδ ◦ Qδ = Qδ ◦ Eδ = Eδ . Since Qδ is a solution to (39), for
each x ∈ X we have that
Proposition 4.1. The collection S (T ) of all subsystems of a
fuzzy transition system T = (A, X, δ) forms a complete L sublattice of F⊗ (A).
Eδ ◦ δx = Eδ ◦ Qδ ◦ δx 6 Eδ ◦ Qδ = Eδ ,
Proof. Note again that S (T ) is the set of all solutions to
system of fuzzy relation inequalities (47). It is easy to check
that the set of all solutions to system (47) is closed under
arbitrary meets, joins and scalar multiplications, and ∅ and
A are solutions to (47), i.e., they belong to S (T ). Thus,
S (T ) is a complete L -sublattice of F⊗ (A).
and hence, Eδ is also a solution to (39).
Let E be an arbitrary fuzzy equivalence on A which is
a solution to (39). By Theorem 3.2, Qδ is the least reflexive
solution to (39), and consequently, Qδ 6 E, which implies
that E ◦ Qδ = Qδ ◦ E = E, Q−1
6 E−1 = E and E ◦ Q−1
=
δ
δ
−1
Qδ ◦ E = E. Now we have that
_
n
Eδ 6 E ◦ Eδ = E ◦
(Qδ ◦ Q−1
)
δ
Theorem 4.2. Let T = (A, X, δ) be a fuzzy transition system
and let α be a fuzzy subset of A. Then the following conditions
are equivalent:
n∈N
_
n
=
E ◦ (Qδ ◦ Q−1
)
= E,
δ
(i) α is a subsystem of T ;
n∈N
(ii) α is forward Vδ -closed;
and therefore, Eδ is the least fuzzy equivalence which is a
solution to (39). Analogously we prove that Eδ is the least
fuzzy equivalence on A which is a solution to (40), (42)
and (43).
Next, let R be an arbitrary fuzzy proximity relation
which is a solution to (41). Then R ◦ δu 6 R and δu ◦ R 6 R,
for every u ∈ X∗ , whence
_ _
R ◦ Qδ = R ◦
R ◦ δu 6 R,
δu =
u∈X∗
n
R ◦ (Qδ ◦ Q−1
= R.
δ )
Hence, Eδ is the least fuzzy proximity relation which is a
solution to (41). Analogously we prove that Eδ is the least
fuzzy proximity relation which is a solution to (44).
Eδ = Qeδ = ((Qrδ )a )t = (Qaδ )t ,
and analogously, Eδ =
_
(iii) α is forward δu -closed, for every u ∈ X∗ ;
(iv) α is forward Qδ -closed;
(v) α is a solution to a fuzzy relation equation
χ ◦ Qδ = χ,
(48)
where χ is an unknown taking values in F (A);
(vi) α can be represented as a linear combination of Qδ -aftersets;
(vii) α is a solution to a fuzzy relation equation
u∈X∗
and similarly, Qδ ◦ R 6 R. The opposite inequalities follow
by reflexivity of Qδ , so we obtain that R ◦ Qδ = Qδ ◦ R =
R. Moreover, due to the symmetry of R, we have that
χ/Qδ = χ,
where χ is an unknown taking values in F (A).
−1
−1
R ◦ Q−1
◦ Q−1
= R−1 = R.
δ = R
δ = (Qδ ◦ R)
9
(49)
Proof. (i)⇒(ii). If α is a subsystem of T , i.e., α ◦ δx 6 α, for
every x ∈ X, then
α ◦ Vδ = α ◦
_
x∈X
and for any b ∈ A we have that
_
(α ◦ Q)(b) =
α(c) ⊗ Q(c, b)
_
(α ◦ δx ) 6 α,
δx =
c∈A
x∈X
and hence, α is forward Vδ -closed.
(ii)⇒(i). Let α be forward Vδ -closed. Then for each
x ∈ X, by δx 6 Vδ it follows that α ◦ δx 6 α ◦ Vδ 6 α, and
consequently, α is forward δx -closed, for each x ∈ X. In the
same way we prove (iv)⇒(i).
(i)⇒(iii). Let α be a subsystem of T , i.e., α is forward
δx -closed, for every x ∈ X. Then α ◦ δe = α, so α is forward
δe -closed, and for an arbitrary u ∈ X+ , if u = x1 · · · xn , for
some x1 , . . . , xn ∈ X, n ∈ N, we have that α ◦ δxi 6 α, for
each i ∈ {1, . . . , n}, and hence,
α ◦ Qδ = α ◦
δu =
u∈X∗
_
=
a∈A
δx ◦ χ 6 χ
_
α(a)(aQ) =
a∈A
_
α=
_
λa (aQ) (b) = α(b).
(x ∈ X),
(50)
Proposition 4.3. Let T = (A, X, δ) be a fuzzy transition system. A fuzzy subset α of A is a reverse subsystem of T if and
only if it is a subsystem of the reverse fuzzy transition system T¯ .
Proof. For all a, b ∈ A and x ∈ X we have α(a) ⊗ δ̄x (a, b) 6
α(b) if and only if δx (b, a) ⊗ α(a) 6 α(b), so α is a reverse
subsystem of T if and only if it is a subsystem of T¯ .
α(a)(aQ) (b),
The next two propositions and a theorem are direct
consequences of the previous proposition, Propositions
4.1 and 5.1, and Theorem 4.2.
λa (aQ),
Proposition 4.4. The collection S r (T ) of all reverse subsystems of a fuzzy transition system T = (A, X, δ) is a complete
L -sublattice of F⊗ (A).
a∈A
where λa = α(a), for each a ∈ A. Therefore, α can be
represented as a linear combination of Q-aftersets.
(v)⇒(iv). Suppose that α can be represented as a linear
combination of Q-aftersets, where Q = Qδ . Then there are
scalars λa ∈ L, a ∈ A, such that
_
c∈A
where χ is an unknown taking values in LA . Moreover, the
following is true:
which means that
α=
λa ⊗ Q(a, b)
Q(a, c) ⊗ Q(c, b)
Again, let T = (A, X, δ) be a fuzzy transition system.
A fuzzy subset α of A will be called a reverse subsystem
of T if it is backward δx -closed, for each x ∈ X, i.e., if
δx (a, b) ⊗ α(b) 6 α(a), for all a, b ∈ A and x ∈ X. Equivalently, α is a reverse subsystem of T if it is a solution to
a system of fuzzy relation inequalities
α(a) ⊗ Q(a, b)
α(a) ⊗ (aQ)(b) =
a∈A
=
_
_
Hence, α ◦ Q = α, and we have shown that (iv) holds.
(v)⇒(vii). Let α be a solution to (48). According to (31),
we have that α 6 α/Qδ . On the other hand, by Proposition
2.1 it follows that α/Qδ 6 α, and thus, α is a solution to (49).
Implication (vii)⇒(v) can be proved in the same way
as (v)⇒(vii).
(α ◦ δu ) 6 α,
_
λa ⊗
a∈A
a∈A
_
=
_
=
u∈X∗
_
λa ⊗ Q(a, c) ⊗ Q(c, b)
a∈A
and thus, α is forward Qδ -closed.
(iv)⇒(v). Let α be forward Qδ -closed, i.e., let α◦Qδ 6 α.
By reflexivity of Qδ it follows that α 6 α ◦ Qδ , and hence,
α ◦ Qδ = α.
(v)⇒(iv). This implication is obvious.
(v)⇒(vi). Let (v) hold. For the sake of simplicity set
Qδ = Q, and take an arbitrary α ∈ S (T ). By (iv) we obtain
that α = α ◦ Q, and for any b ∈ A we have
α(b) = (α ◦ Q)(b) =
=
__
a∈A
Therefore, α is forward δu -closed, for every u ∈ X∗ .
(iii)⇒(iv). If α is forward δu -closed, for every u ∈ X∗ ,
then by (27) it follows that
c∈A a∈A
λa ⊗ (aQ) (c) ⊗ Q(c, b)
c∈A a∈A
α ◦ δu = α ◦ δx1 ◦ · · · ◦ δxn 6 α.
_
=
__
Theorem 4.5. Let T = (A, X, δ) be a fuzzy transition system
and let α be a fuzzy subset of A. Then the following conditions
are equivalent:
(i) α is a reverse subsystem of T ;
(ii) α is backward Vδ -closed;
(iii) α is backward δu -closed, for every u ∈ X∗ ;
λa (aQ),
a∈A
10
(i)⇒(v). Let α be a double subsystem of T . According
to Theorems 4.2 and 4.5, we have that α ◦ Qδ = α and
α ◦ Q−1
= Qδ ◦ α = α, whence
δ
_
n
α ◦ Eδ = α ◦ (Qaδ )t = α ◦
(Qδ ◦ Q−1
δ )
(iv) α is backward Qδ -closed;
(v) α is a solution to a fuzzy relation equation
Qδ ◦ χ = χ,
(51)
where χ is an unknown taking values in F (A);
(vi) α can be represented as a linear combination of Qδ foresets;
(vii) α is a solution to a fuzzy relation equation
Qδ \χ = χ,
n∈N
=
(52)
(x ∈ X),
Under different names, subsystems, reverse subsystems and double subsystems have already been studied
in [11, 14] (see also [12, 13]) in the context of deterministic transition systems. There has been shown that lattices
of subsystems and reverse subsystems are conjugated in
the Boolean algebra of subsets of the underlying set of
states, in the sense that a set of states is a subsystem
if and only if its set-theoretical complement is a reverse
subsystem. Also, the lattice of double subsystems is selfconjugated, which means that a set of states is a double
subsystem if and only if its set-theoretical complement is
a double subsystem. In the fuzzy framework situation is
different. Namely, in the general case complete residuated
lattices do not possess the double negation property, and
we have the following.
(53)
where χ is an unknown taking values in LA .
Proposition 4.6. The collection D(T ) of all double subsystems of a fuzzy transition system T = (A, X, δ) forms a complete L -sublattice of F⊗ (A).
Proof. Since D(T ) = S (T ) ∩ S r (T ), by Propositions 4.1
and 4.4 we obtain that D(T ) is closed under arbitrary
meets, joins and scalar multiplications, and contains ∅
and A. Therefore, D(T ) is a complete L -sublattice of
F⊗ (A).
Theorem 4.8. Let T = (A, X, δ) be a fuzzy transition system
and α a fuzzy subset of A. Then the following is true:
(a) if α is a subsystem, then α¬ is a reverse subsystem;
(b) if α is a reverse subsystem, then α¬ is a subsystem;
(c) if α is a double subsystem, then α¬ is also a double subsystem.
Theorem 4.7. Let T = (A, X, δ) be a fuzzy transition system
and let α be a fuzzy subset of A. Then the following conditions
are equivalent:
(i)
(ii)
(iii)
(iv)
(v)
In addition, if the underlying complete residuated lattice L has
the double negation property, then the opposite implications also
hold.
α is a double subsystem of T ;
α is forward and backward Vδ -closed;
α is forward and backward δu -closed, for every u ∈ X∗ ;
α is Eδ -closed;
α is a solution to a fuzzy relation equation
χ ◦ Eδ = χ,
Proof. For the sake of convenience set Qδ = Q. According
to (5) and (4), for any a ∈ A we have that
(α ◦ Q)¬ (a) = (α ◦ Q)(a) → 0
h_ i
=
α(b) ⊗ Q(b, a) → 0
(54)
where χ is an unknown taking values in F (A);
(vi) α can be represented as a linear combination of Eδ -classes;
(vii) α is a solution to a fuzzy relation equation
χ/Eδ = χ,
=α
(iv)⇒(i). Let α be Eδ -closed. Then α ◦ Qδ 6 α ◦ Eδ 6 α
and Qα ◦α 6 Eδ ◦α = α◦Eδ 6 α, and according to Theorems
4.2 and 4.5, α is a double subsystem of T .
The implication (v)⇒(iv) is obvious, and equivalences
(v)⇔(vi) and (v)⇔(vii) can be proved in the same way as
the corresponding equivalences in Theorem 4.2.
Consider again a fuzzy transition system T = (A, X, δ).
A fuzzy subset α of A will be called a double subsystem of
T if it is both a subsystem and a reverse subsystem of T ,
i.e., if it is both forward and backward δx -closed, for each
x ∈ X. In other words, α is a double subsystem of T if it
is a solution to a system of fuzzy relation inequalities
δx ◦ χ 6 χ
α ◦ (Qδ ◦
n
Q−1
δ )
n∈N
where χ is an unknown taking values in F (A).
χ ◦ δx 6 χ,
_
b∈A
^h
i
=
α(b) ⊗ Q(b, a) → 0
b∈A
^h
i
=
Q(b, a) ⊗ α(b) → 0
(55)
b∈A
=
where χ is an unknown taking values in F (A).
^h
b∈A
i
Q(b, a) → α(b) → 0
^
=
Q(b, a) → α¬ (b) = (Q\α¬ )(a),
Proof. (i)⇔(ii) and (i)⇔(iii). This follows directly from
Theorems 4.2 and 4.5.
b∈A
11
which means that (α ◦ Q)¬ = Q\α¬ . Now, by (6) and (29)
we obtain that
α ◦ Q 6 α ⇒ α¬ 6 (α ◦ Q)¬ ⇔ α¬ 6 Q\α¬
⇔ Q ◦ α¬ 6 α¬ ,
The transition system T is called the direct sum of transition systems Ti (i ∈ I), each Ti is called a direct summand of T , and the partition of A whose components
are different Ai (i ∈ I) is called the direct sum decomposition of T . Clearly, there are no transitions between states
from different direct summands, so everything that happens in the transition system takes place within its direct summands. It is therefore very important to find the
largest possible direct sum decomposition of a given transition system T , i.e., a direct sum decomposition with
the smallest possible summands. This problem was discussed in [11, 14]. Among other things, it was proved
that every deterministic transition system has the greatest direct sum decomposition and its summands are direct sum indecomposable. The results obtained in [11, 14]
for deterministic transition systems are also applicable to
transition systems in general, and also follow from more
general results concerning direct sum decompositions of
quasi-ordered sets given in [12]. Direct summands of a
transition system are precisely double subsystems, and
the summands in the greatest direct sum decomposition
are principal double subsystems, i.e., equivalence classes
of the equivalence relation Eδ . Therefore, computing the
greatest direct sum decomposition of a transition system
T comes down to computing the corresponding equivalence Eδ .
In the fuzzy framework, the fuzzy partition corresponding to the fuzzy equivalence Eδ can also be regarded
as some kind of direct sum decomposition of a fuzzy transition system T = (A, X, δ). Namely, the crisp parts of the
equivalence classes of Eδ form an ordinary partition of A
such that there are not absolutely certain transitions (transitions of degree 1) between states from different blocks
of this partition.
(56)
and hence, if α is a subsystem, then α¬ is a reverse subsystem, i.e., (a) holds. In the same way we prove (b), whereas
(c) follows directly from (a) and (b).
In addition, if the complete residuated lattice L has
the double negation property, then the first implication in
(56) can be turned into equivalence, which means that α is
a subsystem if and only if α¬ is a reverse subsystem. Analogously we show that α is a reverse subsystem if and only
if α¬ is a subsystem, and that α is a double subsystem if
and only if α¬ is a double subsystem.
The next example demonstrates that the double negation property is even a necessary condition for the universal validity of the opposite implications in (a), (b) and (c) of
Theorem 4.8. In other words, if the underlying complete
residuated lattice L does not have the double negation
property, we show that there is a fuzzy transition system
and a fuzzy subset α of its set of states whose complement
α¬ is a double subsystem, but α is neither a subsystem nor
a reverse subsystem.
Example 4.9. Let L be a complete residuated lattice L =
(L, ∧, ∨, ⊗, →, 0, 1) which does not have the double negation property. This means that there is k ∈ L such that
¬¬k k. Consider a fuzzy transition system T = (A, X, δ),
where |A| = 2, X = {x} and the fuzzy transition relation δx
is given by
"
#
0 1
δx =
,
1 1
5. Closures and openings corresponding to subsystems,
reverse subsystems and double subsystems
and a fuzzy subset of A given by α = [¬¬k k]. Then
α¬ = [¬k ¬k], and we have that α¬ ◦ δx = δx ◦ α¬ = α¬ ,
whereas α ◦ δx = δx ◦ α = [k ¬¬k] α. Therefore, α¬ is
a double subsystem, but it is neither a subsystem nor a
reverse subsystem.
Propositions 4.1, 4.4 and 4.6 have shown that the collections of all subsystems, reverse subsystems and double
subsystems of a fuzzy transition system T = (A, X, δ)
form complete L -sublattices of the L -lattice F⊗ (A). In
addition, we have the following.
Let Ti = (Ai , X, δi ) (i ∈ I) be a family of ordinary (crisp)
transition systems with the same input alphabet and pairwise disjoint sets of states, i.e., Ai ∩ A j = ∅, for i , j. Then
we can define a new transition system T = (A, X, δ) by
letting
[
[
Ai ,
δ=
δi .
A=
i∈I
Proposition 5.1. Let T = (A, X, δ) be a fuzzy transition system. Then
(a) S (T ), S r (T ) and S d (T ) are both closure and opening
systems in F (A).
(b) The principal part of S (T ) consists of all Qδ -aftersets.
i∈I
i.e., if δ is regarded as a function from A × X × A to {0, 1},
then for all a, b ∈ A and x ∈ X we have



 δi (a, x, b) if a, b ∈ Ai , for some i ∈ I,
δ(a, x, b) = 


0
otherwise.
(c) The principal part of S r (T ) consists of all Qδ -foresets.
(d) The principal part of S d (T ) consists of all Eδ -classes.
Proof. (a) This follows directly from Propositions 4.1, 4.4
and 4.6.
12
(b) For the sake of simplicity set Qδ = Q and S (T ) =
C . Consider an arbitrary a ∈ A. For each b ∈ A we have
that
_
_
(aQ) ◦ Q (b) =
(aQ)(c) ⊗ Q(c, b) =
Q(a, c) ⊗ Q(c, b)
c∈A
According to the previous theorem, the problem of
computing the closures αsc , αrsc and αdsc and the openings
αso , αrso and αdso of α boils down to computing the fuzzy
quasi-order Qδ and the fuzzy equivalence Eδ . As we have
seen in Section 3, Qδ can be efficiently computed as the
reflexive-transitive closure of the fuzzy relation Vδ , and
Eδ as the reflexive-transitive closure of Vδs = Vδ ∨ Vδ−1 . We
again refer to [23, 25, 26, 53, 87, 88] and the sources cited
there, where one can find more information about efficient
algorithms for computing the transitive closure of a fuzzy
relation.
The following example demonstrates the procedure of
computing the fuzzy quasi-order Qδ , the fuzzy equivalence Eδ , and related closures and openings.
c∈A
= (Q ◦ Q)(a, b) = Q(a, b) = (aQ)(b),
and hence, aQ is a solution to equation (48), i.e., aQ ∈ C . It
is clear that (aQ)(a) = 1, so Ca 6 aQ.
Take an arbitrary α ∈ C = S (T ) such that α(a) = 1.
Then for each b ∈ A we have that
_
(aQ)(b) = Q(a, b) = α(a) ⊗ Q(a, b) 6
α(c) ⊗ Q(c, b)
c∈A
Example 5.3. Let L be the Gödel structure. Consider a
fuzzy transition system T = (A, X, δ), where |A| = 4, X =
{x, y} and the fuzzy transition relations δx and δ y are given
by the following matrices:
= (α ◦ Q)(b) = α(b),
so aQ 6 α, whence it follows that
^n
o
aQ 6
α ∈ C α(a) = 1 = Ca .




0.8 1 0.6 0.8
 1 0.8 0.6 0.8
 1 0.6 0.5 0.9
0.8 1 0.8 0.6




δx = 
.
 , δ y = 
0.3 0.2 0.4 0.8
0.2 0.3 0.8 0.9




0.5 0.3 0.3 1
0.2 0.3 0.8 0.9
Therefore, we have proved that Ca = aQ, for each a ∈ A.
The assertions (c) and (d) can be proved similarly.
Now, it naturally arises the question of computing closures and openings of a fuzzy subset of A that correspond
to the above mentioned closure and opening systems.
Let T = (A, X, δ) be a fuzzy transition system and let α
be a fuzzy subset of A. By αsc , αrsc and αdsc we will denote
the S (T )-closure, S r (T )-closure and S d (T )-closure of
α, respectively, and by αso , αrso and αdso we will denote
the S (T )-opening, S r (T )-opening and S d (T )-opening
of α. In other words, αsc , αrsc and αdsc are respectively the
least subsystem, reverse subsystem and double subsystem
of T containing α, i.e., the least solutions to (47), (50) and
(53) (or, equivalently, to (48), (51) and (54)) containing
α. Similarly, αso , αrso and αdso are respectively the greatest
subsystem, reverse subsystem and double subsystem of
T contained in α, i.e., the greatest solutions to (47), (50)
and (53) (or, equivalently, to (48), (51) and (54)) contained
in α.
The next theorem characterizes the closures αsc , αrsc
and αdsc and the openings αso , αrso and αdso of α.
We easily obtain that


1 0.6 0.8
 1
 1
1 0.8 0.9

Vδ = 
,
0.3 0.3 0.8 0.9


0.5 0.3 0.8 1


1 0.8 0.9
 1
 1
1 0.8 0.9

Qδ = (Vδr )3 = 
,
0.5 0.5 1 0.9


0.5 0.5 0.8 1


1 0.9 0.9
 1
 1
1 0.9 0.9

Eδ = ((Vδs )r )3 = 
.
0.9 0.9 1 0.9


0.9 0.9 0.9 1
For a fuzzy subset α of A given by α = [ 1 0.6 0.8 0.9 ] we
easily obtain that
Theorem 5.2. Let T = (A, X, δ) be a fuzzy transition system
and let α be a fuzzy subset of A. Then
αsc = α ◦ Qδ , αrsc = Qδ ◦ α,
so
α = α/Qδ ,
rso
α
= Qδ \α,
αdsc = α ◦ Eδ ,
dso
α
= α/Eδ .
αsc = α ◦ Qδ = [ 1 1 0.8 0.9 ],
αso = α/Qδ = [ 0.6 0.6 0.8 0.9 ],
αrsc = Qδ ◦ α = [ 1 1 0.8 0.8 ],
αrso = Qδ \α = [ 0.6 0.6 0.6 0.6 ],
αdsc = α ◦ Eδ = [ 1 1 0.9 0.9 ],
αdso = α/Eδ = [ 0.6 0.6 0.6 0.6 ].
It is clear that α is not closed nor open for any of the
considered closure and opening operators.
(57)
It is worth noting that Das in [22] defined a function
α 7→ α♯ , for α ∈ LA , by
_h _ i
α(b) ⊗ δu (b, a) ,
(58)
α♯ (a) =
Proof. By Proposition 2.1, we have (α ◦ Qδ ) ◦ Qδ = α ◦ Qδ ,
and by Theorem 4.2 we get that α ◦ Qδ is a subsystem of
T . Let β be an arbitrary subsystem of T such that α 6 β.
Again according to Proposition 2.1 and Theorem 4.2 we
obtain α ◦ Qδ 6 β ◦ Qδ = β. Thus, α ◦ Qδ is the least subsystem of T containing α, i.e., α ◦ Qδ = αsc .
In the same way we prove the remaining equalities.
b∈A u∈X∗
for each a ∈ A, and proved that this function is a topological closure operator on F (A), i.e., a closure operator which
13
preserves joins (see also [84]). It was also shown in [22]
that closed fuzzy subsets of A with respect to this closure
operator are exactly subsystems of the fuzzy transition
system T = (A, X, δ), and this means that α♯ = αsc . However, in general, for a given fuzzy subset α of A, its closure
α♯ can not be efficiently computed by means of formula
(58). Namely, to compute α♯ (a) by means of formula (58)
we should know all members of the transition monoid
M(T ), which may be infinite, and even if it is finite, its
number of elements could be exponentially larger than
the number of states of T .
The next example shows the case when the number of
fuzzy transition relations δu (u ∈ X∗ ) is infinite.
Example 5.5. Consider again the fuzzy transition system
T = (A, X, δ) from Example 5.3, the same fuzzy subset
α = [ 1 0.6 0.8 0.9 ], and a fuzzy subset β of A given by
β = [ 1 0 0 0 ]. Then
αsc = α ◦ Qδ = [ 1 1 0.8 0.9 ]
< [ 1 1 1 0.9 ] = (α ◦ Vδ )/Vδ ,
(β ◦ Vδ )/Vδ = [ 1 0.6 0.6 0.6 ]
< [ 1 1 0.8 0.9 ] = β ◦ Qδ = βsc ,
which means that γ 7→ γsc and γ 7→ (γ ◦ Vδ )/Vδ are different and mutually incomparable closure operators on the
lattice F (A). In the same way we show that γ 7→ γso and
γ 7→ (γ/Vδ ) ◦ Vδ are different and mutually incomparable
opening operators on the lattice F (A).
Example 5.4. Let L be the product structure. Consider
a fuzzy transition system T = (A, X, δ), where |A| = 3,
X = {x} and the fuzzy transition relation δx is given by the
following matrix:

 0

δx =  1
1
1
0

0 
0  .
1
1
2
2
The above considered closure operator α 7→ α ◦ Qδ is
a direct generalization of the so-called successor operator,
which to any set of states of an ordinary transition system
assigns the set of all states which are accessible from the
states from this set. On the other hand, the closure operator
α 7→ Qδ ◦α generalizes the source operator, which to any set
of states assigns the set of all states from which the states
from this set are accessible. Different generalizations of the
successor and source operators, in the fuzzy framework,
were studied in [61, 65, 84].
An efficient procedure for computing the principal double subsystems of a deterministic transition system was
provided in [11, 14] (see also [12]). It is based on the alternate application of the operators α 7→ αsc and α 7→ αrsc ,
starting from any of them, and stops when the so generated sequence of sets stabilizes. This procedure can be
easily derived from the results obtained here. Namely, for
an arbitrary fuzzy subset α of the set of states of a fuzzy
transition system T = (A, X, δ) we have that
2
As shown in Example 11.2 [44], the transition monoid of
T is infinite, since the transition relations of T can be
represented by
δx2n

 1

=  0
1
2
0
1
1
2

0 
0  ,
1 2n 
(2)
δ2n+1

 0

=  1
1
2
1
0
1
2

0 
0  .
1 2n+1 
(2)
Despite the fact that the number of fuzzy transition relations δu (u ∈ X∗ ) is infinite, we can efficiently compute Qδ
and Eδ , and obtain that


 1 1 0


Qδ = Vδr = δrx =  1 1 0 ,
1 1

1
2
2


 1 1 12 
r
 1 1 1  ,
Eδ = (Vδ ∨ Vδ−1 )r = (δx ∨ δ−1
x ) =
 1 1 2 
1
2
2
(αsc )rsc = Qδ ◦ (α ◦ Qδ ) = (α ◦ Qδ ) ◦ Q−1
δ
a
= α ◦ (Qδ ◦ Q−1
δ ) = α ◦ Qδ ,
and applying the operators α 7→ αsc and α 7→ αrsc alternately n − 1 times, where n is the number of states of T ,
we obtain α ◦ (Qaδ )n−1 = α ◦ Eδ = αdsc . The same happens if
we apply the operator α 7→ αrsc first, and then the operator
α 7→ αsc .
Our final question is how to transfer the concepts of
a subsystems to fuzzy automata. If we make an analogy with deterministic automata, it is natural to define
a subsystem or a fuzzy subautomaton of a fuzzy automaton
A = (A, X, δ, σ, τ) to be any fuzzy subset α of A which is
forward δx -closed, for each x ∈ X, and contains the fuzzy
set of initial states, i.e., σ 6 α. It can be easily seen that
fuzzy subautomata of A form a complete lattice with the
least element σsc = σ ◦ Qδ and the same meets and joins as
in F (A), and the corresponding closure operator is given
by α 7→ (σ ∨ α)sc = σsc ∨ αsc . Similarly, we define α to be a
reverse fuzzy subautomaton of A if it is backward δx -closed,
for each x ∈ X, and contains the fuzzy set of terminal
and then efficiently compute closures αsc , αrsc and αdsc and
openings αso , αrso and αdso , for each α ∈ LA .
Closures and openings with respect to fuzzy quasiorders, represented in the same way as αsc and αso in
Theorem 5.2, have already been studied by Bodenhofer,
De Cock and Kerre [10] and Bodenhofer [8]. In addition,
Bodenhofer in [8] found a way to define closure and opening operators starting from an arbitrary fuzzy relation. He
proved that the function α 7→ (α ◦ R)/R is a closure operator, and the function α 7→ (α/R) ◦ R is an opening operator
on the lattice F (A), for an arbitrary fuzzy relation R on a
set A. However, if we take R to be the fuzzy relation Vδ , we
obtain two operators that are different than the operators
α 7→ αsc and α 7→ αso , and even incomparable with them,
as the following example shows.
14
states, i.e., τ 6 α, and to be a double fuzzy subautomaton
if it is both a fuzzy subautomaton and a reverse fuzzy
subautomaton. Reverse fuzzy subautomata and double
fuzzy subautomata also form complete lattices with the
least elements τrsc = Qδ ◦ τ and (σ ∨ τ)dsc = (σ ∨ τ) ◦ Eδ, and
the same meets and joins as in F (A). The corresponding
closure operators are given by α 7→ (σ ∨ α)rsc = σrsc ∨ αrsc
and α 7→ (σ ∨ τ ∨ α)dsc = σdsc ∨ τdsc ∨ αdsc .
[3]
[4]
[5]
[6]
[7]
[8]
6. Concluding remarks
In this paper we discussed subsystems, reverse subsystems and double subsystems of fuzzy transition systems.
They were characterized as solutions to certain fuzzy relation inequalities and equations. In particular, we proved
that they are respectively eigen fuzzy sets (in the sense of
Sanchez [80]) of Qδ , Q−1
and Eδ , where Qδ is the fuzzy
δ
quasi-order and Eδ is the fuzzy equivalence generated by
the fuzzy transition relations of the considered fuzzy transition system. They were also characterized as fuzzy sets
that can be represented as linear combinations of aftersets
of Qδ , foresets of Qδ and equivalence classes of Eδ . Besides,
we showed that subsystems, reverse subsystems and double subsystems of a fuzzy transition system T form both
closure and opening systems in the lattice of fuzzy subsets
of A, where A is the set of of states of T , and we provide
efficient procedures for computing related closures and
openings of an arbitrary fuzzy subset of A. These procedures comes down to computing the fuzzy quasi-order Qδ
or the fuzzy equivalence Eδ , which can be efficiently computed using the well-known algorithms for computing
the transitive closure of a fuzzy relation.
The obtained results generalize the results concerning
subsystems of fuzzy transition systems over the Gödels
structure given in [22, 61, 64, 84], as well as the results from
[11–14] concerning subsystems, reverse subsystems, double subsystems and direct sum decompositions of crisp
transition systems. They are also closely related to results
from [8, 10] on closure and opening operators defined by
fuzzy relations.
It is worth noting that fuzzy relation equations and
inequalities have been recently very successfully applied
in solving other important problems of the theory of fuzzy
automata. In [20, 21, 86] they were used in the reduction
of the number of states of fuzzy finite automata, and in
[18, 19] (see also [16]) they were employed in the study
of of simulation, bisimulation and equivalence of fuzzy
automata.
[9]
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17