**L**—ordered sets and fuzzy closure operators

- Top of page
- Abstract
- Introduction
**L**—ordered sets and fuzzy closure operators- Fuzzy closure systems on
**L**—ordered sets - Fuzzy closure
**L**—systems on **L**—ordered sets - References

Truth value structure, formally, a set of truth values equipped with some special structure, plays an important role in fuzzy set theory and fuzzy logic. A traditional and common choice is the real interval [0, 1] with an appropriate structure 24. A more general one is the complete residuated lattice, which is significant in fuzzy logic in narrow sense 16, 18.

**Definition 2.1** (5, 16) A complete residuated lattice is a structure **L** = (*L*, ∧ , ∨ , ⊗ , → , 0, 1) such that

- (i)
(*L*, ∧ , ∨ , 0, 1) is a complete lattice with the least element 0 and the greatest element 1;

- (ii)
(*L*, ⊗ , 1) is a commutative monoid, i.e., ⊗ is commutative, associative, and *x* ⊗ 1 = *x* holds for any *x* ∈ *L*;

- (iii)
⊗ and → form an adjoint pair, i.e., for any

*x*,

*y*,

*z* ∈

*L*,

Let **L** be a complete residuated lattice. Given *x*, *y*, *z* ∈ *L*, (*x*_{i})_{i∈I} ⊆ *L*, where *I* is an index set, the following properties will be needed in this paper.

More properties about complete residuated lattice can be found in 5.

**Definition 2.2** (6) Let *X* be a non-empty set. A binary **L**—relation ≈ on *X* is called an **L**—equality if it satisfies that for any *x*, *y*, *z* ∈ *X*, (*x* ≈ *x*) = 1 (reflexivity); (*x* ≈ *y*) = (*y* ≈ *x*) (symmetry); (*x* ≈ *y*) ⊗ (*y* ≈ *z*) ≤ (*x* ≈ *z*) (transitivity); and (*x* ≈ *y*) = 1 implies *x*=*y*.

Let *X* and *Y* be two sets, ≈_{X} and ≈_{Y} the **L**—equalities on *X* and *Y* respectively. A binary **L**—relation *R*: *X* × *Y* → *L* is said to be compatible w.r.t. ≈_{X} and ≈_{Y} if for any *x*_{1}, *x*_{2} ∈ *X* and *y*_{1}, *y*_{2} ∈ *Y*

For the particular situation of an **L**—relation *e* on *X* with an **L**—equality ≈, we say *e* is compatible w.r.t. ≈ for simplicity.

**Definition 2.3** (6) Let *X* be a non-empty set and ≈ an **L**—equality on *X*. An *L*−order on *X* is a binary **L**—relation *e* which is compatible w.r.t. ≈ and satisfies

- (1)
for any *x* ∈ *X*, *e*(*x*, *x*)=1;

- (2)
for any *x*, *y* ∈ *X*, *e*(*x*, *y*) ∧ *e*(*y*, *x*) ≤ (*x* ≈ *y*);

- (3)
for any *x*, *y*, *z* ∈ *X*, *e*(*x*, *y*) ⊗ *e*(*y*, *z*) ≤ *e*(*x*, *z*).

We call ((*X*, ≈), *e*) an **L**—ordered set.

**Example 2.4** (1) Given an **L**—ordered set ((*X*, ≈), *e*), *e*^{−1}: *X* × *X* → *L* is defined by *e*^{−1}(*x*, *y*)=*e*(*y*, *x*) for any *x*, *y* ∈ *X*. Then ((*X*, ≈), *e*^{−1}) is an **L**—ordered set as well and it is called the dual **L**—ordered set of ((*X*, ≈), *e*).

(2) Suppose ((*X*, ≈), *e*) is an **L**—ordered set. For any nonempty subset *M* ⊆ *X*, ((*M*, ≈), *e*) is an **L**—ordered set, where *e* and ≈ are inherited from ((*X*, ≈), *e*).

(3) Let (*X*, ≤) be a crisp partial order set. Choose the truth value structure **L** as the Boolean algebra **2** = {0, 1}. If we define the **L**—equality ≈ on *X* by (*x* ≈ *y*) = 1 iff *x*=*y*, and *e*_{≤}: *X* × *X* → *L* by *e*_{≤} (*x*, *y*) = 1 iff *x* ≤ *y*, then ((*X*, ≈), *e*_{≤} is an **L**—ordered set.

(4) An **L**—set *A* in a given universe *X* is a mapping *A*: *X* → *L*. Given *A*, *B* ∈ *L*^{X}, the subsethood degree *S*(*A*, *B*) of *A* in *B* is defined by . The equality degree *E*(*A*, *B*) between *A* and *B* is defined by . Then ((*L*^{X}, *E*), *S*) is an **L**—ordered set, where *L*^{X} is the collection of all **L**—sets on *X*.

It is noteworthy that an **L**—order defined in Definition Definition 2.3 is equivalent to a fuzzy partial order given by Fan and Zhang (see 10, and also 22, 27). Based on the introduction of **L**—ordered set, the fundamental notions, such as join, meet and complete lattice, can be established as an approach to generalize the classical order theory. In the following, we only recall some basic notions and give a fundamental proposition. One can refer to 6, 27 for further details.

Given an **L**—ordered set ((*X*, ≈), *e*), a pair of operaters on *L*^{X} are defined by

For some *A* ∈ *L*^{X}, we denote by *A*^{u} and by *A*^{l}, and use the notation *A*^{lu} to replace (*A*^{l})^{u}, etc. In addition, *A*^{inf} ∈ *L*^{X} is defined by

and *A*^{sup} ∈ *L*^{X} is defined by

*A* is called an **L**—singleton if there exists *x*_{0} ∈ *X* such that *A*(*x*) = (*x*_{0} ≈ *x*) for any *x* ∈ *X*. Furthermore, we have the following proposition.

**Proposition 2.5**6, 21, 27 Let ((*X*, ≈), *e*) be an **L**—ordered set and *A* ∈ *L*^{X}. The following are equivalent:

- (1)
*A*^{inf} (respectively, *A*^{sup}) is an **L**—singleton.

- (2)
There exists (unique) *x*_{0} ∈ *X* such that *A*^{inf}(*x*_{0})=1 (respectively, *A*^{sup}(*x*_{0})=1).

- (3)
There exists (unique)

*x*_{0} ∈

*X* such that

*A*(

*x*) ≤

*e*(

*x*_{0},

*x*) (respectively,

*A*(

*x*) ≤

*e*(

*x*,

*x*_{0})) for any

*x* ∈

*X*, and

(respectively,

) for any

*y* ∈

*X*.

**Remark 2.6** Notice that Proposition 2.5 provides different ways to define the join and meet of **L**— sets. As a matter of fact, this has been shown in 21 where **L** is a frame as the truth value structure (note: every frame can be viewed as a complete residuated lattice), and the notion of **L**—fuzzy complete lattice introduced there is consistent with Bělohlávek’s completely lattice **L**—ordered set (see 6).

**Definition 2.7** (23) Let ((*X*, ≈), *e*) be an **L**—ordered set. An order-preserving mapping *f*: *X* ↔ *X* is said to be a fuzzy closure (respectively, interior) operator if for any *x* ∈ *X*,

**Example 2.8** Let *X* be a non-empty set and **L** a complete residuated lattice. Recall a mapping *C* : *L*^{X} → *L*^{X} is an **L**—closure operator if for any *A*, *A*_{1}, *A*_{2} ∈ *L*^{X}, (i) *S*(*A*, *C*(*A*)) = 1; (ii) *S*(*A*_{1}, *A*_{2}) ≤ *S*(*C*(*A*_{1}), *C*(*A*_{2})); (iii) *C*(*A*)=*C*(*C*(*A*)). Apparently, *C* is a fuzzy closure operator on ((*L*^{X}, *E*), *S*). In fact, **L**—closure operators are special cases of **L**_{K}—closure operators when *K* = *L*. One can refer to 2 for more details about **L**_{K}—closure operators.

**Remark 2.9** (1) The notion of **L**_{K}—closure operator on **L**—ordered sets was introduced in 4. It can be seen as a generalization of that in 2. It is clear that fuzzy closure operators defined in Definition 2.7 are exactly extreme cases of **L**_{K}—closure operators in 4 when *K* = *L*.

(2) A fuzzy closure operator on an **L**—ordered set is an extension of a classical closure operator on a crisp partial order set. Given an **L**—ordered set ((*X*, ≈), *e*), define ≤_{e} ⊆ *X* × *X* by *x* ≤ _{e}*y*: ⇔ *e*(*x*, *y*) = 1 for any *x*, *y* ∈ *X*. Then *f*: *X* → *X* is a fuzzy closure operator iff *f* is fuzzy order-preserving and a classical closure operator on (*X*, ≤ _{e}).

(3) Every fuzzy closure (respectively, interior) operator on ((*X*, ≈), *e*) is a fuzzy interior (respectively, closure) operator on the dual **L**—ordered set ((*X*, ≈), *e*^{−1}).

As a special case of the fuzzy system closed under *S*_{K}—intersections which is introduced initially in 2, we recall that a system ℘ = {*A*_{i} ∈ *L*^{X}|*i* ∈ *I*} is said to be closed under *S*—intersections if for any *A* ∈ *L*^{X}, it holds that , where for any *x* ∈ *X*,

The following proposition provides not only equivalent characterizations of this notion, but also a basic reference to generalize the notion of fuzzy closure system onto **L**—ordered sets in next section.

**Proposition 2.10** Let *X* be a non-empty set, **L** a complete residuated lattice and ℘ = {*A*_{i} ∈ *L*^{X}|*i* ∈ *I*} a subset of *L*^{X}. Then the following are equivalent:

- (1)
℘ is closed under *S*−intersections.

- (2)
For any

*A* ∈

*L*^{X}, there exists

*A*_{0} ∈ ℘ such that

*S*(

*A*,

*A*_{0}) = 1 and

for any

*j* ∈

*I*.

- (3)
For any *A* ∈ *L*^{X}, there exists *A*_{0} ∈ ℘ such that *S*(*A*, *A*_{0})=1 and *S*(*A*_{j}, *A*_{0}) ⊗ *S*(*A*, *A*_{i}) ≤ *S*(*A*_{j}, *A*_{i}) for any *i*, *j* ∈ *I*.

Proof. (1) ⇒ (2). For any *A* ∈ *L*^{X} let .

Secondly, we show that *S*(*A*, *A*_{i}) ≤ *S*(*A*_{0}, *A*_{i}) for any *A*_{i} ∈ ℘, i.e., for any *A*_{i} ∈ ℘, which is equivalent to *S*(*A*, *A*_{i}) ≤ *A*_{0}(*x*) → *A*_{i}(*x*) for any *x* ∈ *X* iff *S*(*A*, *A*_{i}) ⊗ *A*_{0}(*x*) ≤ *A*_{i}(*x*), i.e., for any *x* ∈ *X*. Indeed, the last inequality is true:

Therefore, for any *i*, *j* ∈ *I*, *S*(*A*_{j}, *A*_{0}) ⊗ *S*(*A*, *A*_{i}) ≤ *S*(*A*_{j}, *A*_{0}) ⊗ *S*(*A*_{0}, *A*_{i}) ≤ *S*(*A*_{j}, *A*_{i}) which implies that *S*(*A*_{j}, *A*_{0}) ≤ *S*(*A*, *A*_{i}) → *S*(*A*_{j}, *A*_{i}). It follows that for any *j* ∈ *I*. In addition, for any *A*_{j} ∈ ℘, since *A*_{0} ∈ ℘, we have

Thus for any *j* ∈ *I*.

(2) ⇒ (3). Since for any *j* ∈ *I*, we have for any *i*, *j* ∈ *I*, *S*(*A*_{j}, *A*_{0}) ≤ *S*(*A*, *A*_{i}) → *S*(*A*_{j}, *A*_{i}) which is equivalent to *S*(*A*_{j}, *A*_{0}) ⊗ *S*(*A*, *A*_{i}) ≤ *S*(*A*_{j}, *A*_{i}).

(3) ⇒ (1). Suppose *A*_{0} satisfies the conditions in (3), we want to show , i.e., for any *x* ∈ *X*, On one hand, since *A*_{0} ∈ ℘,

On the other hand, we need to prove for any *x* ∈ *X* iff *A*_{0}(*x*) ≤ *S*(*A*, *A*_{i}) → *A*_{i}(*x*) for any *i* ∈ *I* iff *S*(*A*, *A*_{i}) ≤ *A*_{0}(*x*) → *A*_{i}(*x*) which is true. Indeed, since *S*(*A*_{j}, *A*_{0}) ⊗ *S*(*A*, *A*_{i}) ≤ *S*(*A*_{j}, *A*_{i}) for any *i*, *j* ∈ *I* and *A*_{0} ∈ ℘,

which implies *S*(*A*, *A*_{i}) ≤ *A*_{0}(*x*) → *A*_{i}(*x*) for any *x* ∈ *X*.

### Fuzzy closure systems on **L**—ordered sets

- Top of page
- Abstract
- Introduction
**L**—ordered sets and fuzzy closure operators- Fuzzy closure systems on
**L**—ordered sets - Fuzzy closure
**L**—systems on **L**—ordered sets - References

In this section, we are going to develop the notion of fuzzy closure system on **L**—ordered sets and investigate the relationship between fuzzy closure systems and closure operators. It is shown that the notion of fuzzy closure system provides an alternative way to think about fuzzy closure operators.

**Definition 3.1** Let ((*X*, ≈), *e*) be an **L**—ordered set. A non-empty subset *M* ⊆ *X* is called a fuzzy closure (respectively, interior) system on *X* if for any *x* ∈ *X*, there exists *m*_{x} ∈ *M* such that

- (1)
*e*(*x*, *m*_{x})=1 (respectively, *e*(*m*_{x}, *x*)=1), and

- (2)
*e*(*n*, *m*_{x}) ⊗ *e*(*x*, *m*) ≤ *e*(*n*, *m*) (respectively, *e*(*m*, *x*) ⊗ *e*(*m*_{x}, *n*) ≤ *e*(*m*, *n*)) for any *m*, *n* ∈ *M*.

We call *m*_{x} the least upper bound (respectively, the greatest lower bound) of *x* in *M*.

**Remark 3.2** (1) Some basic properties of fuzzy closure system are obvious. Given a fuzzy closure system *M* on **L**—ordered set ((*X*, ≈), *e*), the least upper bound of *x* in *M* is unique. Every fuzzy closure system on ((*X*, ≈), *e*) is a fuzzy interior system on the dual **L**—ordered set ((*X*, ≈), *e*^{−1}). Moreover, the least upper bound of *x* in *M* (as a fuzzy closure system) w.r.t. ((*X*, ≈), *e*) is the greatest lower bound of *x* in *M* (as a fuzzy interior system) w.r.t. ((*X*, ≈), *e*^{−1}). Similar properties hold for fuzzy interior systems.

(2) A fuzzy closure system on an **L**—ordered set is an extension of a classical closure system on a crisp partial order set. Let ((*X*, ≈), *e*) be an **L**—ordered set and *M* a non-empty subset *X*. For some *x* ∈ *X*, denote the principal ideal (respectively, filter) generated by *x* w.r.t. ≤_{e} by (respectively, ), i.e., (respectively, ). Then *M* is a fuzzy closure system iff *M* is a classical closure system on (*X*, ≤ _{e}) and *e*(*x*, *m*) ≤ *e*(*x*_{0}, *m*) for any *x* ∈ *X* and *m* ∈ *M*, where *x*_{0} is the minimum upper bound of *x* in *M* w.r.t. ≤_{e}, i.e., . Dually, *N* ⊆ *X* is a fuzzy interior system iff *N* is a classical interior system on (*X*, ≤ _{e}) and *e*(*n*, *x*) ≤ *e*(*n*, *x*_{0}) for any *x* ∈ *X* and *n* ∈ *N*, where *x*_{0} is the maximum lower bound of *x* in *N* w.r.t. ≤_{e}, i.e., .

**Example 3.3** Let *X* be a non-empty set, **L** a complete residuated lattice, and ℘ = (*A*_{i})_{i∈I} a subset of *L*^{X}. By Proposition 2.10, ℘ is a fuzzy closure system on ((*L*^{X}, *E*), *S*) iff ℘ is closedunder *S*−intersections.

**Proposition 3.4** Let ((*X*, ≈), *e*) be an **L**—ordered set and *M* a non-empty subset of *X*. The following are equivalent:

Proof. Suppose *M* is a fuzzy closure system. For any *x* ∈ *X*, let *m*_{x} be the least upper bound of *x* in *M*. It is clear that *e*(*x*, *m*_{x})=1. Furthermore, for any *n* ∈ *M*,

In addition, for any *m* ∈ *M*, since *e*(*n*, *m*_{x}) ⊗ *e*(*x*, *m*) ≤ *e*(*n*, *m*), we have *e*(*n*, *m*_{x}) ≤ *e*(*x*, *m*) → *e*(*n*, *m*). This implies that . Hence, for any *n* ∈ *M*.

Conversely, we show *m*_{0} is exactly the least upper bound of *x* in *M*: Apparently, *e*(*x*, *m*_{0})=1. Moreover, for any *n* ∈ *M*, since , we have *e*(*n*, *m*_{0}) ≤ *e*(*x*, *m*) → *e*(*n*, *m*) for any *m* ∈ *M*. It follows that *e*(*n*, *m*_{0}) ⊗ *e*(*x*, *m*) ≤ *e*(*n*, *m*) for any *m*, *n* ∈ *M*.

**Proposition 3.5** Let ((*X*, ≈), *e*) be an **L**—ordered set and *M* a non-empty subset of *X*. The following are equivalent:

**Proposition 3.6** Let ((*X*, ≈), *e*) be an **L**—ordered set, and *M* a fuzzy closure (respectively, interior) system on *X*. Then there exists (unique) *m*_{x} ∈ *M* for any *x* ∈ *X* such that:

- (1)
*e*(*x*, *m*_{x})=1 (respectively, *e*(*m*_{x}, *x*)=1),

- (2)
*e*(*x*, *m*) ≤ *e*(*m*_{x}, *m*) (respectively, *e*(*m*, *x*) ≤ *e*(*m*, *m*_{x})) for any *m* ∈ *M*, and

- (3)
(respectively,

) for any

*n* ∈

*M*.

Moreover, given *x* ∈ *X*, if there exists *y* ∈ *X* such that

- (1′)
*e*(*x*, *y*)=1 (respectively, *e*(*y*, *x*)=1),

- (2′)
*e*(*x*, *m*) ≤ *e*(*y*, *m*) (respectively, *e*(*m*, *x*) ≤ *e*(*m*, *y*)) for any *m* ∈ *M*, and

- (3′)
(respectively,

) for any

*n* ∈

*M*,

then *y* is exactly the least upper (respectively, greatest lower) bound of *x* in *M*.

**Proposition 3.7** Let ((*X*, ≈), *e*) be an **L**—ordered set, *M* ⊆ *X* a fuzzy closure (respectively, interior) system, and *x*, *y* ∈ *X*. Then the following are equivalent:

- (1)
*x* and *y* have the same least upper (respectively, greatest lower) bound in *M*.

- (2)
*e*(*x*, *m*)=*e*(*y*, *m*) (respectively, *e*(*m*, *x*)=*e*(*m*, *y*)) for any *m* ∈ *M*.

Now we investigate the connection between fuzzy closure (respectively, interior) systems and fuzzy closure (respectively, interior) operators on a common **L**—ordered set.

**Theorem 3.8** Let ((*X*, ≈), *e*) be an **L**—ordered set, *f* a fuzzy closure (respectively, interior) operator and *M* ⊆ *X* a fuzzy closure (respectively, interior) system. Then *M*_{f}:*f*(*X*) is a fuzzy closure (respectively, interior) system on *X* and *f*_{M} is a fuzzy closure (respectively, interior) operator on *X*, where *f*_{M} is defined by associating each *x* ∈ *X* with the least upper (respectively, greatest lower) bound of *x* in *M*. Moreover, and , i.e., mappings and are mutually inverse.

Proof. We only prove the statements related to fuzzy closure systems. The proof for fuzzy interior systems can be obtained analogously.

Suppose *f* is a fuzzy closure operator on *X* and *x* ∈ *X*. It is obvious that *f*(*x*) ∈ *M*_{f}. By Definition 2.7, *e*(*x*, *f*(*x*))=1. For any *m*, *n* ∈ *M*_{f}, since *f* is a fuzzy closure operator, *f*(*m*)=*m* and *f*(*n*)=*n*. Then *e*(*n*, *f*(*x*)) ⊗ *e*(*x*, *m*) ≤ *e*(*n*, *f*(*x*)) ⊗ *e*(*f*(*x*), *m*) ≤ *e*(*n*, *m*). By Definition 3.1, *f*(*x*) is the least upper bound of *x* in *M*_{f} and thus *M*_{f} is a fuzzy closure system on *X*.

On the other hand, suppose *M* ⊆ *X* is a fuzzy closure system. From Remark 3.2 (1), is well-defined. For any *x* ∈ *X*, it is clear that by Definition 3.1. Suppose *x*_{1}, *x*_{2} ∈ *X*, then . Since , by Proposition 3.6, , which implies that . This means is fuzzy order-preserving. Moreover, because and , we have . Therefore, is a fuzzy closure operator on *X*.

Finally, from the above proof, it is clear that mappings and are mutually inverse.

**Corollary 3.9** Let ((*X*, ≈), *e*) be an **L**—ordered set and *f* a mapping on *X*. Then *f* is a fuzzy closure (respectively, interior) operator iff *f*(*X*) is a fuzzy closure (respectively, interior) system.

In the classical setting, Galois connections have a close relation with closure operators and closure systems 9, 15. In formal concept analysis, formal concept lattice is extracted from a formal context based on Galois connection 11, 20, 26. In the fuzzy setting, the notion of fuzzy Galois connection have been introduced in different frameworks and they serve as important tools in fuzzy concept analysis 1, 3, 13, and also 19.

In the following, we are going to investigate the relationship between fuzzy Galois connections and fuzzy closure (interior) systems. The results can be seen as a generalization of those presented by Bělohlávek in 2.

**Proposition 3.12** Let and be two **L**—ordered sets. For any fuzzy Galois connection (φ, ψ) between *X* and *Y*,

The second equation can be proved similarly.

**Remark 3.13** Given a fuzzy Galois connection, the left (respectively, right) adjoint uniquely determine the right (respectively, left) adjoint. Indeed, suppose both (φ, ψ_{1}) and (φ, ψ_{2}) are fuzzy Galois connections between *X* and *Y*. By Definition 3.10 and Proposition 3.11, for any *y* ∈ *Y*. Dually, it holds that for any *y* ∈ *Y*. We thus have ψ_{1} = ψ_{2}.

**Theorem 3.14** Let (φ, ψ) be a fuzzy Galois connection between *X* and *Y*, *f* a fuzzy closure operator on *X* and *g* a fuzzy interior operator on *Y* such that *f*(*X*) and *g*(*Y*) are *e*−isomorphic with ω being the *e*−isomorphism. Put *f*^{(φ,ψ)} = ψφ and *g*^{(φ,ψ)} = φψ. Let φ^{(f,g)}(*x*) = ω(*f*(*x*)) and ψ^{(f,g)}(*y*) = ω^{−1}(*g*(*y*)) for any *x* ∈ *X*, *y* ∈ *Y*. Then

- (1)
*f*^{(φ,ψ)} is a fuzzy closure operator on *X* and *g*^{(φ,ψ)} is a fuzzy interior operator on *Y*; *f*^{(φ,ψ)}(*X*) and *g*^{(φ,ψ)}(*Y*) are *e*−isomorphic;

- (2)
(φ^{(f,g)}, ψ^{(f,g)}) is a fuzzy Galois connection;

- (3)
the correspondences defined by

and

are mutually inverse mappings.

By Theorem 3.8, *f*^{(φ,ψ)}(*X*) is a fuzzy closure system on *X* and *g*^{(φ,ψ)}(*Y*) is a fuzzy interior system on *Y*. Define by ω^{(φ,ψ)}(*x*) = φ(*x*) for any *x* ∈ *f*^{(φ,ψ)}(*X*). It is routine to verify that ω^{(φ,ψ)} is the *e*−isomorphism between *f*^{(φ,ψ)}(*X*) and *g*^{(φ,ψ)}(*Y*).

(2) It is obvious that both φ^{(f,g)} and ψ^{(f,g)} are fuzzy order-preserving.

Suppose *x* ∈ *X*, *y* ∈ *Y*. On one hand,

Therefore, holds for any *x* ∈ *X*, *y* ∈ *Y*. By Definition 3.10, (φ^{(f,g)}, ψ^{(f,g)}) is a fuzzy Galois connection.

(3) By Remark 3.13, in order to show , we only need to prove that . For any *x* ∈ *X*,

In addition, for any *x* ∈ *X*,

Therefore, . The equation can be proved similarly.

**Theorem 3.15** Let ((*X*, ≈), *e*) be an **L**—ordered set. Then *M* ⊆ *X* is a fuzzy closure (respectively, interior) system if and only if there is a fuzzy Galois connection (*g*, *h*) (respectively, (*h*, *g*)) between ((*X*, ≈), *e*) and some fuzzy ordered set ((*Y*, ≈ ′), *e*′) (respectively, between some fuzzy ordered set ((*Y*, ≈ ′), *e*′) and ((*X*, ≈), *e*)) such that *M*=*hg*(*X*).

Proof. We only give the proof for the fuzzy closure systems. The statements about the fuzzy interior systems can be proved analogously.

Suppose *M* is a fuzzy closure system on *X*. Then ((*M*, ≈), *e*) is an **L**—ordered set. Define as for any *x* ∈ *X*, *g*(*x*)=*m*_{x} which is the smallest upper bound of *x* in *M*. Denote as the restriction of id_{X} on *M*, i.e., for any *m* ∈ *M*, *h*(*m*)=*m*. It is obvious that both *g* and *h* are fuzzy order-preserving. Moreover, it is easy to verify that *e*(*x*, *m*)=*e*(*m*_{x}, *m*) for any *x* ∈ *X* and *m* ∈ *M*. This implies that (*g*, *h*) is a fuzzy Galois connection between ((*X*, ≈), *e*) and ((*M*, ≈), *e*). Finally, *M*=*hg*(*X*) is obvious.

The converse immediately follows from Theorem 3.14.

### Fuzzy closure **L**—systems on **L**—ordered sets

- Top of page
- Abstract
- Introduction
**L**—ordered sets and fuzzy closure operators- Fuzzy closure systems on
**L**—ordered sets - Fuzzy closure
**L**—systems on **L**—ordered sets - References

Recall that a fuzzy set *A* in *X* describes the truth degree of the fact that *x* ∈ *X* belongs to *A*. In this sense, fuzzy sets can be interpreted as a generalization of the classical subsets in the fuzzy setting. Apparently, the notion of fuzzy closure system in Definition 3.1 is developed in the classical way. A natural question arises that whether fuzzy closure systems can be defined in term of fuzzy sets. To this end, we propose the following definition.

**Definition 4.1** Let ((*X*, ≈), *e*) be an **L**—ordered set. *P* ∈ *L*^{X} is called a fuzzy closure (respectively, interior) **L**—system if for any *x* ∈ *X*, there exists such that

- (1)
(respectively,

;

- (2)
(respectively,

);

- (3)
(respectively,

) for any

*y*,

*z* ∈

*X* whenever

*P*(

*y*)=

*P*(

*z*)=1.

We call x̃ the upper (respectively, lower) limit of *x* under *P*.

**Remark 4.2**

- (1)
Suppose *P* ∈ *L*^{X} is a fuzzy closure (respectively, interior) **L**—system. For any *x* ∈ *X*, the upper (respectively, lower) limit of *x* under *P* is unique.

- (2)
Obviously, every fuzzy closure **L**—system *P* on **L**—ordered set ((*X*, ≈), *e*) is a fuzzy interior **L**—system on the dual **L**—ordered set ((*X*, ≈), *e*^{−1}). Moreover, for any *x* ∈ *X*, the upper limit of *x* under *P* w.r.t. ((*X*, ≈), *e*) is the lower limit of *x* under *P* w.r.t. ((*X*, ≈), *e*^{−1}). Analogous properties hold for the fuzzy interior **L**—systems.

**Proposition 4.3** Let ((*X*, ≈), *e*) be an **L**—ordered set, *P* ∈ *L*^{X} a fuzzy closure **L**—system and . Then the following are equivalent:

**Proposition 4.4** Let ((*X*, ≈), *e*) be an **L**—ordered set, *P* ∈ *L*^{X} a fuzzy interior **L**—system and . Then the following are equivalent:

Now we are going to study the relationship between fuzzy closure (respectively, interior) systems and fuzzy closure (respectively, interior) **L**—systems on a common **L**—ordered set. For this purpose, we define the following mappings.

Given a fuzzy closure **L**—system *P* ∈ *L*^{X} and a fuzzy closure system *M* ⊆ *X* on **L**—ordered set ((*X*, ≈), *e*), define *M*_{P} ⊆ *X* by *M*_{P} = {*x* ∈ *X*|*P*(*x*) = 1} and define *P*_{M} ∈ *L*^{X} by *P*_{M}(*x*)=*e*(*m*_{x}, *x*) for any *x* ∈ *X*, where *m*_{x} ∈ *M* is the least upper bound of *x* in *M*. For a fuzzy interior **L**—system *Q* ∈ *L*^{X} and a fuzzy interior system *N* ⊆ *X*, define *N*_{Q} ⊆ *X* by *N*_{Q} = {*x* ∈ *X*|*Q*(*x*) = 1} and *Q*_{N} ∈ *L*^{X} by *Q*_{N}(*x*)=*e*(*x*, *n*_{x}) for any *x* ∈ *X*, where *n*_{x} ∈ *N* is the greatest lower bound of *x* in *N*. The theorem below shows that there is a bijective correspondence between fuzzy closure systems and fuzzy closure **L**—systems.

On the other hand, suppose *M* ⊆ *X* is a fuzzy closure system and *x* ∈ *X*. *P*_{M} is well-defined from Remark 4.2 (1). To show *P*_{M} is a fuzzy closure **L**—system on *X*, it is straightforward to check that the least upper bound *m*_{x} of *x* in *M* is exactly the upper limit of *x* under the system *P*_{M}.

It is trivial to check that and .

Finally, we investigate the relationship between fuzzy closure (respectively, interior) **L**—systems and fuzzy closure (respectively, interior) operators. Let ((*X*, ≈), *e*) be an **L**—ordered set, for a fuzzy closure **L**—system *P* ∈ *L*^{X} and a fuzzy closure operator *f*: *X* → *X*, define by associating every *x* ∈ *X* with the upper limit x̃ of *x* under *P* and define *P*_{f} ∈ *L*^{X} by *P*_{f}(*x*)=*e*(*f*(*x*), *x*) for any *x* ∈ *X*. Dually, given a fuzzy interior **L**—system *Q* ∈ *L*^{X} and a fuzzy interior operator *g*: *X* → *X*, define by associating every *x* ∈ *X* with the lower limit x̃ of *x* under *Q* and define *Q*_{g} ∈ *L*^{X} by *Q*_{g}(*x*)=*e*(*x*, *g*(*x*)) for any *x* ∈ *X*. It is clear that , and , . By Theorem 3.8 and Theorem 4.5, we have

From Theorem 3.8, Theorem 4.5, and Corollary 4.7, we have

**Theorem 4.9** The diagram in Figure 1 commutes.

By Theorem 3.8, Theorem 4.6 and Corollary 4.8, one can get a similar commutative diagram corresponding to the fuzzy interior operators, interior systems and interior **L**—systems.