Galois connections are useful to model solutions for both pure and application-oriented problems. Throughout the paper, the general framework is a complete fuzzy lattice over a Heyting algebra. We have established a fuzzy Galois connection between the fuzzy powerset lattice and the set of functions in . Furthermore, the fixed points, or formal concepts, of this fuzzy Galois connection are exactly the fuzzy closure systems and fuzzy closure operators on . The extension of this fuzzy Galois connection to the general framework is discussed but the study of the fixed points is still an open problem.
Galois connections occur in a great variety of mathematical theories and in several instances in
the theory of relations [
The fuzzy extension of the notion of Galois connection was introduced by Bělohlávek [
In [13], the framework of the paper was a complete -fuzzy lattice (, ) , where the infimum and the supremum are denoted by ⊓ and ⊔, respectively and is a complete Heyting algebra. The main discussion was searching for a suitable definition of fuzzy closure system. In that, there are two mappings which transform fuzzy closure operators into fuzzy closure systems and vice versa. These mappings are defined as follows, let Φ ∈ be a fuzzy closure system and c ∶ → be a fuzzy closure operator. Then, cΦ ∶ → is defined as cΦ() = ⨅( ∩ Φ) and Φc() = ( c(), ) .
In this paper, we elaborate on these mappings since their domain and codomain does not have to be the set of fuzzy closure systems and the set of fuzzy closure operators, respectively. Actually, these mappings are well-defined for all isotone functions ∶ → and all fuzzy sets ∈ . The main goal of the paper is studying whether these mappings defined in the most general domains and codomains form a fuzzy Galois connection. As a matter of fact, the fuzzy set Φ can be defined for any function on , but the fuzzy Galois connection found in the results of the paper requires the use of isotone mappings only. This restriction does not afect the main result of the paper since we are interested in including fuzzy closure operators, which are isotone mappings. Then, it is proved that the fixed points of this fuzzy Galois connection satisfy some additional properties, such as extensionality of inflationarity.
The main result of the paper is proving that fixed points of a specific fuzzy Galois connection
are the ones formed by a fuzzy closure system and the fuzzy closure operator induced by it.
Contrary to the main results in Galois connections and FCA [
For the reader’s convenience, we give the main notions and theorems that are used along this paper.
An algebra = (, ≤, 0, 1, →) is a complete Heyting algebra, [16], if (, ≤) is a complete lattice, 0 and 1 are the minimum and maximum elements, respectively, and the following condition holds, for all , , ∈ ∧ ≤ if and only if ≤ → In addition, we have the following properties: ( → ) ∧ ( → ) ≤ ( ∧ ) → ( ∧ ) ∧ ( → ) ≤ ∧ ≤ implies → ≤ → ( ∧ ) ∧ ( ∨ ) = ( ∧ ) ∨ ( ∧ ) ∨ ( ∧ ) = ( ∨ ) ∧ ( ∨ ) (2) (3) (4) (5) (6) for all , , ∈ .
An -set or fuzzy set is a mapping ∶ → from the universe set to the membership values set where () means the degree in which belongs to . An -set is said to be a fuzzy subset of if () ≤ () , for all ∈ . As usual, the core of is the set of elements such that () = 1 and is denoted by ( ) . An -set ∈ is said to be extensional if () ∧ ( ≈ ) ≤ () , for all , ∈ . Finally, a crisp set is considered to be a particular case of -set by using its characteristic mapping ∶ → {0, 1} with () = 1 if ∈ . Operations with -sets are defined pointwise as usual.
Let ∶ × → be a mapping defined by (, ) = ⋀∈ (() → () ). This mapping is called the subsethood degree relation. Whenever is a fuzzy subset of , we have (, ) = 1 , and this is denoted by ⊆ .
The concepts of fuzzy binary relation, its properties, fuzzy poset and fuzzy lattice are used as in the standard definitions, see [ 17, 18, 19]. Throughout this section (, ) will be a fuzzy poset.
The lower and upper cone of a fuzzy set are defined as () = () = ∈ ∈ ⋀ ( () → (, ) ⋀ ( () → (, ) ) ) and
Similarly, the following concepts were introduced in [20],
An element ∈ is an infimum (resp. supremum) of a fuzzy set ∈ if: i. () = 1 (resp. () = 1 ). ii. ⊆ (resp. ⊆ ).
As a consequence, if ∈ is infimum (resp. supremum) of ∈ , then ⊆ (resp. ⊆ ).
As in the classical case, if the infimum (resp. supremum) of a set exists then it is unique and is denoted by ⨅ (resp. ⨆ ). In addition, an element ∈ is a minimum (resp. maximum) of if and only if is an infimum (resp. supremum) of and () = 1 .
Theorem 1. Let ∈ . An element ∈ is the infimum (resp. supremum) of if and only if = (resp. = ).
Definition 2. Given a fuzzy preposet = (, ) operator on if the following conditions hold: i. (, ) ≤ ( c(), c()) , for all , ∈ ii. (, c()) = 1 , for all ∈ iii. ( c(c()), c()) = 1 , for all ∈ .
, a mapping c ∶ → is said to be a closure
A complete fuzzy lattice (, ) is a fuzzy poset such that every fuzzy subset ∈ has supremum and infimum. We denote by ⊥ and ⊤ the bottom and top elements of the complete fuzzy lattice, respectively. The pair ( , ) is an example of complete fuzzy lattice [21], which is called the -powerset lattice of . A set ⊆ is said to be a complete fuzzy sublattice of if, for all fuzzy set ∈ , the elements ⨅ and ⨆ are in .
The focus of this paper is the characterization of closure structures as fixed points of a fuzzy Galois connection. The definition of closure operator is the following.
Conditions i and ii are well-known and are called isotony and inflationarity, respectively. Observe that condition iii could be replaced by (c(c()) ≈ c()) = 1 , and, thus, if is a fuzzy poset, a closure operator is idempotent in a classical sense, i.e., c(c()) = c() for all ∈ .
On the other hand, there is not a unique extension of the notion of closure system to a fuzzy setting. The definition we will use in this paper is the following one, introduced in [ 13]. Definition 3. Let (, ) be a complete fuzzy lattice. We say that an -set Φ ∈ is a fuzzy closure system if it is extensional and ⨅ ∈ (Φ) for all ⊆ Φ .
Fuzzy closure systems and fuzzy closure operators are related concepts. In fact, similarly to the classical case, there is a one-to-one relation among them.
Theorem 2. Let = (, )
be a complete fuzzy lattice. The following assertions hold: i. If Φ is a fuzzy closure system, the mapping cΦ ∶ → defined as cΦ() = ⨅( ∩ Φ) is a closure operator. ii. If c is a closure operator, the fuzzy set Φc defined as Φc() = ( c(), ) is a fuzzy closure system. iii. If Φ is a fuzzy closure system, then Φ = ΦcΦ. iv. If c ∶ → is a closure operator, then cΦc = c.
The following result is a characterization of fuzzy closure systems which is useful in the proofs.
Proposition 1. Let (, ) be a complete fuzzy lattice. A fuzzy set Φ ∈ is a fuzzy closure system if and only if Φ is extensional and min( ∩ Φ) exists for all ∈ .
Fuzzy Galois connections are a main concept in this paper as well. Let us recall the definition. Definition 4 ([19]). Let = ⟨, be two mappings. The pair ( , ) ( , ) ∶ ↼⇀ , if ⟩ and = ⟨, ⟩ be fuzzy posets, ∶ → and ∶ → is called a Galois connection between and , denoted by (, ()) = (, ()) for all ∈ and ∈ . 3. A fuzzy Galois connection between -sets and isotone mappings Consider, on the one hand, the -powerset lattice of , denoted by ( , ) , and, on the other hand, the pair ( , )̃ consisting of the set of (crisp) mappings on and the pointwise -order defined as (̃ 1, 2) = ⋀ ( 1(), 2()) for all 1, 2 ∈ .
∈ Among the mappings in ( , )̃ , the isotone (or “order preserving”) ones play an important role because they reflect the idea of homomorphism betweeen -posets. We denote the set of isotone mappings on (, ) as (Isot( ), )̃ .
Proposition 2. The couple ( , )̃ is a complete fuzzy lattice and (Isot( ), )̃ is a complete fuzzy sublattice.
Consider now the functions c ∶ ( , ) → ( Isot( ), )̃ and Ψ ∶ (Isot( ), )̃ → ( , ) defined as follows: • c assigns to any Φ ∈ the mapping cΦ ∶ → where • Ψ assigns to any isotone mapping ∶ → the -set Ψ ∈ where cΦ() = ⨅( ∩ Φ) for all ∈ . Ψ () = ( (), ) for all ∈ .
The following proposition ensures that the mapping c is well-defined.
Proposition 3. For all Φ ∈ , the mapping c(Φ) is isotone.
Proof. For all , ∈
, we have that (, ) ≤ ⋀((, ) → (, )) ∈ ≤ ⋀(( ∩ Φ)() → ( ∩ Φ)())
∈ ≤ ⋀(( ∩ Φ)() → ( cΦ(), ))
∈ = ( cΦ(), cΦ()) by Def. 1 and (2.43) in [20] by transitivity of
Hence, both Ψ and c are well-defined. In addition, these mappings are related since they form a Galois connection.
Theorem 3. The pair (c, Ψ) forms a fuzzy Galois connection between ( , ) and (Isot( ), )̃ . Proof. Consider Φ ∈ and ∈ ( Isot( ), )̃ . First, we have that Conversely, (Φ,Ψ( )) = ⋀ (Φ() → ( (), )
) ∈ ≤ ⋀ (((, ) ∧ Φ()) → ((, ) ∧ ( (), ) ,∈ ,∈ ≤ ⋀ (( ∩ Φ)() → (( (), ()) ∧ ( (), ) ≤ ⋀ (( ∩ Φ)() → ( (), )
) ,∈ = (̃ , c(Φ)) by Theorem 1 by considering =
by reflexivity ) )
As usual, a fixed point, or a formal concept, of the fuzzy Galois conection is a couple (Φ, ) such that c(Φ) = and Ψ( ) = Φ . The fuzzy Galois connection between these mappings is interesting due to the properties its formal concepts have. The next result proves that the image by c or Ψ satisfy interesting additional properties, such as inflationarity in mappings and extensionality in fuzzy sets.
i. For all Φ ∈ , the mapping c(Φ) is inflationary.
ii. For all ∈ ( Isot( ), )̃ , we have that Ψ( ) is extensional.
Proof. First, given Φ ∈ and ∈ , by Theorem 1, we have that (, cΦ()) = (, ⨅( ∩ Φ)) = ( ∩ Φ) () = ⋀ (( ∩ Φ)() → (, )
) = 1 ∈ On the other hand, given ∈ ( Isot( ), )̃ , for all ∈ , we have that Ψ () ∧ (, ) ∧ (, ) = ( (), ) ∧ (, ) ∧ (, ) ≤ ( (), ) ∧ (, ) ≤ ( (), ) ∧ ( (), ()) ≤ ( (), ) = Ψ ()
by transitivity by isotoncity of by transitivity.
The following theorem is the main result of the paper. It characterizes the formal concepts of the fuzzy Galois connection. Remarkably, every fixed point of this connection is formed exactly by a fuzzy closure system and a fuzzy closure operator.
Theorem 4. The following statements are equivalent: i. The couple (Φ, ) is a fixed point of (c, Ψ). ii. The fuzzy set Φ is fuzzy closure system and = c(Φ).
iii. The isotone mapping is a fuzzy closure operator and Φ = Ψ( ) .
Proof. First, items and being equivalent is a direct consequence of Theorem 2. Furthermore, the equivalence between items and proves directly that the couple (Φ, ) is a fixed point of the fuzzy Galois connection. To prove implies , assume (Φ, ) is a fixed point. Then, we have to prove that Φ is a fuzzy closure system. Recall that fuzzy closure systems are extensional fuzzy sets Φ which satisfy ⨅ ∈ (Φ) , for all ∈ such that ( , Φ) = 1 . Let ∈ such that ( , Φ) = 1 Φ we get = ∩ Φ ⊆ hypothesis this is ( (), ) = ( and let = ⨅ . Then, by Definition 1, we get ⊆ , intersecting ∩ Φ and taking infima we have ( ⨅( ∩ Φ), ⨅ ) = 1 . Using the cΦ(), ) = 1 . Thus, Φ() = Ψ () = 1 and ⨅ ∈ (Φ).
By Lemma 1, Φ is extensional. Hence, Φ is a fuzzy closure system.
Notice that the proof uses strongly the restriction of being in a Heyting algebra. However, in the general residuated lattice case, where, as explored in [14], the natural construction of cΦ is ⨅( ⊗ Φ), the analogous result does not hold. A counterexample of last theorem in the general residuated lattice case is shown below.
Example 1. Let = ({0, 0.5, 1}, ∧, ∨, ⊗, →, 0, 1) be the three-valued Łukasiewicz residuated lattice, and (, )
be the fuzzy lattice with = {⊥, , , , , , ⊤} is described by the following table: and the fuzzy relation ∶ × → ⊥ ⊤ ⊥ 1 0.5 0.5 0.5 0 0 0 1 1 0.5 0.5 0 0 0.5 0.5 1 0.5 1 0 0 1 1 1 1 0.5 1 1 1 1 1 ⊤ 1 1 1 1 1
For the fuzzy set Φ = { /1, /0.5}, the mapping cΦ() = and inflationary, is (⊥) = () = ; () = , () = () = ⨅( ⊗ Φ) = () , which is isotone and () = (⊤) = ⊤ . However, is not a closure operator since it is not idempotent, ( ( ()), ()) = (, ) = 0.5 ≠ 1 . 4. Conclusions and further work This paper continues the line of work which initiated in [13], where fuzzy closure systems were introduced in the Heyting algebra framework. The mappings that take fuzzy closure systems to closure operators and vice versa are studied in a more general setting and are proved to form a fuzzy Galois connection. In addition, the formal concepts of this fuzzy Galois connection are shown to be exactly the fuzzy closure systems and closure operators in .
As a prospect of future work, since we already know the images of the mappings introduced in [14] are not closure operators and fuzzy closure systems in general, this analysis can be extended to the general framework and study whether these mappings form a fuzzy Galois connection as well and, in the case they do, study the nature of its fixed points.
This work has been supported by the State Agency of Research (AEI), the Spanish Ministry of Science, Innovation, and Universities (MCIU), the European Social Fund (FEDER), the Junta de Andalucía (JA), and the Universidad de Málaga (UMA) through the FPU19/01467 (MCIU) internship and the research projects with reference PGC2018-095869-B-I00,TIN2017-89023-P (MCIU/AEI/FEDER, UE) and UMA2018‐FEDERJA‐001 (JA/UMA/FEDER, UE). complete L-ordered sets, in: 2010 Seventh International Conference on Fuzzy Systems and Knowledge Discovery, volume 1, 2010, pp. 216–218. doi:10.1109/FSKD.2010.5569695. [13] M. Ojeda-Hernández, I. P. Cabrera, P. Cordero, E. Muñoz-Velasco, On (fuzzy) closure systems in complete fuzzy lattices, in: 2021 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2021, pp. 1–6. doi:10.1109/FUZZ45933.2021.9494404. [14] M. Ojeda-Hernández, I. P. Cabrera, P. Cordero, E. Muñoz-Velasco, Closure systems as a fuzzy extension of meet-subsemilattices, in: Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP), Atlantis Press, 2021, pp. 40–47. doi:10.2991/ asum.k.210827.006. [15] J. Konecny, M. Krupka, Complete relations on fuzzy complete lattices, Fuzzy Sets and
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