Relational correspondences for L-fuzzy rough approximations defined on De Morgan Heyting algebras

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras by using only a single axiom.


Introduction
Rough sets were introduced by Z. Pawlak in [13] to deal with concepts that cannot be defined precisely in terms of our knowledge.In rough set theory, knowledge about objects U is given in terms of an indistinguishability relation E, which is an equivalence E on U interpreted so that two elements are E-related if we cannot distinguish them in terms of the knowledge E. For each subset, X the lower approximation X E is the set of elements whose E-equivalence class is included in X.The set X E is interpreted as the set of elements that certainly belong to X in view of the knowledge E. The upper approximation X E consists of elements whose E-class intersects with X.The set X E can be seen as the set of elements which possibly belong to X.This means that every vague concept can be approximated from below and above by two sets that are definable by the knowledge E.
In the literature, numerous studies can be found in which equivalences are replaced by an arbitrary binary relation reflecting, for instance, similarity or preference between the elements; see [10,20], for example.Also in such a generalized setting, the lower approximation X R consists of elements which necessarily are in X and X R is the set of elements which possibly are in X, in the view on knowledge R.
Fuzzy sets introduced by L. Zadeh [21] are generalizations of traditional sets such that the setmemberships are expressed by a real number of the interval [0, 1].Similarly, Zadeh defined fuzzy relations as generalizations of traditional binary relations.Soon after Zadeh's paper, J. A. Goguen [4] defined so-called L-fuzzy sets and L-fuzzy relations, in which the [0, 1]-interval is replaced by a complete lattice L having 0 as the smallest and 1 as the greatest elements.Since then, all kinds of structures, such as Heyting algebras or residuated lattices, have been presented as a basis for fuzzy sets [12].
The first approach to integrate rough set theory and fuzzy set theory is the paper by D. Dubois and H. Prade [3], where they introduced fuzzy rough sets.A comprehensive evaluation of the most relevant fuzzy rough set models proposed in the literature can be found in [2].
Correspondences are understood as conditions that connect the properties of relations to the properties of approximation operations.For instance, if R is an arbitrary binary relation on U , then R is symmetric if and only if (X R ) R ⊆ X ⊆ (X R ) R holds for all subsets X of U .Correspondences for binary relations are well-studied in the literature.For instance, reflexive, symmetric, and transitive relations are characterized in [5].In [20], the authors considered serial, reflexive, symmetric, transitive and Euclidean relations.Correspondences for serial, reflexive, mediate, transitive and alliance relations are give in [23].However, there seems to be a problem in the characterization of alliance relations, and we consider this issue in Example 3.27.
As noted above, there are several ways to generalize rough sets to fuzzy rough sets.This means that there are various types of correspondence results in the literature depending on the setting.For instance, in [14] the authors considered L-fuzzy rough approximations in the case L is a complete residuated lattice.They characterized serial, reflexive, symmetric, Euclidean, and transitive Lfuzzy relations.In [18], the authors presented correspondences for reflexive, symmetric, transitive, and Euclidean [0, 1]-fuzzy relations.Similar study of [0, 1]-fuzzy rough sets can be found in [7].Some other recent types of fuzzy rough approximations and their correspondences are considered in [6,15,16,17,19,22], for example.
In [11], the authors consider such L-fuzzy relations that L is a Heyting algebra provided with an antitone involution ′ .This means that L is a so-called De Morgan Heyting algebra.Three new types (mediate, Euclidean, adjoint) of L-fuzzy rough approximation operators were characterized in that paper.In this work, we adopt their approach.We will present a new uniform method in Theorem 3.2 which covers the results in [11], and several others.L. Liu [8] initiated the study of characterizing fuzzy rough set approximations by only one axiom.In [11], the authors provided an axiomatic approach to L-fuzzy rough approximation operators, where L is a De Morgan Heyting algebra.They also presented single axioms to characterize L-fuzzy rough approximation operators corresponding to mediate, Euclidean, adjoint L-fuzzy relations as well as their compositions.In that work, they gave the following open problem: "Using single axioms to characterize L-fuzzy rough approximation operators corresponding to compositions of serial, reflexive, symmetric, transitive, mediate, Euclidean and adjoint L-fuzzy relations."In this work, we present a general solution to this problem, which covers also other types of relations and their compositions.
The paper is structured as follows.In Section 2 we recall from the literature notions and basic results used in this work.More precisely, in Section 2.1, we consider the essential correspondence results related to (crisp) binary relations and rough approximation operations.In this work, we consider fuzzy rough sets on De Morgan Heyting algebras, and Section 2.2 covers the basic facts of these types of algebras.
Section 3 is devoted to the correspondence results characterizing properties of L-fuzzy relations in terms of fuzzy rough approximation operators.In Section 3.1 we present a result which is frequently used in other subsections to prove correspondences.We present correspondence for reflexive and symmetric (Section 3.2), transitive and mediate (Section 3.3), Euclidean and adjoint (Section 3.4), and functional and positive alliance relations (Section 3.5).There are several ways to generalize serial relations to their fuzzy counterparts, and in Section 3.6 we consider three such generalizations.
Finally, in Section 4 we solve the problem given in [11] and some concluding remarks end the work.

Preliminaries
In this section, we first recall correspondences between (crisp) binary relations and rough approximation operations.The second part of this section considers De Morgan Heyting algebras and fuzzy rough approximation operations defined on them.

Binary relations and rough approximation operators
A binary relation on U is a set of ordered pairs (x, y), where x and y are elements of U .If x and y are R-related, this is denoted by x R y or (x, y) ∈ R.
Let us introduce some general properties of binary relations.A binary relation R on U is (i) serial, if for all x ∈ U , there is y ∈ U such that x R y, (ii) reflexive, if x R x for all x ∈ U , (iii) symmetric, if x R y, then y R x for all x, y ∈ U , (iv) transitive, if x R y and y R z, then x R z for all x, y, z ∈ U , (v) mediate, if x R y for some x, y ∈ U , there is z ∈ U such that x R z and z R y, (vi) Euclidean, if x R y and x R z, then y R z for all x, y, z ∈ U .
Next, we recall (see e.g.[5,20]) rough approximation operations defined by arbitrary relations.For any subset X ⊆ U , the lower approximation of X is defined as The upper approximation of X is The following correspondence results can be found in the literature; see [5,20,23], for instance: In this work, we consider similar correspondences in the setting of L-fuzzy rough sets, when a complete De Morgan Heyting algebra is defined on L.
There are two ways to define the operation ′ in L. We can define It is also possible to define In both cases, we have a De Morgan Heyting algebra defined on L.
A complete lattice L satisfies the join-infinite distributive law if for any S ⊆ L and x ∈ L, The dual condition is the meet-infinite distributive law, (MID).It is well known that a complete lattice defines a Heyting algebra if and only if it satisfies (JID).Each De Morgan Heyting algebra L defined on a complete lattice L satisfies both (JID) and (MID), because ′ is an order-isomorphism between (L, ≤) and its dual (L, ≥).In this work, De Morgan Heyting algebras defined on a complete lattice are called complete De Morgan Heyting algebras.
Let L be a complete De Morgan Heyting algebra and U a universe.An L-fuzzy set on U is a mapping A : U → L. We often drop the word 'fuzzy' and speak about L-sets.The family of all L-sets on U is denoted by F L (U ).
The set F L (U ) may be ordered pointwise by setting for The operations ⇒ and ′ are defined for A, B ∈ F L (U ) and x ∈ U by The map 0 : x → 0 is the smallest and 1 : x → 1 is the greatest element of F L (U ), respectively.An L-fuzzy relation R on U is a mapping U × U → L. We often use the term "L-relation" instead of "L-fuzzy relation".The following definition of L-approximations can be found in [11].Definition 2.2.Let L be a complete De Morgan Heyting algebra, R an L-relation on U and A ∈ F L (U ).The upper L-fuzzy approximation and lower L-fuzzy approximation of A are defined by respectively.
If there is no danger of confusion, we may denote U(A) and L(A) simply by UA and LA.In addition, L-fuzzy approximations are called simply L-approximations.
In [11], the following properties of L-approximations generalizing well-known properties of crisp rough approximations are proved.
Proposition 2.3.Let U be a set, L a complete De Morgan Heyting algebra and R an L-relation on U .For {A i } i∈I ⊆ F L (U ), A ∈ F L (U ) and x ∈ U , the following assertions hold: (1) L0 = 0 and U1 = 1; ( For a ∈ L, we define a 'constant' L-set a by a(x) = a for all x ∈ U .
This means that 0 = 0 and 1 = 1.For any x ∈ U , we define a map I x by The idea is that the map I x corresponds to the singleton {x}.For any x, y ∈ U , U(I y )(x) = z (R(x, z) ∧ I y (z)).Because I y (z) = 1 iff z = y, we obtain the following equality, which will be used frequently in this work: (∀x, y ∈ U ) U(I y )(x) = R(x, y). (2.1) It is noted in [11] that each A ∈ F L (U ) can be written in two ways: Note that (I x ) ′ corresponds the set-theoretical complement U \ {x} of the singleton {x}, and the latter equality is clear by Lemma 3.17 of [11].The following facts were also proved in [11].
Lemma 2.4.Let L be a complete De Morgan Heyting algebra and R an L-relation on U .For all a ∈ L and

Correspondence results
In this section, we assume that L is a complete De Morgan Heyting algebra, U is a universe, and R is an L-relation on U .

A general result
In this subsection, we present a general result which can be used to obtain several correspondence results.Let S be a finite combination of rough approximation operators L and U.Because the operators L and U are order-preserving, the operator S is order-preserving.From this it follows that Proof.We prove the claim by induction.If n = 1, then the two cases ā ∧ U(I x ) ≤ U(ā ∧ I x ) and ā ∧ L(I x ) ≤ L(ā ∧ I x ) are clear by Lemma 2.4(2) by setting A = I x .Suppose that the claim holds for all combinations consisting of n L and U operators.Let S be a combination n + 1 operators.Then S = U • S 1 or S = L • S 2 , where S 1 and S 2 are combinations of U and L of length n.
Theorem 3.2.Let S be a finite combination of rough approximation operators L and U.If U(I x ) ≤ S(I x ) for all x ∈ U , then U(A) ≤ S(A) for all A ∈ F L (A).
Proof.Assume that U(I x ) ≤ S(I x ) for all x ∈ U .Then,

Reflexive and symmetric relations
Let us start with the following definition.
The following lemma gives a characterization of reflexivity.
Proof.Suppose that R is reflexive.Then, by Lemma 3.4, I x ≤ U(I x ) for all x ∈ U .We have that for A ∈ F L (U ), Thus, (1) implies (2).Suppose that (2) holds.Then for all x ∈ U , I x ≤ U(I x ), which by Lemma 3.4 yields that R is reflexive.Thus, also (2) implies (1).We prove that (2) and (3) are equivalent.Assume (2) holds.Then for any A binary relation ρ is symmetric whenever, for all x, y ∈ U , x ρ y implies y ρ x.The symmetry condition can be expressed in the form (x ρ c y) ∨ (yρ x), where ρ c denotes the complement of the relation ρ, that is, ρ c = (U × U ) \ ρ.Based on this fact, we present the following definition.
We can now express symmetry in terms of I x and L-approximations.Lemma 3.7.An L-relation R is symmetric if and only if for all x ∈ U , I x ≤ LU(I x ).
Proof.For all x ∈ U , If R is symmetric, then LU(I x )(x) = 1, which means that LU(I x )(x) ≥ I x (x).On the other hand, if LU(I x ) ≥ I x , then LU(I x )(x) = 1 and y (R(x, y) ′ ∨ R(y, x)) = 1.This gives that for each y, R(x, y) ′ ∨ R(y, x) = 1.
We can now write the following characterization of symmetric L-relations.Proposition 3.8.Let L complete De Morgan Heyting algebra and R an L-relation on U .Then, the following are equivalent: (1) R is symmetric; ( Proof.Suppose that R is symmetric.Then, by Lemma 3.7, I x ≤ LU(I x )(x) for each x ∈ U .We have that This means that (1) implies ( 2).If we set A = I x in (2), we get that I x ≤ LU(I x ) for x ∈ U .Hence, R is symmetric.The equivalence of ( 2) and (3) follows easily by the duality of approximation operations.
We end this subsection by the following remark.
Remark 3.9.For a fuzzy relation R, symmetry is typically expressed by a condition (∀x, y ∈ U ) R(x, y) = R(y, x).

(S*)
Let us briefly consider how this relates to the symmetry of Definition 3.6.
On the other hand, consider the 4-element Heyting algebra L := 0 < a, b < 1, where a and b are incomparable.Let the operation ′ be defined by 0 It remains an open problem whether (S*) be characterized in terms of fuzzy rough approximation operations.

Transitive and mediate relations
Obviously, transitivity is equivalent to that (∀x, y ∈ U ) z (R(x, z) ∧ R(z, y)) ≤ R(x, y).Lemma 3.11.An L-relation R is transitive if and only if for all x ∈ U , UU(I x ) ≤ U(I x ).

Euclidean and adjoint relations
A binary relation ρ is Euclidean if x ρ y and x ρ z imply y ρ z.Obviously, x ρ y and x ρ z imply also z ρ y.As noted in [11], this is equivalent to that x ρ z and z ρ c y imply x ρ c y.
Obviously, being Euclidean is equivalent to the condition (euc) Lemma 3.17.An L-relation R is Euclidean if and only if for all x ∈ U , U(I x ) ≤ LU(I x ).
Proof.Let x, y ∈ U .As we have noted, U(I y )(x) = R(x, y) and This means U(I y ) ≤ LU(I y ).
On the other hand, if U(I y ) ≤ LU(I y ), then for all x ∈ U , U(I y )(x) ≤ LU(I y )(x) and thus, Applying De Morgan operation ′ for the both sides of the relation, we obtain that (euc) holds.
The next proposition characterizes Euclidean L-relations in terms of approximations.
Proposition 3.18.Let L be a complete De Morgan Heyting algebra and R an L-relation on U .Then, the following are equivalent: Proof.If R is Euclidean, then U(I x ) ≤ LU(I x ) for all x ∈ U .By Theorem 3.2, UA ≤ LUA for all A ∈ F L (U ).Thus, ( 2) implies (1).By applying A = I x , we see that ( 2) implies (1).The equivalence of ( 2) and ( 3) is obvious.
We recall from [11] the definition of adjoint relations.
We can now characterize adjoint relations in terms of I x and fuzzy approximations operations.
Lemma 3.20.An L-relation R is adjoint if and only if for all x ∈ U , U(I x ) ≤ UL(I x ).
Proof.If R is functional, then U(I x ) ≤ L(I x ) for all x ∈ U .By Theorem 3.2, we obtain UA ≤ LA for all A ∈ F L (U ).Conversely, if UA ≤ LA for any A ∈ F L (U ), then U(I x ) ≤ L(I x ) for all x ∈ U .
'Positive alliance' relations were defined in [23] by stating that a binary relation ρ is a positive alliance if for any elements x, y ∈ U such that x ρ c y, there is z ∈ U satisfying x ρ z, but z ρ c y.
It is clear that each reflexive relation is positive alliance, because if a ρ c b, then a ρ a and a ρ c b hold trivially.
The following facts can be found in [9], but for the sake of completeness we give a proof.
(1) Suppose a ρ c b.Because ρ is serial, there is c such that a ρ c.Now c ρ b is not possible, because that would imply a ρ b, contradicting a ρ c b. Thus, c ρ c b.
On the other hand, it is clear that if ρ is a positive alliance, then it is serial.However, there are positive alliance relations that are not transitive.It is clear that for all x, y ∈ U , y ∈ {x} ρ ⇐⇒ y ρ x.Therefore, As proved in [23], ρ is a positive alliance if and only if ({x} ρ ) c ⊆ (({x} ρ ) c ) ρ for all x ∈ U .Note that the latter condition is equivalent to that ({x} ρ ) ρ ⊆ {x} ρ for all x ∈ U .
For L-relations, we can present the following generalized definition.
Definition 3.28.An L-relation R is said to be a positive alliance if We can now write the following lemma characterizing positive alliance relations in terms of the approximations of the identity functions.But as in the case of (crisp) binary relations, we obviously cannot present a full correspondence result.Lemma 3.29.An L-relation R is a positive alliance if and only if for all x ∈ U , U(I x ) ≥ LU(I x ).
Proof.Let x, y ∈ I.We have Now, U(I x ) ≥ LU(I x ) for all x ∈ U if and only if U(I y )(x) ≥ LU(I y )(x) for all x, y ∈ U if and only if (U(I y )(x)) ′ ≤ (LU((I y )(x)) ′ for all x, y ∈ U , and the equivalence follows from this.

Serial relations
A binary relation ρ on U is serial if for all x ∈ U , there exists y ∈ U such that x R y.It is known that being serial is equivalent to the fact that X ρ ⊆ X ρ for all X ⊆ U .
Our aim of this section is to find a definition for serial L-relations on U such that a relation is serial if and only if In this subsection, we consider three definitions.Let us begin with the following one.Let us consider the L-set 0 defined by x → 0. We have that This means that L(0) ≰ U(0).On the other hand, We have shown that L(I x ) ≤ U(I x ) for all x ∈ U .This means that (3.2) is not equivalent to (3.1).
In case of binary relations, it is also true that a relation ρ is serial if and only if U ρ = U .We end this section by showing that we can have a simple correspondence generalizing this.It is clear that the seriality of Definition 3.30 implies the seriality of Definition 3.36.The converse is not true.For instance, the L-relation R of Example 3.32 is serial in the sense of Definition 3.36, but R(x, y) ̸ = 1 for all x, y ∈ U .
We also proved in Lemma 3.31 that the seriality of Definition 3.30 implies (3.1).Our final example of this subsection shows that seriality of Definition 3.36 does not imply (3.1). and 4 From L-approximations to L-relations In the previous section, we defined the L-approximations LA and UA for any A ∈ F L (U ) in terms of an L-relation R on U .In this section, we consider a converse problem, that is, whether we can define an L-relation of certain type for a dual pair of L-fuzzy operators.As we already noted in Section 2.2, for every L-relation R on U , we have for all x, y ∈ U and A ∈ F L (U ).This provides a 'rule' for defining relations when upper approximations are known.We call any map on F L (U ) as an L-fuzzy operator on U .
Lemma 4.1.Let U be an L-fuzzy operator.For any a ∈ L, A ∈ F L (U ) and {A i } i∈I ⊆ F L (U ) the following are equivalent: (1) Proof.
(1)⇒(2): Suppose that (1) holds.Then, and Combining these two, we obtain that (2)⇒(1): Suppose that (2) holds.Since A = x (A(x) ∧ I x ), we have that In [11], the authors presented the following problem: "Using single axioms to characterize Lfuzzy rough approximation operators corresponding to compositions of serial, reflexive, symmetric, transitive, mediate, Euclidean and adjoint L-fuzzy relations."Next, we solve this problem so that only the parts concerning serial and symmetric relations remains open.
Our following theorem is closely related to Theorem 4.2.
Theorem 4.6.Let U be an L-fuzzy operator on U and let each T k , 1 ≤ k ≤ m, be an L-fuzzy operator on U such that U ≥ T k .Then there exists a unique L-fuzzy relation R on U such that UA = U(A) for all A ∈ F L (U ) if and only if If there is an L-relation R such that UA = UA for all A ∈ F L (U ), then The direction (⇐) is proved as in Theorem 4.2.
Now we can write the following corollary of Theorem 4.6.
Corollary 4.7.Let U and L be dual L-fuzzy operators on U .
(1) There exists a unique reflexive L-relation R on U such that U and L coincide with the upper and lower approximation operators of R, respectively, if and only if for all a ∈ L and {A i } i∈I ⊆ F L (U ).(2) There exists a unique transitive L-relation R on U such that U and L coincide with the upper and lower approximation operators of R, respectively, if and only if for all a ∈ L and {A i } i∈I ⊆ F L (U ). Proof.
(1) By Proposition 3.5, R is reflexive if and only if A ≤ UA for any A ∈ F L (U ) from which the result follows by Theorem 4.6 by setting n = 1 and T 1 (A) = A.
(2) By Proposition 3.12, R is transitive if and only if UUA ≤ UA for any A ∈ F L (U ) from which the result follows by Theorem 4.6 by setting n = 1 and T 1 = UU.
Let U be an L-fuzzy operator on U .Assume that each S j , 1 ≤ j ≤ n, is an L-fuzzy operator on U such that U ≤ S j and suppose each T k , 1 ≤ k ≤ m is an L-fuzzy operator on U such that U ≥ T k .Now we have that

.3)
We can now write the following theorem.Its proof is clear, because (⇒) part follows from (4.3), and (⇐) can be proved as in Theorem 4.2.
Theorem 4.8.Let U be an L-fuzzy operator on U and let each S j , 1 ≤ j ≤ n and T k , 1 ≤ k ≤ m, be L-fuzzy operators on U such that U ≤ S j and U ≥ T k .Then there exists a unique L-fuzzy relation R on U such that UA = U(A) for all A ∈ F L (U ) if and only if for a ∈ L and {A i } i∈I ⊆ F L (U ).
We end this work by the following example of a single condition for a combination of certain relation types.
Corollary 4.9.Let U and L be dual L-fuzzy operators on U .There exists a reflexive, transitive, mediate, Euclidean and adjoint L-relation R on U such that U and L coincide with the upper and lower approximation operators of R, respectively, if and only if for all a ∈ L and {A i } i∈I ⊆ F L (U ).

Some concluding remarks
In this work, we were able to solve the open problem presented in [11] so that only the parts concerning serial and symmetric relations remain open.
One obvious reason for this is that we do not have a condition for symmetry or seriality which is of the suitable form so that the theorems given in Section 4 could be applied.Interestingly, seriality and symmetry are also such that there are some issues how to generalize them as fuzzy relations, at least in case of De Morgan Heyting algebras.

Example 3 .
26.Let U = {a, b, c} and let ρ = {(a, b), (b, b), (c, a), (c, c)}.Then ρ is serial, but not transitive because c ρ a and a ρ b, but c ρ c b. Now we have that: • a ρ c a, and there is b such a ρ b and b ρ c a; • a ρ c c, and there is b such a ρ b and b ρ c c; • b ρ c a, and there is b such b ρ b and b ρ c a; • b ρ c c, and there is b such b ρ b and b ρ c c; • c ρ c b, and there is c such c ρ c and c ρ c b. Hence, ρ is a positive alliance.
Now we can write to following characterization.Lemma 3.37.An L-relation R is serial if and only if U(1) = 1.

Example 3 .
38.Let L be the 4-element lattice 0 < a, b < 1, where a and b are incomparable.If we define 0 ′ = 1, a ′ = a, b ′ = b, and 1 ′ = 0, then we have a complete De Morgan Heyting algebra.Let U = {x, y} and define R(x, x) = R(y, y) = a and R(x, y) = R(y, x) = b.It is now clear that R is serial in the sense of Definition 3.36, because R(x, x) ∨ R(x, y) = a ∨ b = 1 and R(y, x) ∨ R(y, y) = b ∨ a = 1.Let U = {a, b} and define an L-set A : U → L by A(x) = b and A(y) = a.Now