Defining rough sets as core-support pairs of three-valued functions

We answer the question what properties a collection $\mathcal{F}$ of three-valued functions on a set $U$ must fulfill so that there exists a quasiorder $\leq$ on $U$ such that the rough sets determined by $\leq$ coincide with the core--support pairs of the functions in $\mathcal{F}$. Applying this characterization, we give a new representation of rough sets determined by equivalences in terms of three-valued {\L}ukasiewicz algebras of three-valued functions.


Introduction
Rough set defined by Z. Pawlak [1] are closely related to three-valued functions. In rough set theory, knowledge about objects of a universe of discourse U is given by an equivalence E on U interpreted so that x E y if the elements x and y cannot be distinguished in terms of the information represented by E. Each set X ⊆ U is approximated by two sets: the lower approximation X consists of elements which certainly belong to X in view of knowledge E, and the upper approximation X consists of objects which possibly are in X. Let 3 = {0, u, 1} be the 3-element set in which the elements are ordered by 0 < u < 1. For any X ⊆ U , we can define a three-valued function f such that f (x) = 0 if x does not belong to X , that is, x is interpreted to be certainly outside X. We set f (x) = 1 when x ∈ X , meaning that x certainly belongs to X. If x belongs to the set-difference X \ X , which is the actual area of uncertainty, we set f (x) = u.
On the other hand, in fuzzy set theory the 'support' of a fuzzy set is a set that contains elements with degree of membership greater than 0 and the 'core' is a set containing elements with degree of membership equal to 1. Naturally, each 3-valued function can be considered as a fuzzy set, and for f : U → 3, its core C(f ) can be viewed as a subset of U consisting of elements which certainly belong to the concept represented by f , and the support S(f ) may be seen as a set of objects possible belonging to the concept represented by f . Obviously, C(f ) ⊆ S(f ) for any three-valued function f . Note also that different roles of three-valued information, such as vague, incomplete or conflicting information are considered in [2].
We call pairs (A, B) of subsets of U such that A ⊆ B as 'approximation pairs'. The motivation for this name is that X ⊆ X for all X ⊆ U if and only if the relation defining the approximations is serial (see [3], for instance). A relation R on U is serial if each element of U is R-related to at least one element. Therefore, for a serial relation R on U and a subset X ⊆ U , the pair (X , X ), called 'rough set', is an approximation pair, and thus an approximation pair can be seen as a generalization of a rough set (X , X ).
The concept of 'three-way decisions' is recently introduced to relate with the three regions determined by rough approximations; see [4], for example. The idea of three-way decisions is to divide, based on some criteria, the universe U into three pairwise disjoint regions, called the positive (pos), negative (neg), and boundary (bnd) regions, respectively. Obviously, each such a three-way decision forms an approximation pair (pos, pos ∪ bnd) and each approximation pair (A, B) induces a three-way decision, where the positive area is A, the boundary equals the difference B \ A, and B c forms the negative region. More generally, there is a one-to-one correspondence between approximation pairs and 3-valued functions.
The three-valued chain 3 can be equipped with various algebraic structures. It is known that the 3-valued chain 3 forms a complete completely distributive lattice, a Heyting algebra, a 3-valued Lukasiewicz algebra, a semi-simple Nelson algebra, and a regular double Stone algebra, for instance. The operations of these algebras are uniquely defined in 3 and they can be extended pointwise to 3 U , the set of all three-valued functions on U . The set 3 U is canonically ordered pointwise and the set A(U ) of the approximation pairs of U is ordered naturally by the coordinatewise ⊆-relations. The sets 3 U and A(U ) form isomorphic ordered structures (see Section 2). In addition, this means that A(U ) has the above-mentioned algebraic structures 'lifted' from 3 (via 3 U ).
We denote by RS the set of all rough sets {(X , X ) | X ⊆ U } defined by some binary relation R on U . If R is an equivalence, RS is a Stone algebra [5]. In [6] this result was improved by showing that RS forms a regular double Stone algebra. The three-valued Lukasiewicz algebras defined by RS were considered in [7,8,9,10]. P. Pagliani [10] showed how a semisimple Nelson algebra can be defined on RS. In addition, as Stone algebras, three-valued Lukasiewicz algebras, or Nelson algebras, they are subalgebras of A(U ). If R is a quasiorder, RS is a complete polarity sublattice of A(U ) as noted in [11]. Because rough sets are approximations pairs, there is a complete polarity sublattice F of 3 U such that its approximation pairs A(F ) equal RS.
In this work, we obtain sufficient and necessary conditions (C1)-(C3) under which A(F ) = RS holds, where F is a complete polarity sublattice of 3 U and RS is induced by a quasiorder. In the special case of an equivalence R, we have A(F ) = RS exactly when F satisfies (C1)-(C3) and forms a Lukasiewicz subalgebra of 3 U . The latter result uses our earlier result stating that RS forms a 3-valued Lukasiewicz algebra just in case R is an equivalence relation [12]. We also prove that such an F forms a 3-valued Lukasiewicz algebra when it is closed with respect to any of the operations * , + , ▽, △, →, ⇒ defined in 3 U (see Proposition 3.12).
This paper is structured as follows. In the next section, we consider the set 3 U of all 3-valued functions on U and the approximation pairs A(U ) defined by them. We point out that 3 U and A(U ) form isomorphic complete lattices. Also the basic definitions and facts related to rough sets are recalled in this section. In Section 3, we note how 3 U forms a Heyting algebra, a three-valued Lukasiewicz algebra, a semisimple Nelson algebra, and a regular double Stone algebra. The operations on all these algebras are defined pointwise from the operations of 3. Because A(U ) is isomorphic to 3 U , all the mentioned algebras can be defined on A(U ), too. We describe these operations on A(U ) in detail. We end this section by noting that if a complete polarity sublattice F of 3 U is closed with respect to at least one of the operations * , + , ▽, △, →, ⇒ defined in 3 U , then F is closed with respect to all these operations.
It is well-known that there is a one-to-one correspondence between quasiorders and Alexandrov topologies. In Section 4, we consider Alexandrov topologies defined by complete sublattices of 3 U . For a quasiorder ≤, a necessary condition for A(F ) = RS to hold is that the collections C(F ) and S(F ) of the cores and the supports of the maps in F , respectively, form dual Alexandrov topologies. Moreover C(F ) must equal ℘(U ) , the set of lower approximations of subsets of U , and S(F ) needs to coincide with ℘(U ) , the set of upper approximations.
Together with Pagliani the authors of the current work presented in [13] a representation of quasiorderbased rough sets, stating that where S is the set of such elements that they are ≤-related only to itself. This representation appears simple compared to the representation presented here. The fact is that there is already a lot of structural information in each (A, B)-pair of (1), because each such pair is defined by a single quasiorder ≤. But if we just pick an arbitrary collection F of three-valued functions (or approximation pairs), nothing is connecting these functions together. Probably for this reason the conditions under which a complete polarity sublattice F of 3 U is such that A(F ) = RS need to be more complicated than (1). Studying these characteristic properties possibly reveals something new and essential about the nature of rough sets determined by a quasiorder or by an equivalence. Some concluding remarks end the article.

Three-valued functions and approximations
We consider three-valued functions f : U → 3 defined on a universe U , where 3 stands for the threeelemented chain 0 < u < 1. The set of such functions 3 U may be ordered pointwise by using the order of 3: f With respect to pointwise order, 3 U forms a complete lattice such that for any H ⊆ 3 U . The map ⊥: x → 0 is the least element and ⊤: x → 1 is the greatest element of 3 U . It is well-known that 3 is equipped with several operations such as Heyting implication ⇒, polarity ∼, pseudocomplement * , dual pseudocomplement + , possibility ▽ and necessity △ of three-valued Lukasiewicz algebras and Nelson implication →. Any n-ary, n ≥ 0, operation φ on 3 can be 'lifted' pointwise to an operation Φ on the set 3 U by defining for the maps f 1 , . . . , f n ∈ 3 U a function Φ(f 1 , . . . , f n ) in 3 U by setting The operation Φ then satisfies the same identities in 3 U as φ satisfies in 3.
Rough sets are pairs consisting of a lower and an upper approximation of a set. In this work, a generalization of such pairs are in an essential role. Let A, B ⊆ U . We say that (A, B) is an approximation pair if A ⊆ B. We denote by A(U ) the set of all approximation pairs on the set U . The set A(U ) can be ordered Note that A(U ) can be viewed as an instance of where B is a Boolean lattice. It is well known that B [2] is a regular double Stone lattice [14]. Every f ∈ 3 U is completely determined by two sets is an order-isomorphism.
Proof. We first show that ϕ is an order-embedding, that is, . Therefore, also S(f ) ⊆ S(g) and we have proved (C(f ), S(f )) ≤ (C(g), S(g)).
We need to show that ϕ is a surjection. Suppose (A, B) ∈ A(U ). Let us define a function f (A,B) by Now We have now proved that ϕ is an order-isomorphism.
A complete lattice L is completely distributive if for any doubly indexed subset {x i, j } i∈I, j∈J of L, we have i∈I j∈J that is, any meet of joins may be converted into the join of all possible elements obtained by taking the meet over i ∈ I of elements x i, k , where k depends on i.
The power set lattice ℘(U ) is a well-known completely distributive lattice [15]. In ℘(U ) × ℘(U ), the joins and meets are formed coordinatewise, so ℘(U ) × ℘(U ) is a completely distributive lattice. Also a complete sublattice of a completely distributive lattice is clearly completely distributive. Thus, A(U ) and 3 U are completely distributive.
Proof. By Proposition 2.1, the map ϕ: f → (C(f ), S(f )) is an order-isomorphism. Hence, it preserves all meets and joins, and Combining all these, we can write which proves (i), and (ii) is treated analogously.
The set of approximation pairs corresponding to a family F ⊆ 3 U is defined as Obviously, for any F ⊆ 3 U , the ordered sets F and A(F ) are order-isomorphic, whenever F is ordered pointwise and A(F ) coordinatewise. Rough sets were introduced by Z. Pawlak [1]. According to Pawlak's original definition, our knowledge about objects U is given by an equivalence relation. Equivalences are reflexive, symmetric and transitive binary relations. An equivalence E on U is interpreted so that x E y if the elements x and y cannot be distinguished by their known properties. In the literature, numerous studies can be found on rough sets in which equivalences are replaced by different types of so-called information relations reflecting, for instance, similarity or preference between the elements of U (see e.g. [16,17]).
Let U be a set and let R be a binary relation on U . For any x ∈ U , we denote R(x) = {y | x R y}. For all X ⊆ U , the lower and upper approximations of X are defined by respectively. The set X may be interpreted as the set of elements that are certainly in X, because all elements to which x is R-related are in X. Analogously, X can be considered as the set of all elements that are possibly in X, since in X there is at least one element to which x is R-related. Quasiorders are reflexive and transitive binary relations. For instance, a quasiorder R can be interpreted as a specialization order, where x R y may be read as 'y is a specialization of x'. In [18], a specialization order is viewed as 'non-symmetric indiscernibility' such that each element is indiscernible with all its specializations, but not necessarily the other way round. Then, in our interpretation, x ∈ X means that there is at least one specialization y in X, which cannot be discerned from x. Similarly, x belongs to X if all its specializations are in X; this is then interpreted so that x needs to be in X in the view of the knowledge R.
For all X ⊆ U , the pair (X , X ) is called the rough set of X. The set of all rough sets is denoted by Like any set of approximations, RS is ordered coordinatewise: In this work, we consider relations R which are at least reflexive. Then X ⊆ X ⊆ X , and therefore each rough set (X , X ) can be considered as an approximation pair in the above sense. For reflexive relations, RS is not necessarily a lattice. In fact, it is known that there are tolerances, that is, reflexive and symmetric binary relations, such that RS is not a lattice; see [3]. We restrict ourselves to the case in which R is a quasiorder or an equivalence. This has the advantage that the rough sets algebras RS are complete polarity sublattices of A(U ). This also means that there exists a complete polarity sublattice F of 3 U such that its approximation pairs A(F ) are equal to RS.
In this work, we consider the question, what properties of complete polarity sublattice F of 3 U must additionally satisfy so that A(F ) = RS holds, where RS is the collection of rough sets induced by a quasiorder or by an equivalence on U .

Algebras defined on 3 U and A(U )
For an ordered set (P, ≤), a mapping ∼: P → P satisfying ∼∼x = x and x ≤ y implies ∼x ≥ ∼y is called a polarity. Such a polarity ∼ is an order-isomorphism from (P, ≤) to its dual (P, ≥). This means that P is self-dual to itself. Let us define an operation ∼ on ℘(U ) × ℘(U ) by where for any X ⊆ U , X c denotes the complement U \ X of X. We call the pair ∼(A, B) as the opposite of (A, B). Obviously, ∼ is a polarity. Let L be a (complete) lattice with polarity. If S is a (complete) sublattice of L closed with respect to ∼, we say that S is a (complete) polarity sublattice of L.
For any binary relation R on U , the approximation operations and are dual, that is, for X ⊆ U , This implies that for (X , X ) ∈ RS, Therefore, ∼ is a well-defined polarity also in RS.
Remark 3.1. Our study has some resemblance to the study of so-called 'orthopairs' by G. Cattaneo and D. Ciucci [19]. A De Morgan algebra (L, ∨, ∧, ∼, 0, 1) is such that (L, ∨, ∧, 0, 1) is a bounded distributive lattice and ∼ is a polarity. The operation ∼ can be defined also by the identities: which proves the first claim. Since ∼∼f = f , we obtain It is easy to see that ϕ(⊥) = (∅, ∅) and ϕ(⊤) = (U, U ). By Proposition 2.1, it is enough to show that Following A. Monteiro [20], we can define a three-valued Lukasiewicz algebra as an algebra (L, ∨, ∧, ∼, ▽, 0, 1) such that (L, ∨, ∧, ∼, 0, 1) is a De Morgan algebra and ▽ is an unary operation, called the possibility operator, that satisfies the identities: Let us recall from [20] that the following facts hold for all x ∈ L, In addition x ≤ y implies ▽x ≤ ▽y. The necessity operator is defined by The operation △ can be seen as a dual operator of ▽, so △ satisfies the dual assertions of the above. Also △ and ▽ have some mutual connections, for instance: Lukasiewicz algebras satisfy the following determination principle by Gr. C. Moisil (see e.g. [21]): △x = △y and ▽x = ▽y imply x = y.
It is known [9] that if RS is defined by an equivalence relation on U , then it forms a 3-valued Lukasiewicz algebra such that △(X , X ) = (X , X ) and ▽(X , X ) = (X , X ).
Example 3.4. On the chain 3 the operations △ and ▽ are defined as in the following table: x △x ▽x For a map f ∈ 3 U , the functions △f and ▽f are defined pointwise, that is, Also A(U ) forms a three-valued Lukasiewicz algebra in which Proof. Let x ∈ U . Then, Suppose L is a lattice and a, b ∈ L. If there is a greatest element z ∈ L such that a ∧ z ≤ b, then this element z is called the relative pseudocomplement of a with respect to b and is denoted by a ⇒ b. If a ⇒ b exists, then it is unique. A Heyting algebra L is a lattice with 0 in which a ⇒ b exists for each a, b ∈ L. Heyting algebras are distributive lattices and any completely distributive lattice L is a Heyting algebra in which Equationally Heyting algebras can be defined as lattices with 0 and ⇒ satisfying the identities [22]: It is known [21,23] that every three-valued Lukasiewicz algebra forms a Heyting algebra where Example 3.6. The chain 3 is a Heyting algebra in which Also 3 U is a Heyting algebra in which ⇒ is defined pointwise: Since 3 U and A(U ) are isomorphic completely distributive lattices, A(U ) is a Heyting algebra isomorphic to 3 U . Let x = (A, B) and y = (C, D) be elements of A(U ). We may use (3) to infer x ⇒ y. Now It is proved by Monteiro in [20] that every three-valued Lukasiewicz algebra forms a Kleene algebra. Note that x ∧ ∼x ≤ u ≤ y ∨ ∼y for x, y ∈ 3. Obviously, 3 U and A(U ) are isomorphic Kleene algebras via ϕ. Note that already in [24] it is proved that 3 U and ℘(U ) [2] = A(U ) are isomorphic Kleene algebras.
According to R. Cignoli [25] a quasi-Nelson algebra is defined as Kleene algebra (A, ∨, ∧, ∼, 0, 1) where for each pair a, b ∈ A the relative pseudocomplement exists. This means that every Kleene algebra whose underlying lattice is a Heyting algebra forms a quasi-Nelson algebra. In a quasi-Nelson algebra, the element (4) is denoted simply by a → b.
As shown by D. Brignole and A. Monteiro [26], the operation → satisfies the identities: A Nelson algebra is a quasi-Nelson algebra satisfying the identity It is shown in [26] that a Nelson algebra can be defined also as an algebra (L, ∨, ∧, →, ∼, 0, 1), where (L, ∨, ∧, ∼, 0, 1) is a Kleene algebra, and the binary operation → satisfies (N1)-(N5). A Nelson algebra is semisimple if It is known that every three-valued Lukasiewicz algebra defines a semisimple Nelson algebra by setting Similarly, each semisimple Nelson algebra defines a three-valued Lukasiewicz algebra by setting In fact, the notions of three-valued Lukasiewicz algebra and semisimple Nelson algebra coincide [23].
Example 3.7. The Kleene algebra defined on 3 forms also a Nelson algebra in which the operation → is defined as in the following table [27]: . Note also that we can write It can be seen in the above table that the Nelson algebra 3 is semisimple. Therefore, also 3 U forms a semisimple Nelson algebra. Because 3 U and A(U ) are isomorphic as Heyting algebra (recall that if the operation ⇒ exists, it is unique) and as Kleene algebras, by (5) we have that they are isomorphic also as semisimple Nelson algebras.
There are a couple of possibilities how we can derive the outcome of the operation (A, B) → (C, D) in A(U ). We can either use (4) or a → b = ▽∼a ∨ b. It appears that the latter is simpler to apply here. For An algebra (L, ∨, ∧, * , 0) is a p-algebra if (L, ∨, ∧, 0) is a bounded lattice and * is a unary operation on L such that x ∧ z = 0 iff z ≤ x * . The element x * is the pseudocomplement of x. It is well known that x ≤ y implies x * ≥ y * . We also have for x, y ∈ L, Equationally p-algebras can be defined as lattices with 0 such that the following identities hold [28]: Note that (P2) means that 0 * is the greatest element and we may denote it by 1. Therefore, it is possible to include 1 also to the signature of a p-algebra.
An algebra (L, ∨, ∧, * , + , 0, 1) is a double p-algebra if (L, ∨, ∧, * , 0) is a p-algebra and (L, ∨, ∧, + , 1) is a dual p-algebra (i.e. z ≥ x + iff x ∨ z = 1 for all x, y ∈ L). The element x + is the dual pseudocomplement of a. If x ≤ y, then x + ≥ y + . In addition, Note that by definition x ≤ x * * and x ++ ≤ x. Therefore, in a double p-algebra x ++ ≤ x * * . Example 3.8. On 3 the operations * and + are defined as in the following table: For a map f ∈ 3 U the functions f * and f + are defined pointwise, that is, Then, A pseudocomplemented De Morgan algebra is an algebra (L, ∨, ∧, ∼, * , 0, 1) such that (L, ∨, ∧, ∼, 0, 1) is a De Morgan algebra and (L, ∨, ∧, * , 0, 1) is a p-algebra. Such an algebra always forms a double p-algebra, where the pseudocomplement operations determine each other by We say that a double p-algebra is regular if it satisfies the condition x * = y * and x + = y + imply x = y.
T. Katriňák [29] has shown that any regular double pseudocomplemented lattice forms a Heyting algebra such that A p-algebra (L, ∨, ∧, * , 0, 1) is a Stone algebra if L is distributive and for all x ∈ L, x * ∨ x * * = 1.
Example 3.10. As a distributive double p-algebra, 3 forms a double Stone algebra, because x * or x * * equals 1 for any x ∈ 3, and x + or x ++ is 0. From the table of Example 3.8 we can see that (7) holds in 3, meaning that 3 is a regular double Stone algebra. This also implies that 3 U forms a regular double Stone algebra. It is known that every regular double Stone algebra (L, ∨, ∧, * , + , 0, 1) defines a three-valued Lukasiewicz algebra (L, ∨, ∧, ∼, ▽, 0, 1) by setting ▽a = a * * and ∼a = a * ∨ (a ∧ a + ).
Similarly, each three-valued Lukasiewicz algebra defines a double Stone algebra by a * = ∼▽a and a + = ▽∼a.
These pseudocomplement operations determine each other by (6). The correspondence between regular double Stone algebras and three-valued Lukasiewicz algebras is one-to-one; see [30] for details and further references. Note that this means that also regular double Stone algebras and semi-simple Nelson algebras coincide.
Example 3.11. On 3 U the operations * , + , → and ▽ can be defined as follows in terms of the core and support of the functions, cf. Lemmas 3.5 and 3.9. For f, g ∈ 3 U , The following proposition shows how in the presence of ∼, all operations * , + , ▽, △, →, ⇒ are defined in terms of one of them.
Proposition 3.12. Let F be a complete polarity sublattice of 3 U . If F is closed with respect to at least one of the operations * , + , ▽, △, →, ⇒ defined in 3 U , then F is closed with respect to all these operations.
Proof. Let us first note that since F is a sublattice of 3 U , it is distributive. In addition, the least element ⊥ and the greatest element ⊤ of 3 U are in F , because F is a complete sublattice of 3 U .
We have noticed that * and + fully determine each other in the presence of ∼ and together they define ⇒ by (8). When F is closed under * or + of 3 U , the regularity condition (7) holds and Stone identities (9) and (10) are valid. Hence, F forms a regular double Stone algebra. We have seen that each regular double Stone algebra defines a semisimple Nelson algebra and a three-valued Lukasiewicz algebra. Therefore, if F is closed with respect to * or + , it is closed with respect to all of the mentioned operations.
Similarly, ▽ and △ define each other in terms of ∼. As we know, three-valued Lukasiewicz algebras uniquely determine semisimple Nelson algebras and regular double Stone algebras. Therefore, if F is closed with respect to ▽ or △ of 3 U , it is closed with respect to * , + , →. Additionally, ∼, ▽ and △ determine ⇒ by (3).
If F is closed with respect to → of 3 U , then it forms a semisimple Nelson algebra, which in turn defines uniquely a regular double Stone algebra and a three-valued Lukasiewicz algebra. Thus, F is closed with respect to all of the mentioned operations.
Finally, let F be closed with respect to ⇒. Because ⊥ ∈ F , f * is defined by f ⇒ ⊥ for each f ∈ F . From this we get that F is closed with respect to * , + , ▽, △, →.
We end this section by noting that the map ϕ defined in Proposition 2.1 preserves all operations considered in this section. Indeed, let f ∈ 3 U . We have already noted that ϕ(∼f ) = ∼ϕ(f ). Now As we have seen, the operations + , ▽, △, →, ⇒ can be defined in terms of ∨, ∧, ∼, and * , so they are preserved with respect to ϕ.

Alexandrov topologies defined by complete sublattices of 3 U
An Alexandrov topology [31,32] T on U is a topology in which also intersections of open sets are open, or equivalently, every point x ∈ U has the least neighbourhood N (x) ∈ T . For an Alexandrov topology T , the least neighbourhood of x is N (x) = {B ∈ T | x ∈ B}. Each Alexandrov topology T on U defines a quasiorder ≤ T on U by x ≤ T y if and only if y ∈ N (x) for all x, y ∈ U . On the other hand, for a quasiorder ≤ on U , the set of all ≤-closed subsets of U forms an Alexandrov topology T ≤ , that is, B ∈ T ≤ if and only if x ∈ B and x ≤ y imply y ∈ B. Let [x) = {y ∈ X | x ≤ y}. In T ≤ , N (x) = [x) for any x ∈ U . The correspondences T → ≤ T and ≤ → T ≤ are mutually invertible bijections between the classes of all Alexandrov topologies and of all quasiorders on the set U .
Let ≤ be a quasiorder on U . We denote its inverse by ≥. Obviously, also ≥ is a quasiorder and we denote its Alexandrov topology by T ≥ . We say that topologies T 1 and T 2 are dual if The topologies T ≤ and T ≥ are mutually dual. The smallest neighbourhood of a point For the sake of completeness, we prove the following claim.
Lemma 4.1. Let T 1 and T 2 be dual topologies and let ≤ 1 and ≤ 2 be the corresponding quasiorders, respectively. Then This means that there is X ∈ T 2 such that x ∈ X and y / ∈ X. Because T 2 is the dual topology of T 1 , then there is Y ∈ T 1 such that X = Y c . This means that y ∈ Y and x / ∈ Y . Therefore, x / ∈ {Y ∈ T 1 | y ∈ Y }, a contradiction. Thus, x ≥ 2 y holds, and x ≤ 1 y implies x ≥ 2 y. Symmetrically we can show that x ≥ 2 y implies x ≤ 1 y, which completes the proof.
Let us now recall from [11] how Alexandrov topologies relate to rough set approximations. Let ≤ be a quasiorder on U . Then for any X ⊆ U , and In particular, (x] = {x} for all x ∈ U .

Remark 4.2.
There are also other choices for lower-upper approximation pairs defined in terms of a quasiorder ≤ on U . In [33,34] Kumar and Banerjee define the operators L and U by for any X ⊆ U . The sets L(X) and U(X) belong to the same topology T ≤ , whose elements are called 'definable'. These operators can be also be written in form This approach differs significantly from ours, because now the rough set system For a quasiorder ≤ on U , a pair of rough approximations is defined in [18,Theorem 4] for any X ⊆ U by This means that for all X ⊆ U . Obviously, X and X belong to the same Alexandrov topology T ≥ . This also means that {(X, X) | X ⊆ U } does not necessarily form a lattice.
Hence, C(F ) and S(F ) are dual topologies.
(c) Since F is a three-valued Lukasiewicz subalgebra of 3 U , it is also closed with respect to * and + . By Lemma 3.9, S(f ) c = S(f * ) and C(f ) c = C(f + ) for any f ∈ F . This implies that S(F ) and C(F ) are closed with respect to set-theoretical complement. Because they are Alexandrov topologies, they form Boolean lattices. In addition, which implies that C(F ) = S(F ).
Let F be a complete polarity sublattice of 3 U . Then, by Lemma 4.3, C(F ) and S(F ) are dual Alexandrov topologies. Let us define a binary relation ≤ F on U by Let us also introduce the following notation where ≥ F is the inverse relation of ≤ F . Proof. (a) Suppose that x ≤ F y. By definition this is equivalent to that x ∈ C(f ) implies y ∈ C(f ) for all f ∈ F . From this we obtain y ∈ {C(f ) | f ∈ F and x ∈ C(f )}. This means that y belongs to the smallest neighbourhood of x in the Alexandrov topology C(F ). On the other hand, if x F y, then there exists g ∈ F such that g(x) = 1, but g(y) = 1. This then means that x ∈ C(g) and y / ∈ C(g). From this we obtain y / ∈ {C(f ) | f ∈ F and x ∈ C(f )}. We deduce that ≤ F is the quasiorder corresponding to the Alexandrov topology C(F ). Obviously, [x) F is the smallest neighbourhood of the point x in C(F ). Claim (b) can be proved similarly.
(c) Assume x ≤ F y. Since ≥ F is the quasiorder of the Alexandrov topology S(F ), y ≥ F x means that x ∈ {S(f ) | f ∈ F and f (y) ≥ u}. Suppose that f (x) = 0. We must have f (y) u which is equivalent f (y) = 0. On the other hand, if x F y, that is, y F x, then there is g ∈ F such that g(y) ≥ u and g(x) u. The latter means g(x) = 0. Therefore, g(x) = 0 does not imply g(y) = 0.
Remark 4.5. Note that if F is a complete sublattice and a three-valued Lukasiewicz subalgebra of 3 U , then the relation ≤ F is an equivalence. Indeed, suppose that x ≤ F y. Then y belongs to the smallest neighbourhood of x in the Alexandrov topology C(F ). Now C(F ) = S(F ) by Lemma 4.3(c). This means that y belongs to the smallest neighbourhood of x in the Alexandrov topology S(F ), and therefore x ≥ F y. Thus, ≤ F is symmetric.
It is also easy to see that if ≤ F is an equivalence and x ≤ F y, then f (x) = f (y) for all f ∈ F . Indeed, if f (x) = 0, then f (y) = 0 by Lemma 4.4. Similarly, f (x) = 1 implies f (y) = 1. If f (x) = u, then f (y) must be u, because f (y) = 0 or f (y) = 1 and y ≤ F x would imply f (x) = 0 or f (x) = 1, a contradiction.
The following lemma is now clear by (11).  We have already noted that A(U ) is isomorphic to 3 U as a three-valued Lukasiewicz algebra, as a regular double Stone algebra and as a semi-simple Nelson algebra, because ϕ preserves all these operations.
Let us consider the three-element set U = {a, b, c}. The set 3 {a,b,c} can be viewed as a set of 3-valued characteristic vectors of length 3, that is, Obviously, there are 27 such vectors. Let us agree that the first position corresponds to a, the second corresponds to b, and the third corresponds to c. Let us denote the set of these functions by F . Next we construct the Alexandrov topologies C(F ) and S(F ), and the relation ≤ F . We will show that F forms a three-valued Lukasiewicz subalgebra of 3 U , and therefore ≤ F is an equivalence and C(F ) = S(F ) is a Boolean algebra. It is easy to see that F is closed with respect to ∼ of 3 U : and this topology also is equal to C(F ). The topology S(F ) induces an equivalence ≤ F whose equivalence classes are {a} and {b, c}. Obviously, the rough set system defined by ≤ F coincides with RS above.
The above example shows how for each equivalence E on U , we obtain a three-valued Lukasiewicz subalgebra F of 3 U such that in terms of F we can construct the same equivalence E that we started with. On the other hand, we know from the literature [35] that for each complete atomic regular double Stone algebra A there exists a set U and an equivalence E on U such that the rough set system determined by E is isomorphic to A. As we have noted, regular double Stone algebras correspond to three-valued Lukasiewicz algebras. Note that an ordered set with a least element 0 is atomic if every nonzero element has an atom a below it.
Let us consider the two-element set U = {a, b}. Because 3 U is finite, it is atomic. If F is a threevalued Lukasiewicz subalgebra of 3 U , then F is isomorphic to the rough set algebra determined by an equivalence E on some set, not necessarily U . For instance, we can see that 3 {a,b} has 6 different threevalued Lukasiewicz subalgebras: 2, 3, 2 × 2, 2 × 3, 3 × 2, and 3 × 3, but in U it is possible to define only 2 equivalences: the universal relation and the diagonal relation. Therefore, not all complete threevalued Lukasiewicz subalgebras F of 3 {a,b} are such that A(F ) is equal to a rough set system defined by an equivalence on U .
We can ask the following question: Question 4.8. Which three-valued Lukasiewicz subalgebras F of 3 U are such that there is an equivalence E on U whose rough set system is equal to A(F )?
In [12] we proved that if A is a Nelson algebra defined on an algebraic lattice, then there exists a set U and a quasiorder ≤ on U such the rough set Nelson algebra defined by ≤ is isomorphic to A. Recall that an algebraic lattice is a complete lattice such that every element is a join of compact elements. A similar question can be also addressed for Nelson algebras: Question 4.9. Which Nelson subalgebras F of 3 U are such that there is a quasiorder ≤ on U whose rough set system equals A(F )?

Rough sets defined in terms of three-valued functions
Next our aim is to answer Questions 4.8 and 4.9. Let F ⊆ 3 U and x ∈ U . We define two functions In addition, we define an equivalence Θ on F as the kernel of C, that is, Lemma 5.1. Let F be a complete polarity sublattice of 3 U . For all x, y ∈ U , (b) Using Lemmas 2.2 and 4.4, and On the other hand, by Proposition 2.1, This yields completing the proof.
The following lemma describes the rough approximations in terms of cores and supports of maps.
Lemma 5.2. Let F be a complete polarity sublattice of 3 U . If we define the operators and in terms of ≤ F , then for any X ⊆ U , Proof. (a) The set X ∈ S(F ) is the smallest set in S(F ) containing X. On the other hand, is the smallest set in S(F ) containing X. We have that Let F be a complete polarity sublattice of 3 U and x ∈ U . We say that an element x ∈ U is an F -singleton if [x) F = {x}. The following lemma gives a characterisation of F -singletons.
Thus, C(f x ) = {x} and hence x is an F -singleton.
An element x of a complete lattice L is completely join-irreducible if x = S implies x ∈ S. Let us denote by J (L) the set of completely join-irreducible elements of L. A lattice L is spatial if each of its elements is a join of completely join-irreducible elements.
Proposition 5.4. Let F be a complete polarity sublattice of 3 U and x ∈ U . Then F is spatial and Proof. The powerset ℘(U ) forms an algebraic lattice in which finite subsets of U are the compact elements. A product of algebraic lattices is algebraic (see [36,), which implies that ℘(U ) × ℘(U ) is algebraic. A complete sublattice of an algebraic lattice is algebraic [15,Exercise 7.7]. Because A(F ) is a complete sublattice of ℘(U ) × ℘(U ), A(F ) is algebraic. We have already noted in Section 2 that A(U ) is completely distributive. It is known (see e.g. [12] and the references therein) that every algebraic and completely distributive lattice is spatial. Thus, A(F ) is spatial and because F is isomorphic to A(F ), also F is spatial.
Next we need to find the set of completely join-irreducible elements of F . First we show that each 1 and 3 is a chain, we have that g(x) = 1 for some g ∈ G. We obtain g ∈ {f ∈ F | f (x) = 1} and f On the other hand f x = G gives that f x ≥ g. Hence, f x = g ∈ G and f x is completely join-irreducible. In an analogous way, we may show that also f x is completely irreducible.
It is clear that any f ∈ F is an upper bound of Let g be an upper bound of H. We prove that f ≤ g. For this, we assume that f g. This means that there is an element a ∈ U such that f (a) g(a). Because 3 is a chain, we have that f (a) > g(a). We have now three possibilities.
(i) If f (a) = 1 and g(a) = u, then f a ≤ f , but f a (a) = 1 and g(a) = u. Then g is not an upper bound of H, a contradiction. Case (ii), when f (a) = 1 and g(a) = 0 is similar.
(iii) If f (a) = u and g(a) = 0, then f a ≤ f , f a (a) ≥ u, and g(a) = 0. Thus, g is not an upper bound of H, a contradiction. Since each case (i)-(iii) leads to a contradiction, we have that f ≤ g and f is the least upper bound of H. We have that From this it directly follows also that Let us now introduce the following three conditions for a complete polarity sublattice F of 3 U .
Let F be a complete polarity sublattice 3 U . If A(F ) = RS for some quasiorder ≤ on U , then for each rough set (X , X ), there is a map f ∈ F such that (C(f ), S(f )) equals (X , X ). Moreover, for any f ∈ F , the pair (C(f ), S(f )) is in RS. We also have that the Alexandrov topologies coincide, meaning that C(F ) = ℘(U ) and S(F ) = ℘(U ) . Because there is a one-to-one correspondence between Alexandrov topologies, and ≤ is the quasiorder corresponding to the Alexandrov topology ℘(U ) and ≤ F is the quasiorder of C(F ), we have that ≤ and ≤ F are equal. This means that rough set pairs operations can be defined in two ways: either in terms of the rough approximations defined by the quasiorder ≤ F or in terms of the approximation pairs of the maps in F . Proof. (C1) Let x be an F -singleton and f ∈ F . There is X ⊆ U such that C(f ) = X and S(f ) = X . Because [x) = {x}, x ∈ S(f ) = X means that {x} ∩ X = ∅ and x ∈ X. We have [x) = {x} ⊆ X, that is, There is a map g ∈ F such that (C(g), S(g)) = (Z , Z ). Since x ∈ {x} = Z = S(g), we get g(x) ≥ u. Therefore, By the isomorphism given in Proposition 2.1, (C(f x ), S(f x )) ≤ (C(g), S(g)). This means that (C(f x ), S(f x )) is the least rough set such that the second component is {x} .
We have that there are subsets X, Y ⊆ U such that C(f ) = X and S(f ) = Y . Let us denote Suppose that (Z , Z ) ∈ RS is a rough set such that Z = X . Thus, there is f ′ ∈ F that satis- We have shown that (C(f Θ ), S(f Θ )) is the smallest rough set such that its first component equals X .
Let (A , A ) be a rough set such that A = X . Then X ⊆ A gives X ⊆ A . Note that ((X ) , (X ) )) = (X , X ) is a rough set. Therefore, (X , X ) is the smallest rough set such that its first component is X . Hence, (X , X ) = (C(f Θ ), S(f Θ )). Since X ⊆ Y , we obtain which completes the proof.
Let F be a complete polarity sublattice of 3 U . In the following theorem, we denote the rough approximations defined by the quasiorder ≤ F by X and X for any X ⊆ U . Similarly, RS denotes the corresponding rough set system.
is the set of completely join-irreducible elements and each element of RS is a join of some (or none) elements of J (RS). Let us next consider a rough set of the form Using Lemma 5.2, we obtain We have proved in Lemma 5.1 that f x = {h ∈ F | hΘf x }. This gives S(f x ) = S( {h ∈ F | hΘf x }) and we have Combining (12) and (13) Let (X , X ) ∈ RS. As we already noted, each element of RS is a join of elements of J (RS), that is, (X , X ) = i∈I (J i , J i ) for some {(J i , J i ) | i ∈ I} ⊆ J (RS). By the above, every (J i , J i ) is of the form (C(ϕ i ), S(ϕ i )), where each ϕ i belongs to F . We have completing the proof.
We can now write the following theorem answering to Question 4.9. Thus, Using the inverse ϕ −1 of the isomorphism ϕ of Proposition 2.1, we have Similarly, we can show that G belongs to F . For f ∈ F , A(∼f ) = (C(∼f ), S(∼f )) = (S(f ) c , C(f ) c ) = ∼(C(f ), S(f )) ∈ A(F ).
Obviously, F is and its Hasse diagram is given in Figure 1(a). Figure 1(b) contains the Hasse diagram of the corresponding approximations A(F ). Note that elements of sets are denoted simply by sequences of their elements. For instance, {a, b} is denoted ab. Now C(F ) = {∅, {a, b}, {c}, U } and the corresponding quasiorder ≤ F is the equivalence whose equivalence classes are {a, b} and {c}. It is obvious that A(F ) cannot be equal with rough set system RS induced by ≤ F , because RS is isomorphic to the product 2 × 3 and A(F ) is not. Let us now verify that conditions (C1)-(C3) do not hold.
The element c is an F -singleton. Now c ∈ S(f 3 ), but c / ∈ C(f 3 ). Therefore, (C1) does not hold. Let us first compute the map f x = {f | F | f (x) = 1} for each x ∈ U : f a = f b = f 5 ∧ f 7 = f 5 and f c = f 6 ∧ f 7 = f 6 . Now, for example, C(f b ) = C(f 5 ) = {a, b} {b}, meaning that (C2) is not true.

Some concluding remarks
In this work we have answered the question what conditions a collection F of 3-valued functions on U must fulfill so that there exists a quasiorder ≤ on U such that the set RS of rough sets defined by ≤ coincides with the set A(F ) of approximation pairs defined by F . Furthermore, we give a new representation of rough sets determined by equivalences in terms of three-valued Lukasiewicz algebras of three-valued functions.
It is known that for tolerances determined by irredundant coverings on U , the induced rough set structure RS is a regular pseudocomplemented Kleene algebra, but now RS is not a complete sublattice of the product ℘(U )×℘(U ); see [37,38,39]. This means that if F is a collection of three-valued maps such that A(F ) = RS, then obviously F is not a complete sublattice of 3 U . A natural question then is what properties F needs to have to define a rough set system determined by a tolerance induced by an irredundant covering.
Finally, in this work we have considered approximation pairs defined by three-valued functions. But one could change 3 to some other structure. For instance, 3 could be replaced by 4-element lattice introduced in [40], where L4 denotes the lattice F < Both, None < T, where F means 'false', Both means 'both true and false', None means 'neither true nor false', and T means 'true'. In such a setting we could consider, for instance, the approximation pairs formed of level set of functions f : U → L4, that is, f α = {x ∈ U | f (x) ≥ α}, where α belongs to L4.

Declaration of competing interest and authorship conformation
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Both authors have participated in drafting the article and approve the final version.