Lattice Representations with Set Partitions Induced by Pairings

We call a quadruple W := 〈F,U,Ω,Λ〉, where U and Ω are two given non-empty finite sets, Λ is a non-empty set and F is a map having domain U×Ω and codomain Λ, a pairing on Ω. With this structure we associate a set operator MW by means of which it is possible to define a preorder >W on the power set P(Ω) preserving settheoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice L there exist a finite set ΩL and a pairing W on ΩL such that the quotient of the preordered set (P(ΩL),>W) with respect to its symmetrization is a lattice that is order-isomorphic to L. In the second result, we prove that when the lattice L is endowed with an order-reversing involutory map ψ : L → L such that ψ(0̂L) = 1̂L, ψ(1̂L) = 0̂L, ψ(α) ∧ α = 0̂L and ψ(α) ∨ α = 1̂L, there exist a finite set ΩL,ψ and a pairing on it inducing a specific poset which is order-isomorphic to L. Mathematics Subject Classifications: 06A07, 68R05


Introduction
Granular Computing (briefly GrC) is an emerging paradigm which relies on the idea of partitioning a set of objects in some granules depending on some given criteria [29,30,38,39]. Many ideas and methods of GrC have been used in order to investigate discrete mathematical objects, such as matroids, set partitions and ordered structures [21,25,26,36,37].
In particular, a natural partitioning of a finite object set U is given when one uses a corresponding finite attribute set Ω, with respect to which any ordered pair (u, a) ∈ U ×Ω takes a unique value F (u, a) in a given set Λ. The quadruple F, U, Ω, Λ has several names in computer science literature: information system [28], relational data table [34], information table [38], Chu space [20].
In this work we use some algebraic granular computing techniques on quadruples having the aforementioned form, in order to provide two representation results concerning finite lattices. However, due to the fact that in the present paper the above sets U , Ω and Λ have only a formal nature, we will use the more mathematical term pairing instead of the previous ones. Hence, in our specific mathematical viewpoint, we can speak of algebraic GrC methods on pairings (for further results on such topics see also [13,14,17]). Therefore, we shall call a quadruple W := F, U, Ω, Λ , where U and Ω are both finite non-empty sets, Λ is also a non-empty set and F is a map having domain U × Ω and codomain Λ, a pairing on Ω. We may identify the pairing W with the rectangular table with rows labelled by the elements of U , the columns by those of Ω and whose entries are the values F (u, a).
For any A ⊆ Ω we consider the following equivalence relation ≡ A on U : for any u, u ∈ U . Let [u] A be the equivalence class of u with respect to ≡ A and π W (A) := {[u] A : u ∈ U } the set partition on U induced by ≡ A . When we take as pairing W a data table having Ω as its attribute set, the equivalence relation ≡ A becomes a well-known tool of database theory and related fields [28,34].
On the other hand, the relation ≡ A becomes a type of local symmetry relation with respect to a fixed vertex subset A ⊆ V (G) when one interprets a finite simple undirected graph G with vertex set V (G) as a pairing F, V (G), V (G), {0, 1} on V (G), where F (u, a) := 1 if the vertices u, a ∈ V (G) are adjacent and F (u, a) := 0 otherwise. In [13] such a local symmetry relation has been investigated for some basic graph families and, in particular, a classification theorem for the Petersen graph, concerning all subgraphs induced by the vertex subsets A that are minimal with respect to the property that ≡ A agrees with ≡ V (G) has been proved (these vertex subsets have been called symmetry bases of the graph). Again, in [18] the local symmetry relation ≡ A has been studied in its interrelations with a specific type of binary operation • defined on the power set P(V (G)) and whose automorphism group is isomorphic to a subgroup of Aut(G) (for works on similar topics see also [31,35]). Based on the particular graph interpretation introduced in [13], the same applies to the general case. Therefore, we call ≡ A the A-symmetry relation and π W (A) the A-symmetry partition of the pairing W. Moreover, we also consider the equivalence relation ≈ W on the power set P(Ω) defined by the electronic journal of combinatorics 27(1) (2020), #P1. 19 for any A, A ∈ P(Ω), that is equivalent to say that for all u, u ∈ U . We call ≈ W the global symmetry relation of W. Let [A] ≈ P be the equivalence class of A with respect to ≈ W . It is easy to verify that [A] ≈ P is a unionclosed family (see [12] for details). Hence, the global symmetry relation for pairings yields specific models of families in the various mathematical contexts where pairings occur. Now, since [A] ≈ P is a union closed family, it has a maximal element M W (A), that we call the maximum partitioner of A. Then the set operator M W : A ∈ P(Ω) → M W (A) ∈ P(Ω) is a closure operator on Ω, the subset family , it results that the global symmetry relation ≈ W coincides with the symmetrization of the preorder W and that H(W)/ ≈ W is a lattice order-isomorphic to M(W) (see Proposition 14). This preorder satisfies the fundamental property In view of Property (4), we will use the terminology union additive relation induced by W to indicate the preorder W . As a type of dual structure of M(W), the subset family N(W) := {min([A] ≈ P ) : A ∈ M(W)} has been introduced in [14] showing that it is an abstract simplicial complex that is related with M(W) by means of several properties and results. For example, for some types of pairings M(W) and N(W) are respectively the closed subset family and the independent subset family of a matroid (see [14] for details).
Based on recent results obtained in the graph context [13,14] and on other general results linked to theoretical computer science [1,2,12,15,25,26,36,37,39], in this paper we continue the study of the basic property of the set system M(W) and use methods from GrC in relation to order-theory, limited to finite lattices.
To this regard, we prove a representation result for the closure system M(W) and we insert it within a general research perspective involving the interrelations between combinatorics, order theory and topological structures. More in detail, we prove, in a constructive way, that the family of all closed sets of any closure system on some finite set Ω agrees with the family of the maximum partitioners of some pairing W on Ω itself (Theorem 5).
Through the previous result, we will provide two representation results concerning all the preorders on a set Ω satisfying (4) (which we call union additive relations on Ω) and another one concerning finite lattices. On the one hand, we shall show that any union additive relation on Ω agrees with the preorder W induced by some pairing W on Ω itself.
On the other hand, in order to provide a representation theorem for finite lattices in terms of pairings, we shall use a classical result of lattice theory, according to which any finite lattice can be represented as a closure system on the set of all join-irreducible elements of the lattice itself [6]. This is the keypoint of our first representation theorem, since we use the aforementioned result to show that for each finite lattice L there exists a pairing W L on some finite ground set Ω L such that the lattice H(W L )/ ≈ W L is order-isomorphic to L itself (see Theorem 17). Therefore, as a consequence of such a representation theorem, we can reinterpret the order relation in any finite lattice L as the quotient of the preorder relation W , for some W on Ω L . In other terms, we can consider the order relation of any algebraic lattice as the quotient relation of a union additive relation defined on some set Ω L . In this way, we provide an enrichment of the study of the lattice order theory establishing new interrelationships between order and topological properties by means of the union additive property given in (4).
Theorem 17 provides a refinement for the partial order L of the lattice. Indeed, setting for each A, B ∈ P(Ω) and fixing two elements of the lattice x, y ∈ L, it may be easily verified that x L y if and only if γ W (η(x), η(y)) = 1. This fact enables us to compute γ W (η(x), η(y)) even if x and y are non-comparable with respect to the order of the lattice L. In other terms, the function γ W provides extra numerical informations for partial dependencies of subsets, corresponding to two non-comparable nodes of the lattice L and that are not explicit in the lattice itself. Note that each node corresponds to an equivalence class with respect to the equivalence relation ≈ W and the value of γ W corresponding to a pair of subsets belonging to the same equivalence class is clearly 1.
In general, given a pairing W on a finite set Ω with n elements, it is always possible to put within a 2 n × 2 n table T (W) the values γ W (X, Y ). The complete determination of all the entries of the above table enables us to obtain the closure system M(W). In other terms, we may consider the table T (W) as a sort of numerical completion of M(W). The map γ W and some related averages have been broadly used in [16,33] in order to investigate the transmission of symmetry in some basic digraph and graph families.
Eventually, in the last part of the paper we focus our attention on the link between a specific class of lattices endowed with an involutory map and some particular subset families induced by means of pairings. More in detail, we consider a finite lattice L endowed with an order-reversing involutory map ψ : L → L exchanging0 L and1 L and such that ψ(α) ∧ α =0 L and ψ(α) ∨ α =1 L . In this case we say that the pair (L, ψ) is a complemented involutory lattice. In general, the theory of the posets endowed of an involutory map is a well investigated research field [4,11,24,32], which also has links with complex analysis [7] and design theory [9,23] (moreover, for some studies concerning discrete dynamics on particular types of lattices endowed with involutory maps see [8,10]).
In the present context, given a pairing W on Ω and A ∈ P(Ω), we set K W (A) := Ω \ M W (A c ). Subsequently, we say that A ∈ P(Ω) is normally extensible if A = M W (K W (A)). Then we show that for any finite complemented involutory lattice (L, ψ), there exist a the electronic journal of combinatorics 27(1) (2020), #P1.19 finite set Ω L,ψ and a pairing W on Ω L,ψ such that M W (∅) = ∅ and for which the poset of all normally extensible subsets with respect to set theoretical inclusion is a complete lattice order-isomorphic to L (see Theorem 22).
Let us briefly describe the content of our paper. In Section 2 we first recall some basic notions and results on closure systems, lattices, families of subsets and pairings. In Section 3 we provide a constructive proof for the representation result for closure systems on finite sets (Theorem 5) and provide some examples of how our algorithm works. In Section 4 we introduce the notion of union additive relations and give the proof of the representation theorem for such relations. Furthermore, we will also provide a proof for the representation theorem of finite lattices, according to which for any finite lattice L we can find a finite set Ω L and a pairing W L on it such that L is order-isomorphic to M(W L ) (Theorem 17). Section 5 focuses the attention on the representation of complemented involutory lattices through a pairing W on a finite set Ω L,ψ such that the poset of all normally extensible subsets is order-isomorphic to L itself.

Reviews, Notations and Basic Results
Notations. In this paper we denote by Ω a given finite arbitrary set and by P(Ω) the power set of Ω. We use the symbol ⊆ * to denote dual inclusion, that is A ⊆ * B is equivalent to say that B ⊆ A. If n is a positive integer, we denote by Ω n the cartesian product of n copies of Ω. When X ⊆ Ω, we will use both the notations X c and Ω \ X to indicate the complement subset of X with respect to Ω. If f : Ω → Ω is a map between sets, we denote by Im(f ) the image of f . A map ψ is called an involutory map on Ω if ψ 2 = Id Ω , where Id Ω denotes the identity on Ω.
If X is a set and we have two maps ψ : X → X and f : X → P(Ω) such that f (ψ(x)) = Ω \ f (x) for any x ∈ X, we say that f is a ψ-complementary map.
Posets and Lattices. Let P := (P, ) be a poset and X ∈ P(P ) be a non-empty subset. If x, y ∈ P , we also write x < y if x y and x = y. If x, y are two distinct elements of P , we say that y covers x, denoted by x y if x y and there exists no element z ∈ P such that x < z < y.
Let P := (P, ) and P := (P , ) be two posets, α, β ∈ P and f : P → P a map. We say that f is: Let L := (L, L ) be a lattice. For each x, y ∈ L, we denote by x ∨ y the join and by x ∧ y the meet of x and y in L. When they exist, we denote by0 L and1 L the meet and the join of L, respectively. If x ∈ L, we say that x is join-irreducible if x =0 L and x = y ∨ z implies x = y or x = z for any y, z ∈ L. We denote by J(L) the subset of all the electronic journal of combinatorics 27(1) (2020), #P1.19 join-irreducible elements of L. A lattice L = (L, L ) is said complete if there exist both the join and the meet of any subset X ∈ P(L) and we denote them by X and X, respectively.
The most simple example of a complemented involutory lattice is the poset (P(Ω), ⊆) where ψ : P(Ω) → P(Ω) associates with each set its complement. More in general, whenever we fix a set system F ∈ P(P(Ω)) and consider the lattice L F := (X, ⊆), where X consists of ∅, Ω, the members of F and their complements, we obtain a complemented involutory lattice by taking the map ψ associating with each set of F its complement and exchanging ∅ with Ω.
We denote by M n the linear sum 1 ⊕ n ⊕ 1 (see page 17 of [19] for details), where n is the antichain on n elements {1, . . . , n}. Another simple example of a complemented involutory lattice is given by (M n , ψ), with n an even integer, ψ(a i ) = a σ(i) , where σ is a product of disjoint transpositions without fixed elements, ψ(0 L ) =1 L and ψ(1 L ) =0 L .
Set Systems and Set Operators. We call the elements of P(P(Ω)) set systems on Ω.
A set system F on Ω is said: We recall that if F is a closure system on Ω, then the poset (F, ⊆) is a complete lattice (usually called closure lattice of F) in which the meet operation is the subset intersection.
A set operator on Ω is any map σ : P(Ω) → P(Ω). For any set operator σ on Ω, the fixed point set of σ is the following set system of Ω: Moreover, if F is a given set system on Ω we will consider the induced set operator Int F on Ω defined by A set operator σ on Ω is said: • idempotent, if σ(σ(A)) = σ(A) for all A ∈ P(Ω); • a closure operator, if σ is isotone, extensive and idempotent; the electronic journal of combinatorics 27(1) (2020), #P1.19 • a kernel operator, (see [22]) if σ is isotone, intensive and idempotent.
In the next theorem, we recall the well-known bijective correspondence between closure systems and closure operators.
Theorem 1 (Thm 7.3, [19]). Let σ be a closure operator on Ω and F be a closure system on Ω. Then F ix(σ) is a closure system on Ω. and Int F is a closure operator on Ω. Moreover, we have that The next result, whose proof is straightforward, will be useful in what follows.
Proposition 2. Let F be a complement-closed set system on Ω. Then Int F is a closure operator on Ω such that Int F (∅) = ∅.
Set Partitions. If π is a set partition on Ω, we usually denote by {B i : i ∈ I} the block family of π. If u ∈ Ω, we denote by π(u) the block of π which contains the element u.
We will write π ≺ π when π π and π = π . The pair (Π(Ω), ) is a complete lattice which is called partition lattice of the set Ω.
Pairings. We call a quadruple W := F, U, Ω, Λ , where U and Ω are finite non-empty sets, Λ is a non-empty set and F is a map having domain U × Ω and codomain Λ, a pairing on Ω.
For each A ∈ P(Ω) we consider the following equivalence relation ≡ A on U : for any u, u ∈ U . Let [u] A be the equivalence class of u with respect to ≡ A and the set partition on U induced by ≡ A . Let us note that, for any A ∈ P(Ω), we have that In what follows, we will consider the equivalence relation ≈ W on the power set P(Ω) defined by for any A, A ∈ P(Ω), and we will denote by [A] ≈ P the equivalence class of A with respect to ≈ W . The basic properties of the relation ≈ W are recalled in the following result.
Theorem 3 (Proposition 3.2, [12]). We have that: is a closure operator on Ω, whose induced closure system is induces an order isomorphism between the closure lattice M(W) and the poset P sym (W). In particular, P sym (W) is a lattice.
In what follows, we call the elements of M(W) the maximum partitioners of W, the lattice M(W) the maximum partitioner lattice of W and P sym (W) the symmetry partition lattice of W.
Remark 4. There is another lattice which is order isomorphic to the dual closure lattice M(W), whose role is relevant when ones uses micro and macro granular representations induced by information tables (see [14] for details). Such a lattice is obtained by taking firstly the set G(W) := {[A] ≈ W : A ∈ M(W)}, and next by considering the partial order on G(W) defined by: . Then, by Theorem 3 one can deduce that the poset G(W) := (G(W), ) is a lattice which is order-isomorphic to the maximum partitioner lattice M(W) (see [12] for details). Relatively to a specific pairing, we shall visualize the lattice G(W) in Example 7.

Representation Theorem for Closure Systems
The main result of the present section is Theorem 5, which is a representation result for closure systems on Ω by means of pairings on the same ground set. More specifically, for the electronic journal of combinatorics 27(1) (2020), #P1.19 any closure system S on Ω we are able to find a pairing W on Ω such that the set system M(W) of all maximum partitioners of W coincides with S.
The proof of Theorem 5 is constructive and it is based on an algorithmic construction whose underlying idea derives by the simplest case when a closure system is a chain of subsets, as in Example 6.
The main idea of such a proof is based on the fact that the order structure of the given closure lattice on Ω can be entirely described by its maximal chains.
Theorem 5. The map associating with any pairing W on Ω the closure system M(W) on Ω is surjective.
Proof. Let S be a closure system on Ω = {a 1 , . . . , a n } and S := (S, ⊆) its induced closure lattice. Then S has maximum element1 S = Ω and minimum element0 S := E = ∩S. We denote by the covering relation of S.
Let C 1 , C 2 , . . . , C k be all the maximal chains of S. Obviously, any chain C i has bottom E and top Ω.
We use the following notations for the previous chains: . . . Let us note that we have the following partition of U S with integer intervals: Therefore, for any element u ∈ U S there exists a unique index ι u ∈ {1, 2, . . . , k} such that Now we will construct a pairing on Ω for which M(W) = S.
To this purpose, we will define recursively the map F S : U S × Ω −→ N.
the electronic journal of combinatorics 27(1) (2020), #P1.19 Firstly, let us note that the greatest element of any maximal chain is the whole set Ω, therefore for any 1 i k and any a ∈ Ω there exists the following minimum integer number: Now we associate with any ordered pair (u, a) ∈ U S × Ω the following integer set: At this point we define F S recursively, as follows.
Firstly, for any a ∈ E, we set On the other hand, for any a ∈ Ω \ E, we set for all u = 2, . . . , m.
Let us consider the pairing W = F S , U S , Ω, N so constructed. We shall demonstrate that M(W) = S.
For, let us firstly show the inclusion S ⊆ M(W). Let A ∈ S. We distinguish three distinct cases. In fact, by definition of Conversely, let a ∈ M W (∅). Then, in view (iii) of Theorem 3, we have that Let us assume by contradiction that a / ∈ E, and we take v = m 1 ∈ U S , so that ι v = 1. Therefore Then, it results that v / ∈ J v,a since ζ 1,a 2. So, by (10) we have that F S (v, a) = F S (v−1, a), and this contradicts (11). On the other hand, the inclusion M W (∅) ⊆ E also holds. Hence (iii): Let now E A Ω. In view of part (iii) of Theorem 3, it suffices to prove the existence of two elements w, w of U S such that the electronic journal of combinatorics 27(1) (2020), #P1 .19 In fact, if the condition (12) is satisfied for some w, w ∈ U S , by (iii) of Theorem 3 we Then, in order to find two specific elements w, w ∈ U that satisfy the condition (12), we proceed as follows.
Since E A Ω, it follows that any maximal chain of S has length at least three. Let then Thus, by the latter condition and by the definition of both t and w, we get m i−1 + 1 w m i − ζ i,a . In view of (10), since This proves (12) and, hence, we conclude that A ∈ M(W).
It remains to prove the inclusion M(W) ⊆ S. For, let A ∈ M(W). We claim that A ∈ S. As above, we may distinguish three cases.
(ii ): Analogously, if A = Ω, then we get A ∈ S in view of the definition of a closure system.
(iii ): Thus, let E A Ω. Let us consider the set system Let us note that S A is again a closure system on Ω, because Ω ∈ S A and S A is intersection closed. Therefore there exists a minimum element B ∈ S A . At this point, in order to prove that A ∈ S, it suffices to show that B ⊆ A, since in such a case we will get A = B ∈ S.
Let therefore b ∈ B. We must prove that b ∈ A, and this is equivalent to show that Let us show that (13) is equivalent to the following: the electronic journal of combinatorics 27(1) (2020), #P1.19 To see that, it suffices to show that (14) implies (13), since the reverse implication is obvious. To this regard, let b ∈ M W (A) and u, u ∈ U S be such that u ≡ A u . We claim that F S (u, b) = F S (u , b). Without loss of generality, we may assume u < u. In view of (10), the map F S (·, a) is non-decreasing. Therefore, the condition u ≡ A u , or equivalently, In particular, we also have u ≡ A u − 1, therefore we may proceed as before to show that . This shows the equivalence between (13) and (14). Therefore, in order to complete the proof of the theorem, fix an arbitrary element of u ∈ U S such that u ≡ A u − 1. We shall prove that For each maximal chain C i , with 1 i k, let us consider the set system Clearly, Ω ∈ C i,A ; thus, there exists a minimum element in the chain C i containing A. In other terms, for each i = 1, . . . , k, there exists a minimal integer s i such that Now, let us observe that In fact, assume by contradiction that (17) does not hold. In such a case, by (7) we must have necessarily u = m ιu−1 + 1. Moreover, since E A, we can choose an element a ∈ A \ E. Then, relatively to such a choice, by (9) it results that u / ∈ J u,â . Therefore, by (10) we obtain that and this is in contrast with the hypothesis u ≡ A u − 1. Hence (17) holds.
At this point, we can prove (15). In view of (10), it suffices to show that the electronic journal of combinatorics 27(1) (2020), #P1.19 Now, the first of the previous inequality follows immediately by (17), since m ιu−1 + 1 u − 1. Therefore, in order to conclude our proof, it remains to show that In fact, since A ⊆ A ιu,sι u , in view of the definition of the integer ζ ιu,a , we must necessarily have ζ ιu,a s ιu . Furthermore, if it were ζ ιu,a < s ιu , then we infer that where A ιu,t ∈ C ιu , and this contradicts the minimality of the integer s ιu . Therefore, (19) holds.
On the other hand, by (16) we have that b ∈ B ⊆ A ιu,sι u , and hence by (8). Finally, we obtain (18) as a direct consequence of (20) and (21). The closure S has the Hasse diagram represented in Figure 1, therefore S is a chain of length four. The minimum in S is E = {a 1 }, and in such a case we have only a maximal chain C 1 , which is S itself. Therefore, by using the same notations adopted in the proof of Theorem 5, we have that  Now, in the table represented in Figure 2, we show as acts the algorithm which defines the map F S in the proof of Theorem 5. We start from the top of any column a j / ∈ E and we move towards below. When u i ∈ J u i ,a j then we do not change the previous value, otherwise we sum 1 to the above value. We proceed in such a way for any column a j / ∈ E. To the end of this process we obtain the pairing represented in Figure 3. The minimum in S is E = {a 2 }. The lattice associated with S has the Hasse diagram represented in Figure 4. There are three maximal chains from E to Ω: Thus m 0 = 0, m 1 = 4, m 2 = 8 and m = m 3 = 11. Moreover ζ 1,a 1 = 3, ζ 2,a 1 = 4, ζ 3,a 1 = 3, ζ 1,a 3 = 2, ζ 2,a 3 = 2, ζ 3,a 3 = 3, ζ 1,a 4 = 4, ζ 2,a 4 = 3, ζ 3,a 4 = 3, ζ 1,a 5 = 4, ζ 2,a 5 = 4 and ζ 3,a 5 = 2. Then U = {u 1 := 1, u 2 := 2, . . . , u 11 := 11} and, ι 1 = ι 2 = ι 3 = ι 4 = 1, ι 5 = ι 6 = ι 7 = ι 8 = 2, ι 9 = ι 10 = ι 11 = 3.
As has already been done in Example 6, in the table represented in Figure 5, we show as acts the algorithm which defines the map F S given in (10) relatively to the closure system in the present example. At this point we can construct the pairing represented in Figure 6. Moreover, in Figure 7 we have also drawn the Hasse diagram of G(W(S)) (we use string notation to represent the subsets of Ω 5 ).
Let us provide simple consequences of Theorem 5.     Remark 10. A very natural question related to the algorithm introduced in Theorem 5 is about its complexity. It is a hard computational problem and it will not be analyzed in this paper. In order to face it, we could divide it into smaller problems and solve each one of them separately. The first of this problem is known as the Moore problem, in honour to E. H. Moore for its work on closure systems (see [27]). It consists of the computation of the number of closure systems on a given finite set Ω with n elements. Already for n = 7, this number becomes too large and difficult to be found by an enumeration algorithm. By the way, as proved by Alekseev in [3] its asymptotic size is O 2 ( n n 2 ) . The second part of the problem consists of the computation of all chains in any of the previous closure systems. Therefore, in each case, we have to compute the cost of the implementation of the algorithm and, next, to analyze the average complexity or, at least, an estimation of it.
Relatively to a given pairing W on Ω, we now consider three set operators The members of the latter set system will be called normally extensible subsets of the pairing W and will be relevant in the proof of Theorem 22. In the next result, we shall describe some basic properties of the poset E(W) and of the set operator K W which we shall use in the last theorem of the paper. In what follows, we say that a pairing W on Ω is regular if M W (∅) = ∅. Proposition 11. Let A, B, C ∈ P(Ω) and W be a pairing on Ω. Then: (i) K W is a kernel operator on Ω and K W (A) = max{Z ⊆ A : Ω \ Z ∈ M(W)}; (ii) if C is the complement of some maximum partitioner, then M W (C) ∈ E(W); (iii) if A and A c both belong to M(W) then A ∈ E(W); Assume now that W is a regular pairing on Ω. Then: Proof. (i): The set operator K W is clearly intensive and isotone. Furthermore, the idempotence of K W follows by the equalities: To prove the second part of the claim, take Z ⊆ A such that Z c ∈ M(W) and assume that Z = Ω \ M W (B) for some B ∈ P(Ω). We claim that Z ⊆ K W (A). In fact, the (ii): Let C = Ω \ M W (B) for some B ∈ P(Ω). By the above part (i), we get C ⊆ A ∈ E(W).
The fact that φ W is an involutory map follows by the equalities Using the first relation in (23), we get We now prove that the empty set is the only subset contained in M W (A c ) ∩ A and whose complement is a maximum partitioner.
This shows that φ W (A) ∨ A = Ω and the proof concludes here.
We close this section with another consequence of Theorem 5. We shall see that given a complement-closed set system on Ω, we can find a regular pairing whose maximum partitioners agree with the fixed points of Int F . the electronic journal of combinatorics 27(1) (2020), #P1.19 Proposition 12. Let F be a complement-closed set system on Ω. Then there exists a regular pairing W on Ω such that: Proof. Let F be a complement-closed set system on Ω. In view of Proposition 2, we have that the set operator Int F is a closure operator on Ω and, hence, F ix(Int F ) is a closure system on Ω by Theorem 1.

Union Additive Relations and a Related Lattice Representation Theorem
In this section we will introduce the notion of a union additive relation on an arbitrary set Ω, namely a preorder relation on P(Ω) satisfying property (4). We will see that a pairing induces a union additive relation and that, in general, any union additive relation on Ω agrees with the preorder W induced by some pairing W on Ω. We will obtain such a result as a consequence of Theorem 5. Another consequence of the aforementioned theorem consists of the possibility of representing any finite lattice L as the maximum partitioner lattice of some pairing W L on a specific finite ground set Ω L . First of all, we introduce the following basic notion.
Definition 13. We call a binary relation on P(Ω) such that: a union additive relation on Ω.
Theorem 3 enables us to find a union additive relation in the context of pairings. As a matter of fact, we can set: the electronic journal of combinatorics 27(1) (2020), #P1. 19 and it may be easily verified that W is a union additive relation on Ω, which we call the W-union additive relation on Ω. Moreover, we may interpret the relation ≈ W as the equivalence relation induced by the preorder W , i.e.
Consequently, if we consider the preordered set H(W) := (P(Ω), W ), we immediately obtain the following order isomorphisms. By means of Theorem 5, in what follows we will be able to show that any finite lattice is order-isomorphic to the maximum partitioner lattice of some pairing on a specific ground set, and, moreover, that any union additive relation on Ω can be represented by the preorder W induced by some pairing W on Ω. To this regard, we firstly recall some classical notions of database theory that are used to understand the degree of dependency of two attribute subsets of a relational database (see [34] for details).
If A, B ∈ P(Ω), we set and, if U is finite, Some basic properties of the set operator Γ W are described in the next proposition, whose proof is straightforward. Proposition 16. Let A, A , B, B ∈ P(Ω). Then the following hold: At this point, based on a classical representation theorem of lattice theory (see [6]), we can establish the following pairing representation theorem for finite lattices.
Proof. Before tackling the proof, let us recall that any finite lattice L is order-isomorphic to a closure lattice induced by a specific closure system on the set Ω L := J(L), where J(L) is the set of all join-irreducible elements of L (see [6]). Such a closure system consists of the set system of the fixed points of the closure operator φ : P(Ω L ) → P(Ω L ) defined as follows: φ(A) := {x ∈ Ω L : x A}.
Therefore, we set Ω L := J(L). We also denote by S L the resulting closure system and by η : L → S L the order isomorphism between the dual lattice L * and the lattice (S L , ⊆). Then, by Theorem 5, we can construct a pairing W on Ω L such that M(W) = S L . Therefore η W := η is an order isomorphism between L and M(W), and the wanted equivalences become a direct consequence of part (i) of Proposition 16.
The equivalences established in Theorem 17 has some theoretical consequences: it says us that the study of union additive relations between subsets of finite sets is equivalent to the study of order relations on finite lattices. Therefore in the next part of this work we try to investigate the basic theoretical properties of the notion of union additive relations between subsets of finite sets and the direct interrelation of this notion with other classical notions of lattice theory.
We call pairing characteristic of the lattice L, denoted by pc(L), the minimum allowable cardinality of the set Ω L obtained from the thesis of Theorem 17.
Let of Theorem 17, we can consider the inverse order isomorphism η −1 W : M(W) → L and the closure map M W : P(Ω N ) → M(W), so that we obtain the surjective map Let us note that the map ξ W is not canonically determined. In fact, it depends from the order isomorphism η W , which in turn depends on the not uniquely determined closure system S given in the proof of Theorem 17. However, by means of the map ξ W we can formally describe the following equivalences.
Theorem 18. Let L = (L, L ) be a finite lattice, N = pc(L) and X, Y ∈ P(Ω N ). Then the following conditions are equivalent: Proof.
Moreover, let us provide the table whose entries are the values of γ W (X, Y ) for both X, Y varying over P(Ω 3 ).
7/9 7/9 7/9 7/9 7/9 7/9 {a 2 } 1 7/9 1 7/9 7/9 7/9 7/9 7/9 {a 3 } Remark 20. Let us consider the following equivalence relation on the family of all pairings on Ω: two pairings W and W are equivalent whenever their symmetry partition lattices are order-isomorphic. Hence, Theorem 17 ensures that any finite lattice is the symmetry partition lattice of a pairing. In particular, any finite lattice identifies an equivalence class of pairings. Let us note that we have a constructive way in order to associate with a finite lattice a pairing on a suitable set Ω. At this point, by means of Theorem 5, we represent (again in a constructive way) the closure system as a pairing on Ω. However, for pairings, we have defined the set Γ W (A, B) and its numerical counterpart γ W (A, B) in the finite case. Hence, through the equivalences given in Theorem 17, it is possible to see that, whenever we take a finite lattice L with N = pc(L) and W ∈ PL(L), then γ W refines the partial order L of the lattice. In fact, let x, y ∈ L be two elements of the lattice. Then it results that x L y if and only if γ W (η W (x), η W (y)) = 1, hence we can always compute γ W (η W (x), η W (y)) even if x and y are non-comparable. In other terms, the function γ W provides extra numerical informations for partial dependencies of subsets, corresponding to two non-comparable nodes of the lattice L and that are not explicit in the lattice itself. Note that each node corresponds to an equivalence class with respect to the equivalence relation ≈ W and the value of γ W corresponding to a pair of subsets belonging to the same equivalence class is clearly 1. In general, given a pairing W on a finite set Ω with n elements, it is always possible to put within a 2 n × 2 n table T (W) the values γ W (X, Y ). This table is a source of useful the electronic journal of combinatorics 27(1) (2020), #P1.19 informations about the properties of the pairing. Indeed, the complete knowledge of all the entries of the table allows us to reconstruct the closure system M(W), so T (W) represents a numerical completion of M(W).

Pairing Representation of Finite Complemented Involutory Lattices
In this section, we prove a second representation theorem for particular types of lattices.
To be more detailed, we will consider finite lattices endowed with an order-reversing involutory map ψ : L → L exchanging0 L and1 L and such that ψ(α) ∧ α =0 L and ψ(α) ∨ α =1 L and, next, we show that they are represented by pairings W on some finite ground set Ω L,ψ so that the poset of the normally extensible subsets of W is orderisomorphic to the starting lattice L.
In Theorem 21 we shall see that starting from a finite complemented involutory lattice, a finite set Ω n , a complement-closed family on Ω n and an order-preserving ψ-complementary map f : L → (F, ⊆), we may construct a new finite set, a complement-closed family on it and an order-preserving ψ-complementary map from L to such a family with some specific properties of preservation of pairs of elements of the lattice.
Proof. Let us firstly assume that f is {α, β}-preserving. In such a case, we take Ω = Ω n and g = f . Moreover, since the image of a finite complemented involutory lattice by an order-preserving ψ-complementary map is a complement-closed family, we can also take G = Im(g). Thus, the claim holds when f is {α, β}-preserving. Therefore, we may assume that f is not {α, β}-preserving. Then, five possible cases may occur: Then we define a map h 1 : L → P(Ω n+1 ) as follows: for any γ ∈ L.
If Γ 1 = ∅, we set g := h 1 , Ω := Ω n+1 and G := Im(g). Then, G is a complement-closed set system on Ω. Moreover, g(α) g(β), i.e. g is {α, β}-preserving. Hence the set Ω, the set system G and the map g satisfy the conditions of the statement and the proof concludes here.
Then, reasoning as for the map h 1 , it may be easily proved that h 2 is order-preserving and that if h 1 is {λ, µ}-preserving, then also h 2 is {λ, µ}-preserving. Moreover h 2 (α) h 2 (β). Set As before, note that if γ / ∈ Γ 2 , then while if γ ∈ Γ 2 , then we get Furthermore, note that Γ 2 Γ 1 because β 1 ∈ Γ 1 \ Γ 2 . At this point, we can reiterate the previous construction in order to obtain a sequence of functions h i and of subsets Γ i of L such that each h i is order-preserving and {α, β}-preserving and Γ i Γ i−1 for any index i 2.
As L is finite, there must be some index k, and hence a subset Γ k which must be empty. In other terms, for each γ ∈ L, it must be At this point, we get the thesis taking g := h k , Ω := Ω n+1 and G := Im(g) ⊆ P(Ω).
(C2): The proof is the same as that given in (C1), after reversing the role of α and β.
(C3): Also in this case, we take Ω n+1 , h 0 , β 0 and h 1 as in the case (C1). Then, as α β, we easily prove as in (C1) that h 1 is a ψ-complementary map. Therefore, also in the present case we can define the sequence of subsets Γ i and the sequence of functions h i as in (C1). Again, let k be the minimum integer for which Γ k = ∅, so that h k (γ) = Ω n+1 \ h k (ψ(γ)).
Note that if γ / ∈ Γ 2 , then it may be easily shown that while if γ ∈ Γ 2 , we have that Furthermore, notice that Γ 2 Γ 1 since α 1 ∈ Γ 1 \ Γ 2 . Iterating the above construction, we obtain a sequence of maps h i and of subsets Γ i of L such that each h i is an orderpreserving map and also an {α, β}-preserving map and Γ i Γ i−1 for each index i 2. As L is finite, we will find an index s for which Γ s = ∅. We get the thesis taking g := h s , Ω := Ω n+2 and G = Im(g) ⊆ P(Ω). (C4): Just set h 0 := f and repeat the construction given in (C3). (C5): Taking both the cases (C3) and (C4), we showed that the thesis follows when α β and f (α) ⊆ f (β). Therefore, if α β and f (β) f (α), we may proceed as in (C4) reversing the role α of β.
In the next result, we prove that for any finite complemented involutory lattice there exists a pairing on a certain finite set Ω L,ψ whose normally extensible family is isomorphic to L itself. Proof. The idea of the proof is the following: first of all, we want to show the existence of a finite set Ω L,ψ and of a complement-closed family F on Ω L,ψ such that L is orderisomorphic to (F, ⊆); secondly, we want to show the existence of some regular pairing W on Ω L,ψ such that F = E(W). To this regard, we will divide our proof in two steps: (i): We shall construct a finite set Ω L,ψ and a complement-closed family F on Ω L,ψ such that an isomorphism f : L → (F, ⊆) exists. Firstly, set Λ 0 := ∅, F 0 := {∅} and f 0 : L → F 0 is the constant map on L such that f (γ) = ∅ for each γ ∈ L. Note that F 0 is a complementclosed family on Λ 0 and that f 0 is trivially an order-preserving map from L onto (F 0 , ⊆). If |L| = 1, the thesis has been proved just taking Ω L,ψ := Λ 0 and G := {∅} and g := f 0 as our isomorphism. Therefore, in what follows, we may assume that l := |L| 2. Let us consider the following subset of pairs of L: Clearly, the number of elements of H 0 is finite and equals l 2 . Fix now (α 0 , β 0 ) ∈ H 0 . In view of Theorem 21 there exist a finite set Ω, a complement-closed family G on Ω and an order-preserving, ψ-complementary and {α 0 , β 0 }-preserving map g : L → (G, ⊆). Set Λ 1 := Ω, F 1 := G and f 1 := g and define the following subset of pairs of L: Notice that H 1 H 0 as (α 0 , β 0 ) ∈ H 0 \ H 1 . At this point, fix (α 1 , β 1 ) ∈ H 1 . In view of Theorem 21 there exist a finite set Ω, a complement-closed family G on Ω and an orderpreserving, ψ-complementary and {α 1 , β 1 }-preserving map g : L → (G, ⊆). Set Λ 2 := Ω, F 2 := G and f 2 := g and define the following subset of pairs of L: the electronic journal of combinatorics 27(1) (2020), #P1.19 Notice that H 2 H 1 as (α 1 , β 2 ) ∈ H 2 \ H 1 . Furthermore, let us notice that if (λ, µ) ∈ L 2 \ H 2 and λ = µ, then the map f 1 is {λ, µ}-preserving and Theorem 21 ensures that f 2 is also {λ, µ}-preserving. Therefore, we may iterate the above construction to obtain a strictly decreasing sequence of subsets of L 2 which, in view of the finiteness of L 2 , must terminate with H k = ∅ for some integer k. Take Ω L,ψ := Λ k , F := F k and f := f k . Clearly, in view of Theorem 21, it follows that f is order-preserving and that F = Im(f ). Moreover, f is both injective and surjective, since it is {α, β}-preserving for each α, β ∈ L.
The fact that f is an {α, β}-preserving map also ensures that f −1 is also order-preserving, i.e. f is an isomorphism between the posets L and (F, ⊆). Thus, the isomorphism f : L → (F, ⊆) induces on (F, ⊆) a natural lattice structure. (ii): At this point, we want to show the existence of a regular pairing W on Ω L,ψ such that F = E(W). Let us consider the set operator Int F : P(Ω L,ψ ) → P(Ω L,ψ ). In view of Proposition 2, it results that Int F is a closure operator such that Int F (∅) = ∅ and, hence, Proposition 12 ensures the existence of a regular pairing W on Ω L,ψ such that M(W) = F ix(Int F ). Furthermore, it results that Int F = M W in view of Theorem 1. We must prove that F = E(W). We shall firstly show the inclusion F ⊆ E(W). To this regard fix Z ∈ F. Proposition 12 ensures that K W (Z) = Z, so we get i.e. Z ∈ E(W). On the other hand, we shall demonstrate that E(W) ⊆ F. So, let Z ∈ E(W). Clearly, Z ∈ M(W). Now, since F is a complete lattice, it follows that Furthermore, we have that We claim that Z = Z * . In view of Proposition 12, we easily note that (Z * ) c ∈ M(W). Furthermore, let A ⊆ Z be such that A c ∈ M(W). We claim that To this regard, just notice that A c = M W (A c ) = Int F (A c ) = {Y ∈ F : A c ⊆ Y }, whence, passing to the complements, we infer the validity of (35), as wanted. Clearly, in view of (34), we easily deduce that Y ⊆ Z * for each subset Y ∈ F which is also contained in A. In particular, this implies that A ⊆ Z * . In other terms, we proved that Z * is the greatest subset of Z whose complement is a maximum partitioner. So, K W (Z) = Z * in view of (i) of Proposition 11. As K W (Z) = Z * ∈ F, we get the following identities: Z = N W (Z) = M W (Z * ) = Z * whence we deduce that Z ∈ F. So, F = E(W) and the proof concludes here.

Conclusions
In this paper we continued the investigations of some links between granular computing (GrC), closure systems and algebraic order theory. To this regard, the notion of pairing becomes a basic tool of our analysis.
In the perspective of the present work, a pairing W = F, U, Ω, Λ has been considered as a purely mathematical interpretation of the classical notions of information system [28] and relational data table [34]. We used the set partitions induced by the map F : U × Ω −→ Λ in order to define a specific closure system M(W) (whose members are called maximum partitioners), and a related closure operator M W on the set Ω. By means of such an operator, one can describe the classical functional dependency between attribute subsets of Ω, when this set is interpreted as an attribute set of some information system. Now, since several constructions used in database theory have a natural lattice structure (see [34] for details), it is natural to investigate the links between lattices and the above closure system induced by pairings. These links may be obtained by means of a binary relation between subsets of Ω, which we called union additive relation. Any can be represented as a preorder W induced by some pairing W on the same ground set Ω. This implies that pairings are concrete models describing the union additive relations. Then, in this paper we proved three main representation results.
The first of such results is Theorem 5. In this theorem we showed that any closure system S on the set Ω can be described as the family of all maximum partitioners M(W) of some pairing W on Ω. The proof given in Theorem 5 is constructive, and by means of it we can explicitly provide a pairing W on Ω such that M(W) = S.
The second representation result is Theorem 17, which concerns finite lattices and that is proved by means of Theorem 5. In such a theorem we showed that any finite lattice L coincides with the maximum partitioner lattice of some pairing on the set Ω L of all join-irreducible elements of L. As a direct consequence, we see that the partial order of any finite lattice can be refined by means of a numerical map γ W associated with the closure system M(W). In fact, by means of the map γ W we can also establish a partial dependency level between non-comparable nodes of the lattice.
Eventually, in Section 5 we studied a specific class of finite lattices, which we called complemented involutory lattices, i.e. lattices L = (L, L ) endowed with an orderreversing involutory map ψ : L → L exchanging0 L and1 L and such that ψ(α) ∧ α =0 L and ψ(α) ∨ α =1 L . Also for such types of lattices we provided a representation result in terms of pairings, which is Theorem 22. In this theorem we found a finite set Ω L,ψ and a pairing W on Ω L,ψ such that the poset induced by the set system E(W) = F ix(M W • K W ) (where K W is the set operator defined by K W (A) := Ω \ M W (A c )) is order-isomorphic to L.
As a future research perspective, we can start from the observation that there is not a uniquely determined way to associate a pairing with a given closure system S. Therefore one could study the relations between all pairings W on Ω for which M(W) = S, in terms of their corresponding maps γ P . Closely related to the previous question is the problem of determing a sort of canonical pairing associated with the lattice L.
the electronic journal of combinatorics 27(1) (2020), #P1. 19 By the way, we think that the aforementioned theorems represent a useful starting point to frame in an unifying perspective the abstract analysis of lattices with motivations related to the functional dependency in database theory.