Chu spaces [1–5], formal concept analysis (FCA) [6–12], automata [13, 14] and rough sets [15–18] are some of the well-known relational structures in the literature with numerous applications. Among these structures, Chu spaces were first introduced by Barr [1] as a general framework for several types of mathematical structures. Such spaces have been shown to be capable of describing many mathematical structures (cf., [4, 5]) in connection with the concurrency description in computer science. The theory of FCA was first introduced by Wille [2] and was developed based on a formal context given by a binary relation between a set of objects and a set of attributes. FCA has been applied in many disciplines, such as software engineering, knowledge discovery, and information retrieval. In the area of data mining, FCA has been used primarily to extract a hierarchy of mined information from voluminous data [19–23]. In another direction, as the simplest mathematical model in computational theory, finite-state automata not only lay the theoretical foundations of computer science but also are closely related to other fields, such as neural networks and model theory. Categories have also appeared in some areas of theoretical computer science, and in particular, many articles have considered the categorical approach to automata theory. The theory of rough sets introduced by Pawlak [16] has been shown to be useful in studying intelligent systems with insufficient and incomplete information. Finally, several generalizations of rough sets have been established in [15, 17, 18], using an arbitrary relation in place of the equivalence relation.

After Lotfi Zadeh introduced the theory of fuzzy sets in 1965, the above relational structures were fuzzified to handle ambiguous and insufficient information. The details are summarized below.

• • Fuzzy Chu spaces: Papadopoulos and Syropoulos [3] proposed the concept of fuzzy Chu spaces, which have been shown to be capable of describing many fuzzy mathematical structures. Interestingly, the adjointness condition in the case of fuzzy Chu spaces is equivalent to the extension principle of fuzzy set theory.

• • Fuzzy formal context analysis: As one of the most essential and contemporary perspectives on FCA, the study of fuzzy formal contexts has drawn the attention of numerous academics. In [8], the authors developed concept lattice theory in the fuzzy framework. The concept derivation operators are specified as either an implication operator or a t-conorm. Belohlavek [6, 7] further explored the fuzzy concept lattice under the condition that the truth values are derived from a complete residuated lattice. Some researchers [9–11] studied fuzzy concept lattices between a crisp set and a fuzzy set that does not include the implication operator. In [24], three kinds of variable threshold concept lattices were discussed based on the Galois connection which complements Belohlavek’s derivation operators. In [58], the theory of fuzzy concept lattices was studied based on generalized fuzzy rough approximation operators using the Lukasiewicz implicator.

• • Fuzzy automata: Fuzzy automata and languages have been studied as methods for bridging the gap between the precision of computer languages and vagueness. These studies were initiated by Santos [14], Wee [26], and Wee and Fu [27] and further developed by other researchers (cf., [28–30]). Fuzzy automata and languages with membership values in different lattice structures have attracted considerable attention from researchers working in this area (cf., [28–33]).

• • Fuzzy rough sets: Dubois and Prade [34] proposed the fuzzy version of rough sets in which fuzzy relations play a key role instead of crisp relations. Recently, the combinations of fuzzy sets and rough sets were investigated using various fuzzy logic operations and binary fuzzy relations in [35–43]. From a categorical point of view, Banerjee and Chakraborty [44] started the study of categories of rough sets, whereby the category (ROUGH) of rough sets was introduced. Li and Yuan [45] further defined a category (RSC) of rough sets based on Iwiński’s I-rough sets [46]. Most recently, the internal algebras of RSC, ROUGH, related categories, and their generalizations were explored in [47].

In this paper, we confine ourselves to establishing the relationship among fuzzy approximation spaces, fuzzy Chu spaces, and fuzzy concept analysis. The motivation for this study is that (i) each fuzzy topological space can be represented as a fuzzy Chu space, and this representation preserves products and co-products; (ii) fuzzy approximation operators can induce a fuzzy topological space if and only if the fuzzy relation is reflexive and transitive, and seemingly the fuzzy approximation space can be linked to fuzzy Chu space. This paper shows that every fuzzy approximation space can be represented as a fuzzy Chu space. Furthermore, this representation preserves products and co-products. The primary motivations for this study were to enhance the theoretical understanding of fuzzy systems, provide novel methodologies for dealing with fuzziness, and bridge the gap between different mathematical approaches to fuzziness.

The paper is organized as follows. In Section 2, we recall some basic properties of residuated lattices. Some categorical frameworks for fuzzy rough sets are introduced, and the relationship among them is discussed in Section 3. In the next section, the relationship between generalized L-fuzzy approximation space and L-fuzzy formal context analysis is established, and three types of L-fuzzy concept lattices are constructed based on generalized L-fuzzy rough approximation operators. The categorical frameworks for L-fuzzy FCA are further studied. Section 5 shows that each L-fuzzy approximation space can be represented as an L-fuzzy Chu space, and this representation preserves the product and co-product. Finally, in Section 6, conclusions are drawn.

In this section, we recall some concepts related to residuated lattices and Zadeh’s fuzzy forward operators. For details on residuated lattices and Zadeh’s fuzzy forward operators, we refer to the works in [48–55]. We begin with the following:

Definition 2.1

A residuated lattice is an algebra L ≡ (L,∧,∨, ⊗,→, 0, 1) such that

• (i) (L,∧,∨, 0, 1) is a bounded lattice with the least element 0 and the greatest element 1;

• (ii) (L,⊗, 1) is a commutative monoid; and

• (iii) ∀a, b, c ∈ L; a ⊗ b ≤ c iff a ≤ b → c, i.e., (→,⊗) is an adjoint pair on L.

A residuated lattice L is complete if it is complete as a lattice.

Proposition 2.1

Let L be a complete residuated lattice. Then for a, b, c ∈ L,

• 1. a ≤ b ⇒ a ⊗ c ≤ b ⊗ c,

• 2. 1 ⊗ a = a ⊗ 1 = a,

• 3. a ⊗ 0 = 0 ⊗ a = 0,

• 4. a ⊗ (∨_i∊Ib_i) = ∨_i∊I (a ⊗ b_i),

• 5. 1 → a = a,

• 6. 0 → a = 1.

In this paper, an L-fuzzy set that takes values from a fixed complete residuated lattice L is identified with its membership function. For a nonempty set X, L^X denotes the collection of all L-fuzzy subsets of X, and for all a ∈ L, a(x) = a denotes a constant L-fuzzy set.

Definition 2.2

Let L be a residuated lattice. A negation in L is a unary operation ¬ defined by ¬a = a → 0, ∀ a ∈ L. L is said to be regular if a = ¬(¬a), ∀ a ∈ L.

Proposition 2.2

Let L be a complete regular residuated lattice. Then for a, b, c ∈ L, we have

• a → b = ¬(a ⊗ ¬b),

• ¬a → ¬b = b → a.

Definition 2.3

Let X be a nonempty set. The following are induced operations of intersection ∧, union ∨, multiplication ⊗, implication→, and negation ¬ on L^X:

(A∧B)(x)=A(x)∧B(x),(A∨B)(x)=A(x)∨B(x),(A⊗B)(x)=A(x)⊗B(x),(A→B)(x)=A(x)→B(x),(¬A)(x)=¬A(x).

Under the assumption of the completeness of L, we consider an intersection and a union of an arbitrary family of L-fuzzy sets.

At the end of this section, we recall the following definition from [56].

Definition 2.4

Let φ : X → Y be a map, the Zadeh fuzzy forward operator φ^→ : L^X → L^Y is defined as follows:

φ→(A)(y)=∨φ(x)=yA(x),∀A∈LX,∀y∈Y.

In this section, some categories of L-fuzzy rough sets are proposed, and the relationship among them is investigated.

Let A be an L-fuzzy set, then a crisp set A⁻¹[0, 1] is defined as A⁻¹[0, 1] = {x : A(x) ∈ [0, 1]}.

Definition 3.1

The category FRSC is defined as follows:

• (a) Objects are pairs (A₁, A₂), where A₁ and A₂ are two L-fuzzy sets such that A₁ ≤ A₂.

• (b) f : (A₁, A₂) → (B₁, B₂) is a morphism if f:A2-1[0,1]→B2-1[0,1] is a map and f^→(A₁) ≤ B₁.

Before stating the next theorem, we recall the definition of a topological category from [57].

Definition 3.2

A concrete category C with a forgetful functor U : C → SET is a topological category if a system of objects A_i ∈ obj(C) and X ∈ SET. Then for any system of maps g_i : X → U(A_i), there exists an initial lift with the following properties:

• (i) an object A ∈ obj(C), such that U(A) = X,

• (ii) a system of K-morphism f_i : A → A_i, such that U(f_i) = g_i,

• (iii) for each B ∈ obj(C), a map w : U(B) → X, a system of C-morphism t_i : B → A_i such that g_i ο w = U(t_i), there exists a unique K-morphism h : B → A such that U(h) = w and f_i ο h = t_i.

Theorem 3.1

The category FRSC is a topological category.

Proof: Let U : FRSC → SET be a functor. For (A₁, A₂) ∈ obj − FRSC, U(A₁, A₂) = A₂⁻¹[0, 1]. Again, let f be a morphism of FRSC. Then U(f) = f. Now, let {(A_1i, A_2i) : i ∈ I} be a system of FRSC-objects and f_i : X → A_2i⁻¹[0,1] be maps, where X ∈ SET is set. Then define two L-fuzzy sets B₁, B₂ ∈ L^X such that for x ∈ X, B₂(x) = 1 and B₁(x) = ∧_i∊IA_1i(f_i(x)). Now, for y_i ∈ A_1i⁻¹[0,1], fi→(B1)(yi)=∨fi(x)=yiB1(x)≤Ai1(yi), for all i ∈ I. Thus, f_i : (B₁, B₂) → (A_1i, A_2i) are the FRSC-morphisms, where U(f_i) = f_i. Again, let (C₁, C₂) be an object of FRSC. Then U(C1,C2)=C2-1[0,1] and for a map w:C2-1[0,1]→X and a morphism t_i : (C₁, C₂) → (A_1i, A_2i) such that the diagram in Figure 1 commutes, i.e., f_i ο w = U(t_i) = t_i. Now, we have to prove h : (C₁, C₂) → (B₁, B₂) is a FRSC-morphism such that U(h) = w.

For yi∈A1i-1[0,1],

ti→(C1)(yi)≤A1i(yi),(fi∘h)→(C1)(yi)≤A1i(yi),(fi→∘h→)(C1)(yi)≤A1i(yi),∨fi(x)=yih→(C1)(x)≤A1i(yi),h→(C1)(x)≤∧i∈IA1i(fi(x)),h→(C1)(x)≤B1(x).

Thus, h is a FRSC-morphism. Since U(h) = w implies h = w.

Thus h is a unique morphism. Hence the proof.

Before proceeding, we define a new type of category of L-fuzzy rough set which is denoted by FROUGH.

Definition 3.3

The category FROUGH is defined by

• (a) Objects are pairs (X, R, A), where X is a crisp set; A is an L-fuzzy set on X; and R : X ×X → L is an L-fuzzy reflexive relation.

• (b) f : (X, R, A) → (Y, S, B) is a morphism if f : X → Y is a map and f^→ ο R(A) ≤ S(B), where

  R_(A)(x)=∧x′∈X(R(x,x′)→A(x′)),R¯(A)(x)=∨x′∈X(R(x,x′)⊗A(x′)).

Theorem 3.2

Let L be a complete regular residuated lattice. Then FRSC and FROUGH are equivalent.

Proof: Define a functor G from FRSC to FROUGH, then G maps an object of FRSC (A₁, A₂) to the object of FROUGH (X, R, A), where X={(x,1),(x,2):x∈A2-1[0,1]}; A = {((x, 1), A₂(x)), ((x, 2), 0)} ∀ (x, 1), (x, 2) ∈ X and R is defined below.

R((x,i),(y,j))={1,if x=y, i=j=1,0,if x≠y, i=j=1,¬A1(x),if i=1, j=2,0,if i=2, j=1,1,if i=2, j=2.

For i = 1,

R¯(A)(x,i)=∨(y,j){R((x,i),(y,j))⊗A(y,j)}=∨{1⊗A(x,1),if x=y, j=1,0⊗A(y,j),if x≠y, j=1,¬A1(x)⊗A(y,j),if j=2=∨{A(x,1),if x=y, j=1,0,if x≠y, j=1,¬A1(x)⊗0,if j=2=∨{A(x,1),if x=y, j=1,0,if a≠x, j=1,0,if j=2=A(x,1)=A2(x).

For i = 2,

R¯(A)(x,i)=∨(y,j){R((x,i),(y,j))⊗A(y,j)}=∨{0⊗A(y,j),if j=1,1⊗A(y,j),if j=2=∨{0⊗A(y,1),if x=y, j=1,1⊗0,if x≠y, j=2=0.

Therefore, we can say R̄(A) ≅ A₂.

Again, for i = 1,

R_(A)(x,i)=∧(y,j){R((x,i),(y,j))→A(y,j)}=∧{1→A(x,1),if x=y, j=1,0→A(y,j),if x≠y, j=1,¬A1(x)→A(y,j),if j=2=∧{A(x,1),if x=y, j=1,1,if x≠y, j=1,¬A1(x)→0,if j=2=∧{A(x,1),if x=y, j=1,1,if x≠y, j=1,¬¬A1(x),if j=2=∧{A2(x),if x=y, j=1,1,if x≠y, j=1,A1(x),if j=2=A1(x).

For i = 2,

R_(A)(x,i)=∧{0→A(y,j),if j=1,1→A(y,j),if j=2=∨{0→A(y,j),if j=1,1→0,if j=2=∨{1,if j=1,0,if j=2=0.

Thus, R(A) ≅ A₁.

Consider FRSC-arrow f : (A₁, A₂) → (B₁, B₂) and G(f) = f : (X, R, A) → (Y, S, B).

Now, the morphism in FRSC is:

f→(A1)≤B1,f→(R_(A))≤S_(B),f→∘R_(A)≤S_(B),

which is an arrow in FROUGH-category.

Conversely, let (X, R, A) and (Y, S, B) be two objects of FROUGH, where R and S are L-fuzzy reflexive relations on X and Y, respectively. Then (R(A), R̄(A)) and (S(B), S̄(B)) are objects of FRSC. Since R and S are reflexive L-fuzzy relations, then R(A) ≤ R̄(A) and S(B) ≤ S̄(B)). Let f : X → Y be a FROUGH-morphism such that

f→(R_(A))≤S_(B),f→(A1)≤B1.

Here, f : A⁻¹[0, 1] → B⁻¹[0, 1] can be written as f : R̄(A)⁻¹ [0, 1] → R̄(B)⁻¹[0, 1]. Thus f is a FRSC-morphism.

Next, we recall the following concept of topos theory from [45].

Definition 3.4

A topos is a category C satisfying the five conditions:

• (i) Finite products exist in C.

• (ii) There is a terminal object M in C such that for each object A, there is one and only one morphism from A to M, which is denoted as !, and M is denoted as 1.

• (iii) For any objects A and B and morphisms from A to B, an equalizer exists in C.

• (iv) Exponentials exist in C.

• (v) There is a sub-object classifier in C.

Now, we have the following:

Theorem 3.3

The category FRSC has all the topos properties except one, for it has no sub-object classifier.

Proof: (i) Let (A₁, A₂) and (B₁, B₂) be two objects in the category FRSC. Let C₁ = A₁ × B₁, C₂ = A₂ × B₂ and for x ∈ A₂⁻¹[0, 1], y ∈ B₂⁻¹[0, 1], C₂(x, y) = A₂(x) ∧ B₂(y), and C₁(x, y) = A₁(x) ∧ B₁(y). Thus, (C₁, C₂) is an FRSC-object. Now, arrow p₁ : (C₁, C₂) → (A₁, A₂) is a map from C2-1[0,1] to A2-1[0,1] such that p₁(x, y) = x. Now, p1→(C1)(x)=∨p1(x,y)=xC1(x,y)≤A1(x). Thus p₁ is an FRSC-morphism. Again arrow p₂ : (C₁, C₂) → (B₁, B₂) is a map from C2-1[0,1] to B2-1[0,1] such that p₂(x, y) = y and p2→(C1)≤B1. Thus, p₂ is an FRSC-morphism. Then {(C₁, C₂), p₁, p₂} is a finite product of (A₁, A₂) and (B₁, B₂).

(ii) Let f, g : (A₁, A₂) → (B₁, B₂) be two morphisms. Then f,g:A2-1[0,1]→B2-1[0,1] are two maps and f^→(A₁) ≤ B₁ and g^→(A₁) ≤ B₁. Now, D2-1[0,1]={x∈A2-1[0,1]|f(x)=g(x)} and D1-1[0,1]=(A1-1∪D2-1)[0,1]. Again D2={(x,A2(x)),x∈D2-1[0,1]} and D1={(x,A1(x)),x∈D1-1[0,1]}. Then {(D₁, D₂), e} is an equalizer of f, g : (A₁, A₂) → (B₁, B₂) where e:D2-1[0,1]→A2-1[0,1] is a mapping and e(x) = x, ∀x∈D2-1[0,1]. Thus, an equalizer exists in FRSC.

(iii) There is a terminal object in FRSC. In fact, 1 = M = (0, 0) is a terminal object, where 0 is an L-fuzzy set with membership value 0 for all x ∈ X.

(iv) Let (A₁, A₂) and (B₁, B₂) be two objects in FRSC. Also, let G₁ and G₂ be two L-fuzzy sets such that G2-1[0,1]={f|f:A2-1[0,1]→B2-1[0,1]is a mapping}; G₁⁻¹[0, 1] = {f ∈ G₂|f^→(A₁) ≤ B₁} and G₂(f) = ∧_{x∊A1⁻¹ [0,1]}f^→(A₁) (f(x)) and G₂(f) = G₁(f). Now, we define ev : (G₁, G₂) × (A₁, A₂) → (B₁, B₂), where ev : G₂⁻¹[0, 1] × A₂⁻¹[0, 1] → B₂⁻¹[0, 1] is a map such that (f, x) ↦ f(x) and (G₂×A₂)(f, x) = G₂(f) ∧ A₂(x). Now, for f(x) ∈ B₁,

ev→(G1×A1)=∨ev(f′,x′)=(f,x)(G1×A1)(f′,x′)=(G1×A1)(f,x)=G1(f)∧A1(x)=∧x′{f→(A1)(f(x′))}∧A1(x)≤B1(f(x))∧A1(x)≤B1(f(x)).

Thus ev is an FRSC-morphism.

Let (H₁, H₂) be an object in FRSC and F : (H₁, H₂) × (A₁, A₂) → (B₁, B₂) be an FRSC-morphism. Again, let F̄ : H₂ → G₂ such that F̄ : H₂⁻¹[0, 1] → G₂⁻¹[0, 1] is a map and for y ∈ H₂⁻¹, y ↦ F̄(y). Also, for x ∈ A₂⁻¹[0, 1], F̄(y)(x) = F(y, x) and G₁(F̄(y) = ∧_{x∊A2⁻¹ [0,1]}B₁(F(y, x)).

F¯→(H1)(F¯(y))=∧x∈A2-1[0,1]F¯(y)→(A1)(F¯(y)(x))=∧x∈A2-1[0,1]F¯(y)→(A1)F(y,x)≤∧x∈A2-1[0,1]B1(F(y,x))≤G1(F¯(y)).

Thus, F̄ is an FRSC-morphism, and the diagram in Figure 2 commutes, i.e., ev ο (F̄ × Id_A2) = F, and the morphism F̄ satisfies the condition of uniqueness. Therefore {(G₁, G₂), ev} is an exponential of (A₁, A₂) and (B₁, B₂).

(iv) Example to show that FRSC has no sub-object classifier. Assume that T : 1 → Ω is a sub-object classifier. Let A = (0, 0) and B = (1, 2) be two objects of FRSC, then f : 0⁻¹[0, 1] → 2⁻¹[0, 1] is a mono-morphism and consequently there is an unique morphism X_f : B → Ω such that Figure 3 is a pullback. In Figure 3, there is a unique morphism g : B → A. It follows that 1 = g^→(1) ≤ 0. This is a contradiction. Therefore, there is no sub-object classifier in FRSC.

Remark

Since the categories FRSC and FROUGH are equivalent and the category FRSC has all the topos properties except one, it has no sub-object classifier. Then the category FROUGH also has all the topos properties except one for it has no sub-object classifier.

In this section, we construct L-fuzzy concept lattices based on generalized L-fuzzy rough approximation operators. Throughout this section, L is a regular residuated lattice. We recall the following definition from [58].

Definition 4.1

Let (P, ≤) and (Q, ⊑) be two partially ordered sets. Then (f, g) is a Galois connection between P and Q, where f : P → Q and g : Q → P are two functions and for each p ∈ P and q ∈ Q, p ≤ g(q) if and only if q ⊑ f(p).

A Galois connection may be defined as an order-reversing or order-preserving function. They connections were initially defined for order-reversing functions between powersets by Birkhoff [59], who called them “polarities.” Subsequently, Ore [60] extended Birkhoff’s notion to arbitrary posets and called them “Galois connexions.” For the work presented in this paper, we use the order-reversing function for the Galois connection, which is common in L-fuzzy formal context analysis.

Definition 4.2

A triple (X, Y, R) is referred to as an L-fuzzy formal context, where X is a nonempty finite set of objects; Y is a finite set of attributes; and R : X × Y → L is an L-fuzzy relation.

Next, we recall the following definition from [58].

Definition 4.3

Let (X, Y, R) be an L-fuzzy formal context. Then for A ∈ L^X and B ∈ L^Y , two functions ↑: L^X → L^Y and ↓: L^Y → L^X are called Birkhoff operators, where

A↑=∧x∈XA(x)→R(x,y),B↓=∧y∈YB(y)→R(x,y).

Now, (↑, ↓) is a Galois connection, where L^X and L^Y have the usual point-wise order.

Remark

Let L_L(X, Y, R) = {(A, B) ∈ L^X × L^Y , A = B^↓, B = A^↑} and (A₁, B₁) ≤ (A₂, B₂) if and only if A₁ ≤ A₂. Then L_L(X, Y, R) is a complete L-fuzzy complete lattice.

Before proceeding, we recall the following from [56].

Definition 4.4

Let X and Y be two non-empty sets and R be an L-fuzzy relation between X and Y . Then (X, Y, R) is a generalized L-fuzzy approximation space. For A ∈ L^X, the generalized L-fuzzy upper and lower approximation operators are defined as follows:

R¯(A)(y)=∨x∈X{R(x,y)⊗A(x)},R_(A)(y)=∧x∈X{R(x,y)→A(x)}.

Definition 4.5

Let (X, Y, R) be an L-fuzzy formal context. Then for all A ∈ L^X and B ∈ L^Y , we can define the L-fuzzy approximation operators as follows:

R¯(A)(y)=∨x∈X{R(x,y)⊗A(x)},R_(A)(y)=∧x∈X{R(x,y)→A(x)}.R′¯(B)(x)=∨y∈Y{R(x,y)⊗B(y)},R′_(B)(x)=∧y∈Y{R(x,y)→B(y)}.

Now, we have the following propositions:

Proposition 4.1

Let (X, Y, R) be an L-fuzzy formal context. For A_i ∈ L^X and B_i ∈ L^Y , we have

• (i) R(¬A) = ¬R̄(A), R′_(¬B)=¬R′¯(B)

• (ii) For A₁ ≤ A₂, R(A₁) ≤ R(A₂) and R̄(A₁) ≤ R̄(A₂).

• (iii) For B₁ ≤ B₂, R′(B₁) ≤ R′(B₂) and R′¯(B1)≤R′¯(B2).

• (iv) R(∧_i∊IA_i)=∧_i∊IR(A_i) and R̄(∨_i∊IA_i)=∨_i∊IR̄(A_i).

• (v) R′(∧_i∊IB_i) = ∧_i∊IR′(B_i) and R′¯(∨Bi)=∨i∈IR′¯(Bi).

• (vi) R′¯(R_(A))≤A≤R′_(R¯(A)),R¯(R′_(B))≤B≤R_(R′¯(B)).

• (vii) R_(R′¯(R_(A)))=R_(A) and R̄(R′(R̄(A))) = R̄(A). R′(R̄(R′(B))) = R′(B) and R′¯(R_(R′¯(B)))=R′¯(B).

Next, we define the following:

Definition 4.6

Let (X, Y, R) be an L-fuzzy formal context. For A ∈ L^X and B ∈ L^Y , define two operators ⇑: L^X → L^Y and ⇓: L^Y → L^X such that ⇑ (A)(y) = R(¬A)(y) and ⇓ (B)(x) = R′(¬B)(x).

The set of all images of ⇑ is denoted as ⇑ (L^X), and the set of all images of ⇓ is denoted as ⇓ (L^Y ).

Proposition 4.2

Let (X, Y, R) be an L-fuzzy formal context. Then the pair (⇑, ⇓) is a Galois connection between L^X and L^Y .

Proof: We only need to prove that, A ≤ B^⇓ if and only if B ≤ A^⇑.

For x ∈ X,

A≤B⇓⇔A(x)≤B⇓(x)⇔A(x)≤R′_(¬B)(x)⇔A(x)≤∧y∈Y(R(x,y)→¬B(y)).

For any y ∈ Y ,

⇔A(x)≤(R(x,y)→¬B(y))⇔¬A(x)≥¬(R(x,y)→¬B(y)⇔¬A(x)≥R(x,y)⊗B(y)⇔R(x,y)⊗B(y)≤¬A(x)⇔B(y)≤R(x,y)→¬A(x).

If this is true for all x ∈ X, then

⇔B(y)≤∧x∈X(R(x,y)→¬A(x))⇔B(y)≤A⇑(y)⇔B≤A⇑.

Hence, (⇑, ⇓) is a Galois connection.

Proposition 4.3

Let (X, Y, R) be an L-fuzzy formal context and (⇑, ⇓) be a Galois connection between L^X and L^Y . Then

• (1) ⇑⇓⇑=⇑ and ⇓⇑⇓=⇓.

• (2) A ∈⇓ (L^Y ) if and only if A =⇓⇑ (A). Again B ∈⇑ (L^X) if and only if ⇑⇓ (B).

• (3) ⇑:⇓ (L^Y ) →⇑ (L^X) and ⇓:⇑ (L^X) →⇓ (L^Y ) are order-reversing bijections. Also ⇑ (L^X) and ⇓ (L^Y ) are antiisomorphic partially ordered sets.

Definition 4.7

Let (X, Y, R) be an L-fuzzy formal context. For A ∈ L^X, B ∈ L^Y , then

• (1) the pair (A, B) is called an object-oriented L-fuzzy formal concept if A=R′¯(B) and B = R(A);

• (2) the pair (A, B) is called a property oriented L-fuzzy formal concept if A = R′(B) and B = R̄(A); and

• (3) the pair (A, B) is called an L-fuzzy formal concept if A =⇓ (B) and B =⇑ (A).

The set of all object-oriented L-fuzzy formal, property oriented L-fuzzy formal, and L-fuzzy formal concepts are denoted by L_o(X, Y, R), L_p(X, Y, R), and L_q(X, Y, R), respectively. Also if (A₁, B₁) ≤ (A₂, B₂) if and only if A₁ ≤ A₂. Then L_o((X, Y, R), L_p((X, Y, R) and L_q((X, Y, R) are all posets.

Now, we have the following:

Theorem 4.1

Let (X, Y, R) be an L-fuzzy formal context. Then for A ∈ L^X and B ∈ L^Y,

• (1) (R′¯(R_(A)),R(A)) is an object-oriented L-fuzzy formal concept;

• (2) (R′(R̄(A)), R̄(A)) is a property oriented L-fuzzy formal concept;

• (3) (A^⇑⇓, A^⇑) is an L-fuzzy formal concept.

• (4) (R′¯(B),R_(R′¯(B))) is an object-oriented L-fuzzy formal concept;

• (5) (R′(B), R̄(R′(B))) is a property oriented L-fuzzy formal concept;

• (6) (B^⇓, B^⇓⇑) is an L-fuzzy formal concept.

Proof: (1) Let P=R′¯(R_(A)) and Q = R(A). Then R_(P)=R_(R′¯(R(A)))=R_(A)=Q. Again, P=R′¯(R_(A))=R′¯(Q). Hence (R′¯(R_(A)),(R_(A)) is an object-oriented L-fuzzy formal concept.

(2) Let P = R′(R̄(A)) and Q = R̄(A). Then P = R′(R̄(A)) = R′(Q). Again, R̄(P) = R̄(R′(R̄(A))) = R̄(A) = Q. Hence (R′(R̄(A)), R̄(A)) is a property-oriented L-fuzzy formal concept.

(3) Let P = A^⇑⇓ and Q = A^⇑. Then P = A^⇑⇓= Q^⇓ and P^⇑ = A^⇑⇓⇑ = A^⇑ = Q. Hence (A^⇑⇓, A^⇑) is an L-fuzzy formal concept.

Similarly, we can prove (4), (5), and (6).

Theorem 4.2

Let (X, Y, R) be an L-fuzzy formal context. Then L_o(X, Y, R), L_p(X, Y, R), and L_q(X, Y, R) are all complete lattices. Let {(A_i, B_i) ∈ L_o(X, Y, R)} be an arbitrary subset of L_o(X, Y, R). Then the infimum and supremum are respectively

∧i∈I (Ai,Bi)=(R′¯(R_(∧i∈IAi)),∧i∈IBi),∨i∈I(Ai,Bi)=(∨i∈IAi,R_(R′¯(∨i∈IBi))).

Let {(A_i, B_i) ∈ L_p(X, Y, R)} be an arbitrary subset of L_p(X, Y, R). Then the infimum and supremum are respectively

∧i∈I (Ai,Bi)=(∧i∈IAi,R_(R′_(∧i∈IBi))),∨i∈I(Ai,Bi)=(R′_(R¯∨i∈IAi)),∨i∈IBi).

Let {(A_i, B_i) ∈ L_q(X, Y, R)} be an arbitrary subset of L_q(X, Y, R). Then the infimum and supremum are respectively

∧i∈I(Ai,Bi)=(∧i∈IAi,(∨i∈IBi)⇓⇑),∨i∈I(Ai,Bi)=((∨i∈IAi)⇑⇓,∧i∈IBi).

Proof: To prove that L_o(X, Y, R) is a complete lattice, we only need to prove that an arbitrary subset of L_o(X, Y, R) has both an infimum and supremum. Now we prove that (R′¯(R_(∧i∈IAi)),∧i∈IBi)∈Lo(X,Y,R),(∨i∈IAi,R_(R′¯(∨i∈IBi)))∈Lo(X,Y,R), and they are the infimum and supremum of {(A_i, B_i)|∀i ∈ I, (X_i, B_i) ∈ L_o(X, Y, R)} respectively.

Now, R_(R′¯(R_(∧i∈IAi)))=R_(∧i∈IAi)=∧i∈I(R_(Ai))=∧i∈I(Bi).

Again R′¯(∧i∈I(Bi))=R′¯(∧i∈IR_(Ai))=R′¯(R_(∧i∈I(Ai)).

Thus, (R′¯(R_(∧i∈IAi)),∧i∈IBi)∈Lo(X,Y,R). To prove that it is the infimum, let (C, D) ∈ L_o(X, Y, R) be an arbitrary lower bound of {(A_i, B_i)|∀i ∈ I, (X_i, B_i) ∈ L_o(X, Y, R)}. Then for all i ∈ I,

C≤Ai,C≤∧i∈I(Ai),R_(C)≤R_(∧i∈I(Ai)),R′¯(D)≤R′¯R_(∧i∈I(Ai)),C≤R′¯R_(∧i∈I(Ai)).

Hence, (R′¯(R_(∧i∈IAi)),∧i∈IBi) is the infimum.

Now, R_(∨i∈IAi)=R_(∨i∈IR′¯(Bi))=R_(R′¯(∨i∈IBi)).

Again R′¯R_(R′¯(∨i∈IBi))=R′¯(∨i∈IBi)=∨i∈IR′¯(Bi)=∨i∈IAi.

Thus, (∨i∈IAi,R_(R′¯(∨i∈IBi)))∈Lo(X,Y,R). Also it is the supremum of {(A_i, B_i)|∀i ∈ I, (X_i, B_i) ∈ L_o(X, Y, R)}.

Similarly, it can be proven that (∧_i∊IA_i, R̄(R′(∧_i∊IB_i))) and (R′(R̄(∨_i∊IA_i)), ∨_i∊IB_i) are the infimum and supremum of {(A_i, B_i)|∀i ∈ I, (X_i, B_i) ∈ L_q(X, Y, R)} respectively.

Also (∧_i∊IA_i, (∨_i∊IB_i)^⇓⇑) and ((∨_i∊IA_i)^⇑⇓, ∧_i∊IB_i) are the infimum and supremum of {(A_i, B_i)|∀i ∈ I, (X_i, B_i) ∈ L_q(X, Y, R)}, respectively.

Remark

The set of all pairs of L-fuzzy formal concepts L_q(X, Y, R) is isomorphic to ⇓ (L^Y ) and anti-isomorphic to ⇑ (L^X).

Theorem 4.3

Let (X, Y, R) be an L-fuzzy formal context. Then

• (1) (A, B) ∈ L_o(X, Y, R) if and only if (¬A, B) ∈ L_L(X, Y, ¬R),

• (2) (A, B) ∈ L_p(X, Y, R) if and only if (A, ¬B) ∈ L_L(X, Y, ¬R),

• (3) (A, B) ∈ L_q(X, Y, R) if and only if (A, B) ∈ L_L(X, Y, ¬R).

Proof: (1) Let (A, B) ∈ L_o(X, Y, R). Then, A=R′¯(B) and B = R(A). Now for y ∈ Y,

(¬A)↑(y)= ∧x∈X{¬A(x)→¬R(x,y)}=∧x∈X{R(x,y)→A(x)}=R_(A) =B,B↓(x)=∧y∈Y{B(y)→¬R(x,y)}=∧y∈Y{¬(R(x,y)⊗B(y))}=¬∨y∈Y{(R(x,y)⊗B(y))}=¬R′¯(B)=¬A.

Hence (¬A, B) ∈ L_L(X, Y, ¬R).

Conversely, let (¬A, B) ∈ L_L(X, Y, ¬R). Then (¬A)^↑ = B and B^↓ = ¬A.

R_(A)(y)=∧x∈X{R(x,y)→A(x)}=∧x∈X{¬A→¬R(x,y)}=(¬A)↑(y)=B(y).

Again,

R′¯(B)(x)=∨y∈Y{R(x,y)⊗B(y)}=∨y∈Y{¬(B(y)→¬R(x,y))}=¬∧y∈Y{(B(y)→¬R(x,y))}=¬(B↓)(x)=¬(¬A)(x)=A(x).

Hence, (A, B) ∈ L_o(X, Y, R).

Similarly, we can prove (2) and (3).

Next, we define a category of L-fuzzy formal contexts.

Definition 4.8

Let F₁ = (X₁, Y₁, R₁) and F₂ = (X₂, Y₂, R₂) be two L-fuzzy formal contexts. Also, let (⇑₁, ⇓₁) and (⇑₂, ⇓₂) be Galois connections of F₁ and F₂, respectively. Then a morphism from F₁ to F₂ is defined as a pair of one-to-one functions (f, g⁻¹) : F₁ → F₂ such that f : L^X₁ → L^X₂ and g⁻¹ : L^Y₂ → L^Y₁ with ⇑₁= g⁻¹ο ⇑₂ οf and ⇓₂= fο ⇓₁ οg⁻¹.

The set of all L-fuzzy formal contexts with morphisms, as defined above, forms a category. For future reference, we shall denote the category as FCI (formal context interchanges).

Before proceeding with the next theorem, we recall the definition of detecting ordering map from [58].

Definition 4.9

A mapping h from a pre-ordered set (X, ≤) to a pre-ordered set (Y, ≤) is called detect ordering if for all x₁, x₂ ∈ X, h(x₁) ≤ h(x₂) implies x₁ ≤ x₂.

Next, we study in detail the behaviors of the morphisms of the category FCI defined for FCA.

Theorem 4.4

Let F₁, F₂ ∈ obj(FCI) and (f, g⁻¹) be morphisms between F₁ and F₂ in the category FCI. Then

• (1) For A₁, B₁ ∈ L^X₁ , ⇑₁ (A₁) =⇑₁ (B₁) if and only if ⇑₂ (f(A₁)) =⇑₂ (f(B₁)). Again for C₂, D₂ ∈ L^Y₂, ⇓₂ (C₂) =⇓₂ (D₂) if and only if ⇓₁ (g⁻¹(C₂)) =⇓₁ (g⁻¹(D₂)).

• (2) Each of f and g⁻¹ adheres to the fixed L-subsets set up by the appropriate Galois connection. More precisely, for A₁ ∈ L^X₁, ⇓₁⇑₁ (A₁) = A₁ if and only if ⇓₂⇑₂ (f(A₁)) = f(A₁); and for B₂ ∈ L^Y₂, ⇑₁⇓₁ (B₂) = B₂ if and only if ⇑₁⇓₁ (g⁻¹(B₂)) = g⁻¹(B₂).

• (3) f :⇓ (L^Y₁) →⇓ (L^Y₂) and g :⇑ (L^X₂) →⇑ (L^X₁) are bijection maps.

• (4) One of the maps f and g detects the ordering on image sets if and only if another one is order-preserving on image sets.

• (5) Object-oriented L-fuzzy formal concepts in F₁ are mapped to object-oriented L-fuzzy formal concepts in F₂; i.e., if (A₁, g⁻¹(B₂)) ∈ L_o(F₁), then (f(A₁), B₂) ∈ L_o(F₂), provided f(¬A₁) = ¬f(A₁), for all A₁ ∈ L^X₁.

• (6) Property-oriented L-fuzzy formal concepts in F₁ are mapped to property-oriented L-fuzzy formal concepts in F₂; i.e., if (A₁, g⁻¹(B₂)) ∈ L_p(F₁), then (f(A₁), B₂) ∈ L_p(F₂), provided g⁻¹(¬B₂) = ¬g⁻¹(B₂), for all B₂ ∈ L^Y₂.

• (7) L-Fuzzy formal concepts in F₁ are mapped to L-fuzzy formal concepts in F₂; i.e., if (A₁, g⁻¹(B₂)) ∈ L_q(F₁), then (f(A₁), B₂) ∈ L_q(F₂).

Proof: (1) For A₁, B₁ ∈ L^X₁, if ⇑₁ (A₁) =⇑₁ (B₁), then g⁻¹(⇑₂ (f(A₁))) =⇑₁ (A₁) =⇑₁ (B₁) = g⁻¹(⇑₂ (f(B₁))). Since g⁻¹ is a one-to-one map, then ⇑₂ (f(A₁)) =⇑₂ (f(B₁)). Conversely, let ⇑₂ (f(A₁)) =⇑₂ (f(B₁)) implies g⁻¹(⇑₂ (f(A₁))) = g⁻¹(⇑₂ (f(B₁))), which in turn implies ⇑₁ (A₁) =⇑₁ (B₁).

Similarly, we can prove for C₂, D₂ ∈ L^Y₂ that ⇓₂ (C₂) =⇓₂ (D₂) if and only if ⇓₁ (g⁻¹(C₂)) =⇓₁ (g⁻¹(D₂)).

(2) For A₁ ∈ L^X₁, let ⇓₁⇑₁ (A₁) = A₁. Then, ⇓₂⇑₂ (f(A₁)) = f ⇓₁g ⇑₂ (f(A₁)) = f ⇓₁⇑₁ (A₁) = f(A₁).

Conversely, let ⇓₂⇑₂ (f(A₁)) = f(A₁), implies f ⇓₁⇑₁ (A₁) = f(A₁). Since f is a one-to-one map, then ⇓₁⇑₁ (A₁) = (A₁).

Similarly, we can prove for B₂ ∈ L^Y₂ that ⇑₁⇓₁ (B₂) = B₂ if and only if ⇑₁⇓₁ (g⁻¹(B₂)) = g⁻¹(B₂).

(3) By the definition of FCI, f is a one-to-one function. Then f :⇓₁ (L^Y₁) →⇓₂ (L^Y₂) is one-to-one. By Proposition 4.3 (3), ⇓₂ is a bijection map on image sets and ⇓₂= fο ⇓₁ οg⁻¹. It is well-known that the last function in the factorization of an onto function must be onto. Hence, f :⇓₁ (L^Y₁) →⇓₂ (L^Y₂) is a bijective function.

Similarly, it can be shown that g⁻¹ :⇑₂ (L^X₂) →⇑₁ (L^X₁) is a bijective map.

(4) Let f :⇓₁ (L^Y₁) →⇓₂ (L^Y₁) be a detect-ordered map. Then, we have to show that g⁻¹ :⇑₂ (L^X₂) →⇑₁ (L^X₁) is an order-preserving map. Let B₂, B′₂ ∈⇑₂ (L^X₂), and B₂ ≤ B′₂.

B2≤B′2,⇓2(B2)≥⇓2(B′2),f⇓1g-1(B2)≥f⇓1g-1(B′2),⇓1g-1(B2)≥⇓1g-1(B′2),⇑1⇓1g-1(B2)≤⇑1⇓1g-1(B′2),g-1(B2)≤g-1(B′2).

Thus g⁻¹ is order-preserving.

Again, let f be an order-preserving map. Then, we must show that g⁻¹ is a detect-ordered map on image sets. For B₂, B′₂ ∈⇑₂ (L^X₂), let g⁻¹(B₂) ≤ g⁻¹(B′₂).

g-1(B2)≤g-1(B′2),⇓1(g-1(B2))≥⇓1(g-1(B′2)),f∘⇓1(g-1(B2))≥f∘⇓1(g-1(B′2)),⇓2(B2)≥⇓2(B′2),⇑2⇓2(B2)≤⇑2⇓2(B′2),B2)≤B′2.

Thus, g⁻¹ is a detect-ordered map on image sets.

Similarly, it can be shown that if g⁻¹ is an order-preserving map, then f is a map that detects the ordering on image sets. Again, if g is detected as an ordered map, then f is an orderpreserving map on image sets.

(5) Let (A₁, g⁻¹(B₂)) ∈ L_o(F₁). Then A1=R1′¯(g-1(B2)) and g⁻¹(B₂) = R1¯(A1).

f(A1)=f∘R1′¯(g-1(B2)) =f(¬⇓1(g-1(B2))) =¬f(⇓1(g-1(B2))) =¬⇓2(B2) =¬¬R2′¯(B2) =R2′¯(B2),g-1(B2)=R1_(A1) =⇑1(¬A1) =g-1∘⇑2∘f(¬A1) =g-1∘⇑2(¬f(A1)) =g-1∘R2_(f(A1)).

Since g⁻¹ is a one-to-one function, then B₂ = R₂(f(A₁)).

Thus (f(A₁), B₂) ∈ L_o(F₂).

(6) Let (A₁, g⁻¹(B₂)) ∈ L_p(F₁). Then A1=R1′_(g-1(B2)) and g-1(B2)=R1¯(A1).

f(A1)=f∘R1′_(g-1(B2)) =f(⇓1(¬g-1(B2))) =⇓2(¬B2) =R2′_(B2),g-1R2¯(f(A))=g-1(¬⇑2(f(A1))) =¬g(⇑2(f(A1))) =¬⇑1(A1) =¬¬R1¯(A1) =R1¯(A1) =g(B2).

Since g⁻¹ is a one-to-one function, then R2¯(f(A))=B2. Thus (f(A₁), B₂)) ∈ L_p(F₂).

(7) Let (A₁, B₁) ∈ L_q(F₁). Then A₁ =⇓ (B₁) and B₁ =⇑ (A₁).

⇑2(f(A1))=(g∘⇑)(A1) =g(B1),⇓2(g(B1))=(⇓2∘g)(B1) =(f∘⇓1)(B1) =f(⇓1(B1)) =f(A1).

Thus (f(A₁), g(B₁)) ∈ L_q(F₂).

Theorem 4.5

If (A, g⁻¹(B)) ∈ L_q(F₁) implies (f(A), B) ∈ L_q(F₂), then (f, g) is a morphism of FCI.

Proof: Let (A₁, g⁻¹(B₂))∈L_q(F₁), then A₁=⇓₁ (g⁻¹(B₂)) and g⁻¹(B₂) =⇑₁ (A₁). Again (f(A₁), B₂) ∈ L_q(F₂), then f(A₁) =⇓₂ (B₂) and B₂ =⇑₂ (f(A₁)). Now ⇑₁ (A₁) = g⁻¹(B₂) = g⁻¹(⇑₂ (f(A₁))). Thus, ⇑₁= g⁻¹ο ⇑₂ οf. Again ⇓₂ (B₂) = f(A₁) = f(⇓₁ (g⁻¹(B₂))). Thus ⇓₂= fο ⇓₁ οg⁻¹. Hence, (f, g) is a morphism of the category FCI.