Representation theorem for finite intuitionistic fuzzy perfect distributive lattices

In this paper, we extend some results obtained by A. Amroune and B. Davvaz in [1]. More precisely, we will develop a representation theory of intuitionistic fuzzy perfect distributive lattices in the finite case. To that end, we introduce the notion of intuitionistic fuzzy perfect distributive lattices and the one of fuzzy perfect Priestley spaces. In this way, the results of A. Amroune and B. Davvaz are extended to the intuitionistic fuzzy perfect case and the equivalence between the category of finite intuitionistic fuzzy perfect Priestley spaces the dual of the category of finite intuitionistic fuzzy perfect distributive lattices is proved.


Introduction
The representation theorems appeared in the thirties of the last century; M. Stone [19] proved that every Boolean algebra is isomorphic to a set of {I a : a ∈ A} (where I a denotes the set of prime ideals of A not containing a).The representation theorem for distributive lattices proved by Birkhoff [3]; asserts that any finite distributive lattice L is isomorphic to the lattice of the ideals of the partial order of the join-irreducible elements of L. In series of papers, Priestley [15,16], gave a theory of representation of distributive lattices.The duality is central in making the link between syntactical and semantic approaches to logic, also in theoretical computer science this link is central as the two sides correspond to specification languages and the space of computational states.This ability to translate faithfully between algebraic specification and spatial dynamics has often proved itself to be a powerful theoretical tool as well as a handle for making practical problems decidable.Topological duality for Boolean algebras [18] and distributive lattices [19] is a useful tool for studying relational semantics for propositional logics.Canonical extensions [10,11,12,13], provide a way of looking at these semantics algebraically.Priestley's duality for bounded distributive lattices has enjoyed growing attention and has been variously applied in the international literature since its inception in 1970.Since their first appearance in 1971, [22] fuzzy relations have known a vast development, as well many notions and results of the ordered sets theory have been extended to fuzzy ordered sets.One of the most important approaches and theories treating generalization of ordinary fuzzy relations was the concept of intuitionistic fuzzy sets introduced by Atanassov in [2].In [20], Venugopalan introduced a definition of a fuzzy ordered set (foset) (P, µ) and he extended this concept to obtain a fuzzy lattice in which he defined a (fuzzy) relation as a generalization of equivalence.In 2009 Chon defined a fuzzy lattice as a fuzzy relation and developed some basic properties of fuzzy lattices, [7].In [1], Amroune and Davvaz gave a representation theory of fuzzy distributive lattices in the finite case.In this paper, we extend some results of [1,15,16], more precisely, we give a representation theory of perfect intuitionistic fuzzy distributive lattices in the finite case.This paper is organized as follows: In the next section, basic definitions and notions are presented.In the third section, we give and prove the main result using a definition of intuitionistic fuzzy perfect ordering relation.

Preliminaries
In the following we recall some definitions of the intuitionistic fuzzy sets, intuitionistic fuzzy relations [2], [5].
Definition 2.1.[6] Let L * = { (a 1 , a 2 ) ∈ [0, 1] 2 : a 1 + a 2 ≤ 1 } and ≤ L * be an order in L * defined by ∀(a 1 , a 2 ), (b Let X be a given non-empty set.An intuitionistic fuzzy set in X is an expression A given by x ∈ X (We will denote by A = (µ A , ν A )).The numbers µ A (x) and ν A (x) denote respectively the degree of membership and the degree of non-membership of the element x in the set A. We will denote by IFS S (X) the set of all intuitionistic fuzzy sets on X.In particular, 0 and 1 denote the intuitionistic fuzzy empty set and the intuitionistic fuzzy whole set in X defined by 0 (x) = (0, 1) and 1 (x) = (1, 0) for each x ∈ X, respectively.Obviously, when ν A (x) = 1 − µ A (x) for every x in X, the set A is fuzzy set.We will denote the set of all IFSs in X by IFS(X).Definition 2.2.The following expressions are defined in [2] for all intuitionistic fuzzy sets A, B in X;

A = B if and only if A ⊆ B and B
An intuitionistic fuzzy relation (for short, IFR) R is an intuitionistic fuzzy subset of X ×Y given by the expression In particular, if R is an intuitionistic fuzzy relation from X to itself, then R is called a binary intuitionistic fuzzy relation on X, and we will denote the set of all intuitionistic fuzzy relations on X by: IFR(X).

transitive if and only if for all
A reflexive, perfect antisymmetric intuitionistic and transitive intuitionistic fuzzy relation is called an intuitionistic fuzzy perfect partial ordering relation.An intuitionistic fuzzy perfect partial order relation R is an intuitionistic fuzzy perfect total order relation if and only if (µ R (x, y) > 0 and ϑ R (x, y) < 1) or (µ R (y, x) > 0 and ϑ R (y, x) < 1) for all x, y ∈ X.A set equipped with an intuitionistic fuzzy perfect partial order relation is called an intuitionistic fuzzy perfect poset.The height of R, denoted by h(R), is defined by: h(R) = ∨ ≤ L * {(x,y)∈X 2 :x̸ =y} R (x, y) .

Intuitionistic fuzzy perfect lattices
In this section, we first extend the concept of fuzzy lattices studied in [7], to intuitionistic fuzzy case.Hence, we extend some results in this direction.The fo1lowing definition introduce the intuitionistic fuzzy lattice as a relational structure.Definition 2.4.Let (X, R) be an intuitionistic fuzzy perfect poset and let A be a subset of X.An element u ∈ X is said to be an upper bound for A if and only if µ R (a, u) > 0 and ϑ R (a, u) < 1, for all a ∈ A. An upper bound u 0 for A is the least upper bound of A if and only if µ R (u 0 , u) > 0 and ϑ R (u 0 , u) < 1, for every upper bound u of A. An element l ∈ X is said to be a lower bound for A if and only if µ R (l, a) > 0 and ϑ R (l, a) < 1, for all a ∈ A. A lower bound l 0 of A is the greatest lower bound of A if and only if µ R (l, l 0 ) > 0 and ϑ R (l, l 0 ) < 1, for every lower bound l of A.
The least upper bound and the greatest lower bound of a set {x, y} are denoted by x ∨ y and x ∧ y respectively.Definition 2.5.Let (X, R) be an intuitionistic fuzzy perfect poset.(X, R) is an intuitionistic fuzzy perfect lattice if and only if x ∨ y and x ∧ y exist for all x, y ∈ X.
Proposition 2.2.Let (X, R) be an intuitionistic fuzzy perfect lattice and x, y, z ∈ X.Then Proof.Straightforward.

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The following definition and theorem give a characterizations of intuitionistic fuzzy perfect distributive lattices.
Theorem 2.1.Let (X, R) be an intuitionistic fuzzy perfect totally ordered set.Then (X, A) is an intuitionistic fuzzy perfect distributive lattice. Proof.Straightforward.
Definition 2.7.Let (X, R, ∧, ∨) be an intuitionistic fuzzy perfect lattice, and F be a nonempty crisp subset of X. F is a filter of (X, R, ∧, ∨) if for all x, y ∈ X, it holds that Definition 2.8.Let (X, R, ∧, ∨) be an intuitionistic fuzzy perfect lattice and F be a filter of (X, R, ∧, ∨).Then F is called prime filter if F is proper (F ̸ = X) and for all x, y ∈ X, x ∨ R y ∈ F imply x ∈ F or y ∈ F.

Intuitionistic fuzzy lattices isomorphisms
Next, we extend the concept of fuzzy lattices isomorphism studied in [20], to intuitionistic fuzzy case.

Let f : L → M be a monotone function between intuitionistic fuzzy perfect lattices. Then f is called a lattice homomorphism if for any x, y
If f is a bijection, then f is said to be intuitionistic fuzzy lattices isomorphism.
This proposition is an intuitionistic fuzzy version of [4, Proposition 1.3.9].

F is a prime lter;
2. There is a surjective lattice homomorphism f : This corollary is an intuitionistic fuzzy version of [4,Corollary1.3.13 ].
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Intuitionistic fuzzy perfect Priestley spaces
In this section we extend the notion of increasing (decreasing) subset introduced by [20] and some results obtained by A. Amroune and B. Davvaz in [1] in the intuitionistic fuzzy perfect case.Let (X, R) be an intuitionistic fuzzy perfect ordered set.A subset E of X is called increasing if for all x belongs to E and µ R (x, y) > 0 and ϑ R (x, y) < 1 (y is an upper bound of x), then y belongs to E (a decreasing set is defined in a dually).An intuitionistic fuzzy ordered space is a triplet (X, τ, R) , where X is a non empty set, τ is a topology on X and R is an intuitionistic fuzzy perfect order on X.An intuitionistic fuzzy perfect ordered space (X, τ, R) is called perfect totally order disconnected if for x, y ∈ X, µ R (x, y) = 0 and ϑ R (x, y) = 1, there exists an increasing τ−clopen U and a decreasing τ−clopen V such that U ∩V = / 0 with x ∈ U and y ∈ V .We recall that a τ−clopen set in a topological space is a set which is both open and closed.An intuitionistic fuzzy perfect ordered space (X, τ, R) is called an intuitionistic fuzzy perfect Priestley space if it is compact and perfect totally order disconnected.
2. Let f : X → X ′ be a function between intuitionistic fuzzy perfect Priestley spaces.Then f is called a Priestley spaces homomorphism if is monotone and continuous.If f is a bijection, then f is said to be intuitionistic fuzzy perfect Priestley spaces isomorphism.

Duality for intuitionistic fuzzy perfect distributive lattices
In this section, we extend the concept of Priestley duality for distributive lattices studied in [1,15,16], to intuitionistic fuzzy case.Throughout this section, all fuzzy intuitionistic lattices are perfect distributive lattices and homomorphisms preserve first (0) and last (1) elements.If (A, ∨, ∧, R) is an intuitionistic fuzzy perfect distributive lattice, then its dual space is defined by: T (A) = (X, τ, R 1 ), where X is the set of 0 − 1 homomorphisms from A onto {0, 1} , and τ be the topology induced on X by the product topology on {0, 1} A and R 1 is an intuitionistic fuzzy perfect order adequately chosen on X.Indeed, R 1 is defined by R, see Lemma 3.1.If δ = (X, τ, r) is an intuitionistic fuzzy perfect Priestley space, then its dual is defined by: (L (δ ) , ∨, ∧, r 1 ), where L (δ ) = {Y ⊆ X|Y is increasing and τ−clopen} and r 1 is an intuitionistic fuzzy perfect order adequately chosen.Lemma 3.1.If (A, ∨, ∧, R) is an intuitionistic fuzzy finite perfect distributive lattice, then there exists two intuitionistic fuzzy orders R 1 , R 2 such that: otherwise.where the symbol ∧ stands for an infimum with respect to the intuitionistic fuzzy relation R. We show that R 1 is an intuitionistic fuzzy perfect order.We have µ R

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Hence, R 1 is perfect antisymmetric relation.In order to verify the transitivity of R We use the following truth table, where the proposition The only case for investigating is f −1 (1) ⊂ g −1 (1) and g −1 (1) ⊂ h −1 (1).By the transitivity of R, for every a, b, c in ) .
Then for all f , g, h Hence, R 1 is an intuitionistic fuzzy perfect order and by [15], [16] ) is an intuitionistic fuzzy perfect Priestley space. ( International Scientific Publications and Consulting Services and otherwise.for all H, D ∈ L(T (A)), where the symbol ∧ stands for an infimum with respect to the fuzzy relation R. First, we show that R 2 is an intuitionistic fuzzy perfect order.Since, µ R 2 (H, D) In order to verify the transitivity, we use the following truth table, where the proposition and Table 4: (P').
First, if one of the three elements H, D, E is empty, then the transitivity is a trivial fact.
If H ̸ = ϕ and D ̸ = ϕ and E ̸ = ϕ , the only case that need investigation is when is the least upper bound of {H, D} we have four cases: Similarly, we prove that H ∧ R 2 D = H ∩ D, its known that H ∪ D and H ∩ D are increasing and τ−clopens.This shows that (L(T (A)), ∨, ∧, R 2 ) is a fuzzy distributive lattice.
Proof.(1) (i) If h(r) = (0, 1), then X is an antichain and we can write r 1 as follows: r 1 is an intuitionistic fuzzy relation.It is easy to show that r 1 is an intuitionistic fuzzy perfect order, and A∨ r 1 B = A∪B, A ∧ r 1 B = A ∩ B exists for every A and B from L(δ ), and A ∪ B, A ∩ B are increasing and τ-clopens sets of L(δ ), where (L(δ ), ∨, ∧, r 1 ) is an intuitionistic fuzzy perfect distributive lattice.If h(r) ̸ = 0, then X is not an antichain, setting M = (M 0 , M 1 ) such that M = ∧ ≤ L * {r (x, y) /x, y ∈ X, x ̸ = y and r (x, y) ̸ = (0, 1)}.Then, M ̸ = (0, 1) and we can take r 1 = (µ r 1 , ϑ r 1 ) such that for every A and B from L(δ ) Similar to the previous lemma, r 1 is an intuitionistic fuzzy perfect order and we can assume that A ∨ r 1 B = A ∪ B and A ∧ r 1 B = A ∩ B, for every A and B from L(δ ), where (L(δ ), ∨, ∧, r 1 ) is an intuitionistic fuzzy perfect distributive lattice.
(2) To proof the second assertion, let r 2 = (µ r 2 , ϑ r 2 ) , such that International Scientific Publications and Consulting Services otherwise.where the first infimum ∧ is in the sense of the intuitionistic fuzzy perfect ordering relation r and the second infimum ∧ is in the sense of the intuitionistic fuzzy relation r 1 .Note that r 2 is well defined: A, where the symbol ∧ stands for an infimum with respect to the intuitionistic fuzzy perfect ordering relation r 1 , it exists because L(δ ) is a lattice and a = ∧A 1 , where the symbol ∧ stands for an infimum with respect to the intuitionistic fuzzy relation r.
Then, a exists because A 1 is a finite increasing τ-clopen, if A 1 has two minimal elements x, y, then r (x, y) = (0, 1), there exists an increasing τ−clopen U and a decreasing τ− clopen V such that U ∩V = / 0 with x ∈ U and y ∈ V .It is easy to see that A 1 ⊂ U, then U ∩V ̸ = / 0 contradiction.By definition r 2 is an intuitionistic fuzzy relation.It is easy to show that r 2 is an intuitionistic fuzzy perfect ordering relation.Furthermore, By [15], [16] (T (L(δ )), τ, r 2 ) is an intuitionistic fuzzy perfect Priestley space.
The following, shows that the category of finite intuitionistic fuzzy perfect Priestley spaces is equivalent to the dual of the category of finite intuitionistic fuzzy perfect distributive lattices.

Lemma 3.3. Let A be an intuitionistic fuzzy perfect distributive lattice. The map F
. Therefore F A is prime filter.Suppose that a ̸ = b, it follows R (a, b) = (0, 1) or R (b, a) = (0, 1) .If R (a, b) = (0, 1) then, there exist a prime filter F such that a ∈ F and b / ∈ F, it follows that there exists a surjection otherwise.and the symbol ∧ stands for an infimum with respect to the fuzzy relation R. Note that F A (x) ̸ = ϕ for all x ∈ A.
has minimal element z ̸ = x, then r (x, z) = (0, 1), there exists an increasing τ−clopen U and a decreasing τ−clopen V such that U ∩V = / 0 with x ∈ U and z ∈ V .It is easy to see that It follows that the map F A is an intuitionistic fuzzy perfect lattice isomorphism.
• f is a homomorphism of intuitionistic fuzzy perfect Priestley space, i.e., a continuous and increasing map .
Proof.For all The continuity of T ( f ) follows from the fact that for every a ∈ A 1 , for all Y ∈ L(δ ) is an isomorphism of intuitionistic fuzzy perfect Priestley space, i.e., a bijection, continuous and increasing map.
To prove the injectivity, let To prove that G δ is continuous, let Z a τ−clopen of T (L (δ )) .Then, there exist y ∈ L (δ ) such that Y = F L(δ ) (y) .

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Hence, G δ is continuous.
To prove that G δ is increasing it suffices to show that r(x, y) otherwise.

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Then its dual is: T and its bidual is: , where R 2 is given by: And r 2 will be given by: and the isomorphism G X is defined by:G X : X → T (L (X)), where Example 3.3.Let (X, τ, r) be a Priestley space, where X = {x, y, z,t} and r is given by: The isomorphism G X is defined as follows: G X : X → T (L(X))

Conclusion
In this paper, we have proposed a way to represent finite intuitionistic fuzzy perfect distributive lattices.This, by constructing adequate intuitionistic fuzzy perfect orders.In this context, a theory of representation of finite intuitionistic fuzzy perfect distributive lattices in the finite case is presented.The main result extends the one obtained in [1] and shows that the category of finite intuitionistic fuzzy perfect Priestley spaces is equivalent to the dual of the category of finite intuitionistic fuzzy perfect distributive lattices.

Theorem 3 . 3 .Example 3 . 1 .
The dual of the category of intuitionistic fuzzy distributive perfect lattices is equivalent to the category of intuitionistic fuzzy perfect Priestley spaces.Proof.Lemma 3.3, Lemma 3.5, Theorem 3.1 and Theorem 3.2 establish the functorial isomorphisms.Let (A, ∨, ∧, R) be an intuitionistic fuzzy perfect distributive lattice, where A = {a, b, c, d, e, f } and R is an intuitionistic fuzzy perfect relation defined by: International Scientific Publications and Consulting Services