Poset valued intuitionistic fuzzy sets

Abstract

In this paper poset valued intuitionistic fuzzy sets are introduced and investigated (for posets in general). It is proved that in case of arbitrary finite posets, represented as a poset of a convenient distributive lattice, a natural version of the definition is obtained, using join-irreducible elements. Two families of cuts and their properties are investigated. Necessary and sufficient conditions under which two families of subsets are families of cuts of a poset valued intuitionistic fuzzy set are given.

Keywords

Poset-valued intuitionistic fuzzy sets cut-sets representation by cuts

1 Introduction

One of the generalizations of the notion of the fuzzy set (introduced by Zadeh in 1965) is obtained by considering two functions instead of one: the membership and the non-membership function. Using this idea, the concept of intuitionistic fuzzy set is introduced by K. Atanassov in [1 , 4]. An intuitionistic fuzzy set on a set X is an ordered triple (X, μ, ν), where μ and ν are both functions from X to the [0, 1] real interval, such that for every x ∈ X, μ (x) + ν (x) ≤1.

Further generalizations of intuitionistic fuzzy sets are obtained by replacing a codomain of the membership and non-membership functions by a richer structure (e.g. a complete lattice, which follows the notion of a lattice valued fuzzy set introduced by Goguen in 1967). In this way, Atanassov and Stoeva defined a lattice valued intuitionistic fuzzy set (an intuitionistic L-fuzzy set) [3], with the codomain being a special type of a complete lattice, on which an involutive order reversing unary operation can be defined.

In this context, if L is a complete lattice with $N : L \to L$ , an order reversing unary operation, then a lattice valued intuitionistic fuzzy set (LIFS) is the ordered triple: (X, μ, ν) where μ and ν are functions μ : X → L, ν : X → L, such that for all x ∈ X,

$μ (x) ⩽ N (ν (x)) .$ (1)

In the papers [7] and [11] the disadvantages of this definition are presented, as e.g. that the requirement of existence of an involutive order reversing unary operation for a lattice is rather a strong condition, therefore the conclusion is that a LIFS could not be defined for a large class of lattices. Therefore, some new definitions of lattice valued intuitionistic fuzzy sets have been proposed in papers [6, 7] and [11].

The definition proposed in [6] uses a linearization function and the one proposed in [7] a lattice homomorphism (as a special case of linearization function) as tools for connection membership and non-membership functions.

However, further disadvantages of these definitions were pointed out in [11], together with a proposal for the most general and most convenient concept of lattice valued intuitionistic fuzzy sets.

Besides these, there are several papers in which the relationship between intuitionistic fuzzy sets and some other extensions of fuzzy sets theory have been investigated, out of which we mention here the paper [5].

In the investigation of several types of fuzzy sets (in which the codomain is a kind of ordered structure), cut sets are a very useful tool to give the connection of fuzzy structures and crisp structures of the corresponding type. Cut sets of fuzzy sets are ordinary sets, cut relations are ordinary relations which preserve some properties of the fuzzy structures: e.g. cuts of fuzzy equivalence relations are ordinary equivalences, cuts of fuzzy groups are ordinary groups and similar.

A lot of properties of fuzzy structures can be investigated just using a related family of cut sets. These properties are known as “cutworthy” properties (see e.g., [15]).

In the paper [14] a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a poset. The co-domain for this type of preference relations is a special type of poset, which is called a weakly orthocomplemented posets and which have at least one fixed point under involution.

In this paper a new definition of a poset valued intuitionistic fuzzy set in which a codomain can be an arbitrary poset is given. Since every lattice is a poset, this approach is applicable also in the lattice setting. A motivation for the new approach is threefold. First: to generalize the notion of a lattice valued intuitionistic fuzzy sets. Second: to enable definition of (poset valued and lattice valued) intuitionistic fuzzy sets with all types of ordered sets as co-domains. Three: to develop the theoretical basis which will enable exploitation of the new notion e.g. in preference relations and in other areas.

The advantages of this definition are numerous. Since every lattice is also a poset, this procedure can be applied to every lattice. Hence, all lattice-codomains can be considered in this setting and no extra order reversing unary operation is needed to mimic complements. In this approach natural complements are obtained by coordinatization using join-irreducible elements. Hence, all the disadvantages from the definition from [3] are removed. Moreover, there is no necessity to introduce external functions, like linearization functions proposed in [6] and a lattice homomorphism proposed in [7]. Finally, this definition is more general than the definition from [11], since the definition from [11] enable only lattice co-domains. One great advantage of the new definition is that it is superior over the definition from the paper [14] since the definition from [14] requires some special type of posets, while the new definition is suitable for all posets.

Moreover, in case when the poset is finite, it can be represented as a subset of {0, 1}ⁿ using join-irreducible elements of posets, and then, our representation yields a natural definition of this concept for finite posets.

The section Preliminaries contains basic notions of posets, necessary for our approach, including a material on join-irreducible elements. The section Poset-valued intuitionistic fuzzy sets contains the main results of the paper.

2 Preliminaries

In this part, basic notions and properties of posets are given, as well as the representation of finite posets as subsets of a Boolean algebra {0, 1}ⁿ, using complete join-irreducible elements of the poset (see e.g. [25] for the dual representation by meet-irreducible elements).

A more comprehensive presentation on posets can be found e.g. in [8].

A partially ordered set, poset, is an ordered pair of a nonempty set and a relation (P, ⩽), where the relation “ ⩽ ” is reflexive, antisymmetric and transitive relation on P. A poset (P, ⩽) is sometimes denoted by the underlying set P only.

If (P, ⩽) is a poset, then (P, ⩾) is also a poset, where the two ordering relations are connected as follows:

x ⩾ y if and only if y ⩽ x. Poset (P, ⩾) is said to be dual to poset (P, ⩽). Sometimes the relation ⩾ is used when dealing with poset (P, ⩽).

If there is a least (bottom) element of a poset, then it is denoted by B and if there is a greatest (the top) element of P, it is denoted by T.

If P is a poset, then Q ⊆ P is the down-set if from p ∈ Q and q ⩽ p, it follows that q ∈ Q. The dual notion is called up-set: it is a subset Q ⊆ P, such that for p ∈ Q and p ⩽ q, it follows that q ∈ Q. A special kind of down-set for p ∈ P is the principal filter generated by p, denoted by p↑: $p ↑ = {x ∣ x \in P & p ⩽ x} .$

Dually, the principal ideal generated by p, denoted by p↓ is $p ↓ = {x ∣ x \in P & p ⩾ x} .$

A function f : P → Q from a poset (P, ⩽) to poset (Q, ⩽) is isotone if from x ⩽ y it follows that f (x) ⩽ f (y), for all x, y ∈ P. If f is also an injection, it is called an order embedding from P into Q, and if it is bijective it is an order isomorphism.

Let (P, ⩽) be a poset. For an element a ∈ P, if a ¬ = B (if the least element in P exists), a is a completely join-irreducible element, if from a = ⋁ I, for every I ⊂ P, it follows that a = x_i for some i ∈ I. It is easily proved by induction, that in every poset, every element is a join of completely join-irreducible elements.

Equivalently, every element in a poset is a join of all completely join-irreducible elements below it (or equal to it).

Let $J$ be a set of all completely join-irreducible elements of a finite poset P.

Let $| J | = n$ and let f be a bijection $f : {1, . . ., n} \to J$ .

Now, a mapping from P → {0, 1}ⁿ is considered, defined by F (x) = (a₁, …, a_n), where a_i = 1 if and only if f (i) ⩽ x.

If P has the bottom element B, then F (B) = (0, …, 0).

Mapping F is well defined, since every element is uniquely represented by the join of completely join-irreducible elements. Hence, if a poset P has n completely join-irreducible elements, then the poset P is represented as a subset of {0, 1}ⁿ.

A collection $F$ of subsets of a nonempty set X union of which is $F$ such that it is closed under the componentwise intersections, i.e., if $⋂ {Y ∣ Y \in F & x \in Y} \in F$ is true for every x ∈ X is called a point closure system or a centralized systemon X. A centralized system is a generalization of a closure system (which is a collection of subsets closed under all intersections).

Let X be a nonempty set, and P a poset, then a mapping μ : X ⟶ P is a poset-valued fuzzy set (P-valued fuzzy set) [22].

For p ∈ P, a subset of X denoted by μ_p, and defined by μ_p = {x ∣ x ∈ X & μ (x) ⩾ p}, is called a cut or p-cut of a fuzzy set. Basic properties of cuts can be found e.g. in [20 –22].

The following lemma is a well known theorem of synthesis of poset valued fuzzy sets [22].

Lemma 1. Let $P$ be a centralized system of subsets of X.

$μ : X ⟶ P$ , where the codomain is $(P, \supseteq)$ , defined by: $μ (x) : = ⋂ {p ∣ p \in P & x \in p}$ is a P-valued fuzzy set on X, and for every p such that $p \in P$ , p = μ_p.□

The following theorem from [26] gives necessary and sufficient conditions under which a family of subsets on a set is a family of cuts of a poset-valued fuzzy set.

Theorem 1. [26] Let $F$ be a family of subsets of a nonempty set X and let (P, ⩽) be a poset. Then, there is a fuzzy set μ : X ⟶ P such that $F$ is its collection of cuts if and only if the following are satisfied:

$F$ is closed under centralized intersections and $⋃ F = X$ .

There is an isotone function E : Y ⟶ P from the poset (Y, ⊇) to (P, ≤), where $Y = {Z_{x} ∣ x \in X and Z_{x} = ⋂ {f ∣ f \in F & x \in f}}$ , such that for every r such that r ∈ P, , ⋃ E^-1(↑ r ∩ E (Y)) ∈ $F$ , and the mapping $P ⟶ F$ , defined by Φ (r) = ⋃ E^-1(↑ r ∩ E (Y)) is ’onto’.□

Remark. The function E appearing in Theorem 1 is not only isotone, but also an order embedding from (Y, ⊇) to (P, ⩽).

Corollary 1. Let (P, ⩽) be a poset, X a nonempty set and $F$ a collection of some of its subsets whose union is X and which is a point closure system. Let also there exists an order embedding φ from the poset $(F, \supseteq)$ into (P, ⩽), such that for every r ∈ P, $↑ r \cap φ (F) = ↑ φ (f), for some f \in F .$ (2)

Then there is a P-valued fuzzy set μ : X ⟶ P, such that $F$ coincides with its collection of cut sets. □

3 Poset-valued intuitionistic fuzzy sets

Let X be a nonempty set and P be an arbitrary poset. Let μ : X ⟶ P and ν : X → P be two functions from X to P.

Then (X, μ, ν) is a poset valued intuitionistic fuzzy set on set X if for each x ∈ X, $μ (x) ↓ \cap ν (x) ↓ \subseteq S,$ where S is {B} if P has the bottom element B, and S =∅ otherwise.

As usual for intuitionistic fuzzy sets, the function μ is called the membership function, where μ (x) represents belonging of an element x to the intuitionistic fuzzy set and the function ν is called the non-membership function, ν (x) representingnon-belonging of an element x to the intuitionistic fuzzy set.

The notions of related cut sets (two natural types) for poset valued intuitionistic fuzzy sets are defined in the sequel.

For each p ∈ P, two types of cut sets of (X, μ, ν) are defined as follows: $\begin{matrix} μ_{p} & = & {x ∣ x \in X & μ (x) ⩾ p} and \\ ν_{p} & = & {x ∣ x \in X & ν (x) ⩽ p} . \end{matrix}$

By $M_{P}$ and $N_{P}$ two families of cut sets of (X, μ, ν) are denoted:

$M_{P} = {μ_{p} ∣ p \in P}$ and $N_{P} = {ν_{p} ∣ p \in P}$ .

Sometimes, elements from $M_{P}$ are called μ-cuts and elements from $N_{P}$ are called ν-cuts.

In the following proposition, the properties of both types of cuts are listed, some of which are consequences of known facts of poset-valued fuzzy sets. Most of the proofs are easy and omitted. In the proofs for properties of family $N_{P}$ , the known fact that the ν-cuts are the usual cuts of a poset valued fuzzy set, as a mapping from X to the dual poset (P, ⩾) is used.

Proposition 1. Let P be a poset, X a non-empty set, and (X, μ, ν) is a poset valued intuitionistic fuzzy set. Then, the following is valid.

1. If there is the bottom element B of poset P, then μ_B = X. If there is the top element T of poset P, then ν_T = X.

2. If p ⩽ q, then μ_q ⊆ μ_p, and ν_p ⊆ ν_q .

μ (x) =⋁ {p ∣ p ∈ P & x ∈ μ_p} ;

ν (x) = ⋀ {p ∣ p ∈ P & x ∈ ν_p} .

This means that the infimum and the supremum on the right side of the equalities exist and the equalities are valid.

4. If P₁ ⊆ P, is a subposet of P which has the supremum, then $⋂ (μ_{p} ∣ p \in P_{1}) = μ_{⋁ {p ∣ p \in P_{1}}}$ and similarly, if P₁ has the infimum, then $⋂ (ν_{p} ∣ p \in P_{1}) = ν_{⋀ {p ∣ p \in P_{1}}} .$

5. $M_{P}$ and $N_{P}$ ordered by inclusion are centralized systems.

6. ⋃ {μ_p ∣ p ∈ P} = X; ⋃ {ν_p ∣ p ∈ P} = X

7. For x, y ∈ X, μ (x) ≤ μ (y) if and only if μ_μ(y) ⊆ μ_μ(x) and ν (x) ≤ ν (y) if and only if ν_ν(x) ⊆ ν_ν(y) .

8. For every p ∈ P, μ_p ∩ ν_p ⊆ ν^-1(B) , if there is the bottom element B in poset, otherwise, μ_p ∩ ν_p = ∅.

Proof. The statement 3. follows from the fact that x ∈ μ_μ(x) and x ∈ ν_ν(x). For the proof of all other statements can be found in [20–].

Now, only 8 have to be proved. Take p ∈ P and suppose that x ∈ μ_p ∩ ν_p. Then, p ⩽ μ (x) and ν (x) ⩽ p, i.e., ν (x) ⩽ μ (x). Now $μ (x) ↓ \cap ν (x) ↓ = ν (x) ↓ \subseteq S,$ where S is {B} if P has the bottom element B, and S =∅ otherwise. This means that ν (x) ↓ ⊆ {B}, hence ν (x) = B, or such x does not exist. Therefore, x ∈ ν^-1(B) and μ_p ∩ ν_p ⊆ ν^-1(B) . □

In the following part the new introduced definition will be justified analyzing the case of finite codomains (it is also possible to be done in case of spatial posets: the ones in which every element is a join of join-irreducible elements).

In the sequel, every finite poset P is considered as a subset of ${0, 1}^{J}$ , where the set of join-irreducible elements of a poset is an index set $J$ with the cardinality n.

For the sake of simplicity, a finite set of cardinality $J = n$ is ordered, so sometimes elements from ${0, 1}^{J}$ are considered as ordered n-tuples.

Let X be a non-empty set. Now, considering a poset P as a poset of ${0, 1}^{J}$ , in the following proposition it is proved that a poset valued intuitionistic fuzzy set on set X is an ordered triple (X, μ, ν), where μ and ν are mappings from X to $P \subseteq {0, 1}^{J}$ , such that $μ (x) (i) + ν (x) (i) ⩽ 1,$ for all x ∈ X and all $i \in J$ .

In other words, it is shown that the sum of values of membership function and non-membership function on each coordinate is less or equal to 1.

In case when the poset is a lattice, this definition is somehow related to earlier proposed definitions of lattice valued intuitionistic fuzzy sets (see the paper [11]).

Theorem 2. Let X be a non-empty set, (P, ≤) a finite poset with the set of join-irreducible elements $J$ , represented by a subset of ${0, 1}^{J}$ and let μ : X → P and ν : X → P be two arbitrary functions. Then (X, μ, ν) is a poset valued intuitionistic fuzzy set on set X if and only if μ (x) (i) + ν (x) (i) ≤1, for all x ∈ X and all $i \in J$ .

Proof. Let P be a finite poset, represented by a subset of ${0, 1}^{J}$ , where the set of join-irreducible elements of a poset is $J$ with the cardinality n. Since the set $J$ is ordered, elements of P are considered as ordered tuples, and elements from $J$ as coordinates.

Now, let (X, μ, ν) be an ordered triple, where μ : X → P and ν : X → P are two functions from X to P. Since P is represented by a subset of ${0, 1}^{J}$ , each element of P is considered as a mapping from $J$ to {0, 1}. Therefore, for every x ∈ X, μ (x) and ν (x) are mappings from $J$ to {0, 1}, and with μ (x) (i) ν (x) (i) values of μ (x) and ν (x) are denoted for $i \in J$ (in other words, coordinate i of values of μ (x) and ν (x)). In this denotation, $μ (x) (i) + ν (x) (i) ⩽ 1, for all x \in X and all i \in J .$

This means that for some $i \in J$ , if μ (x) (i) =1, then ν (x) (i) =0.

This means that if a completely join-irreducible element i, i ⩽ μ (x), then, inotleqslantν (x) and vice versa. Hence, $μ (x) ↓ \cap ν (x) ↓ \cap J = \emptyset$ . Therefore, if there is a bottom element B is poset P, then μ (x) ↓ ∩ ν (x) ↓ ⊆ {B}, otherwise μ (x)↓ ∩ ν (x) ↓ = ∅, which proves that (X, μ, ν) is a poset valued intuitionistic fuzzy set.

To prove the converse, if (X, μ, ν) is a poset valued intuitionistic fuzzy set, i.e., if μ (x)↓ ∩ ν (x) ↓ = ∅ or μ (x) ↓ ∩ ν (x) ↓ ⊆ {B}, then $μ (x) ↓ \cap ν (x) ↓ \cap J = \emptyset$ . This means that there are no join-irreducible elements at the same time below μ (x) and ν (x), which means that at each coordinate either μ (x) =1 and ν (x) =0, or μ (x) =0 and ν (x) =1, or both elements are 0. Therefore, μ (x) (i) + ν (x) (i) ⩽1 for every x and every i. □

This proposition shown that the definition of a poset valued intuitionistic fuzzy set is very natural. It also has an advantage that it is applicable on any poset (and also on a lattice, giving a new framework of lattice valued intuitionistic fuzzy sets).

The following example illustrates the introduced notions.

Example. Let (P, ⩽) be given in Fig. 1. The join-irreducible elements of this poset are denoted by filled circles (all the elements here, except e, are join-irreducibles). There are five join-irreducible elements indexed, as follows:

$\begin{matrix} 1 & 2 & 3 & 4 & 5 \\ a & b & c & d & f \end{matrix}$

and the following mapping from P to {0, 1}⁵ is obtained:

$\begin{matrix} a & b & c & d & e & f \\ 10000 & 01000 & 00100 & 11010 & 11100 & 01101 \end{matrix}$

Let X = {x, y, z, u, v, w} and let μ and ν be defined by: $μ (x) = (\begin{matrix} x & y & z & u & v & w \\ d & d & f & a & b & c \end{matrix}); ν (x) = (\begin{matrix} x & y & z & u & v & w \\ c & c & a & b & c & a \end{matrix}) .$

Obviously, the triple (X, μ, ν) is a poset valued intuitionistic fuzzy set (by Theorem 2).

Cuts of this fuzzy set, are as follows:

μ_d = {x, y} , ν_d = {z, u, w}

μ_e =∅, ν_e = {x, y, z, u, v, w}

μ_f = {z}, ν_f = {x, y, u, v}

μ_a = {x, y, u}, ν_a = {z, w}

μ_b = {x, y, z, v}, ν_b = {u}

μ_c = {z, w}, ν_c = {x, y, v}.

It can be checked that Proposition 1 is true, in particular the intersection of μ_i∩ ν_i = ∅ for every i ∈ {a, b, c, d, e, f} . □

In the following necessary and sufficient conditions under which two families of subsets of a set X are families of the cut sets of a poset valued intuitionistic fuzzy set are given.

Theorem 3. Let X be a set and let P be a poset. Let $M$ and $N$ be two families of subsets of a set X.

There is a poset valued intuitionistic fuzzy set on X, with the codomain P, such that $M$ and $N$ are its families of cut sets if and only if the following conditions are satisfied:

$M$ and $N$ are closed under centralized intersections and the union of each of the family is X.

There is an isotone function E : Y ⟶ P from the poset (Y, ⊇) to (P, ⩽), where $Y = {Z_{x} ∣ Z_{x} = ⋂ {f ∣ f \in M & x \in f}}$ , such that for every r ∈ P, ⋃E^-1(↑ r ∩ E (Y)) ∈ $M$ , and the mapping $Φ : P ⟶ M$ , defined by Φ (r) = ⋃ E^-1(↑ r ∩ E (Y)) is ’onto’.

There is an isotone function G : Y ⟶ P from the poset (W, ⊆) to (P, ⩽), where $W = {T_{x} ∣ T_{x} = ⋂ {f ∣ f \in N & x \in f}}$ , such that for every r ∈ P, ⋃G^-1(↑ r ∩ G (W)) ∈ $N$ , and the mapping $Ψ : P ⟶ F$ , defined by Ψ (r) = ⋃ G^-1(↑ r ∩ G (W)) is ’onto’.

For every x ∈ X, E (Z_x) ↓ ∩ G (T_x) ↓ ⊆ S, where S is {B} if P has the bottom element B, and S =∅ otherwise.

If the poset P is finite, then the condition (4) can be interchanged by the following condition:

For every x ∈ X, E (Z_x) (i) + G (T_x) (i) ≤1 on each coordinate i, where poset P is represented as a subset of {0, 1}ⁿ.

Proof. Suppose that $M$ and $N$ are two families of cuts of a poset valued intuitionistic fuzzy set on X, with the codomain P. Then, the condition 1. is true by Proposition 1, 5., the conditions 2. and 3. are valid by Theorem 1 and the condition 4. is valid by the definition of poset valued intuitionistic fuzzy set.

In order to prove the converse, a mapping μ : X ⟶ P is defined by μ (x) : = E (Z_x) and a mapping ν : X ⟶ P is defined by ν (x) : = G (T_x). In the same way as in the proof of Theorem 1 in [26], it is proved that μ : X ⟶ P is a poset valued fuzzy set with the family of cuts $M$ . Similarly, it is proved that ν : X ⟶ P_d, where P_d is a dual poset to P, is a poset valued fuzzy set with the family of cuts $N$ . Now, the ordered triple (X, μ, ν) is considered, which therefore has families of cuts μ and ν. By condition 4, (X, μ, ν) is a poset valued intuitionistic fuzzy set on X.

In case when poset P is finite, conditions (4) and (5) are equivalent, by Theorem 2.□

4 Conclusion

A new definition of a poset valued intuitionistic fuzzy set is introduced, by using a representation of poset as a subset of distributive lattice. In this way, it is possible to deal with the membership and non-membership functions and their relationship without entering into wide and complicated topic of defining complements in posets. Moreover, in this framework, every poset can be a codomain of a intuitionistic fuzzy set (without the requirement that the poset is bounded). This definition and the results can be further generalized to infinite spatial posets (the ones in which every element is the join of some join-irreducible elements).

Footnotes

Acknowledgments

The research of the second author was supported by the Serbian Ministry of Science and Technological Development, Grant No. 174013.

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