fuzzy adjunctions and fuzzy morphological operations based on implications

Fuzzy Adjunctions and FuzzyMorphological Operations Basedon ImplicationsY. Shi,1,∗ M. Nachtegael,1,† D. Ruan,1,2,‡ E. E. Kerre1,§1Fuzziness and Uncertainty Modelling Research Unit,Department of Applied Mathematics and Computer Science,Ghent University, Krijgslaan 281 (S9), 9000 Gent, Belgium2Belgium Nuclear Research Centre (SCK•CEN), 2400 Mol, Belgium

The concept of adjunction plays an important role in mathematical morphology. If the morpho-logical operations, dilation and erosion form an adjunction in a complete lattice, then they, aswell as the closing and opening constructed by them, will fulfill certain required properties inan algebraic context. In the context of fuzzy mathematical morphology, which is an extensionof binary morphology to gray-scale morphology based on fuzzy set theory, we use conjunctionsand implications to define fuzzy dilations and fuzzy erosions. In this paper, we investigate whenthese pairs of dilations and erosions form a fuzzy adjunction, which is also defined by an impli-cation. We find that the so-called adjointness between a conjunction and an implication plays animportant role here. Finally, we develop a theorem stating that a conjunction that is adjoint withan implication cannot only be generated by an R-implication but also by other implications. Thisallows the easy construction of fuzzy adjunctions. C© 2009 Wiley Periodicals, Inc.

1. INTRODUCTION

Mathematical morphology is an important theory developed in image process-ing to analyze the geometric features of n-dimensional images. These images can bebinary images, which are represented as subsets of R

n or gray-scale images whichare represented as R

n → [0, 1] mappings.1 Morphological operations are the basictools in mathematical morphology. They transform an image A by using anotherimage B, which is called the structuring element. Dilation and erosion are the twobasic morphological operations. Another two important morphological operations,closing and opening, can be constructed via dilation and erosion. If a dilation and

∗Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].‡e-mail: [email protected].§e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 24, 1280–1296 (2009)C© 2009 Wiley Periodicals, Inc. Published online in Wiley InterScience

(www.interscience.wiley.com). • DOI 10.1002/int.20385

FUZZY ADJUNCTIONS AND FUZZY MORPHOLOGICAL OPERATIONS 1281

an erosion form an adjunction in a complete lattice, then a group of properties ofthem will be fulfilled, as well for the closing and opening constructed by them.2

When extending binary morphology to gray-scale morphology, fuzzy sets inR

n are proper representations of gray-scale images. Fuzzy dilation and fuzzy erosioncan be defined via a conjunction and an implication, respectively.3 Moreover, animplication can be defined as a fuzzy adjunction in the complete lattice F (Rn),which is the set of all fuzzy sets on R

n. In this paper, we analyze for different fuzzydilations defined by a conjunction and fuzzy erosions defined by an implication whenthey form adjunctions defined by an implication. In Section 2, we give some basicnotions of conjunctions and implications. And then we define fuzzy morphologicaloperations via conjunctions and implications. In Section 3, we extend a classicaladjunction, to a fuzzy adjunction, which is generated by an implication. We analyzeunder which conditions a fuzzy dilation, which is generated by a conjunction, anda fuzzy erosion, which is generated by an implication, form an adjunction andthen outline the importance of the adjointness between the conjunctions and theimplications.

2. FUZZY MORPHOLOGICAL OPERATIONS DEFINED BYCONJUNCTIONS AND IMPLICATIONS

2.1. Conjunctions and Implications

DEFINITION 1. A mapping N : [0, 1] → [0, 1] is a fuzzy negation if N is decreasingand satisfies N(0) = 1 and N(1) = 0. A fuzzy negation is called strong if ( ∀x ∈[0, 1])(N(N(x)) = x).

DEFINITION 2. A mapping C : [0, 1]2 → [0, 1] is a conjunction if it has increasingfirst and second partial mappings and satisfies C(0, 0) = C(0, 1) = C(1, 0) = 0 andC(1, 1) = 1. A conjunction is a seminorm if (∀x ∈ [0, 1])(C(1, x) = C(x, 1) = x). Aseminorm is a triangular norm (t-norm for short) if C is commutative and associative.

The four most famous t-norms are

• TM(x, y) = min(x, y) (minimum)• TP(x, y) = xy (product)• TL(x, y) = max(x + y − 1, 0) (Łukasiewicz t-norm)

• TD(x, y) ={

min(x, y) x = 1 or y = 10 otherwise.

(drastic product)

Minimum, product, and Łukasiewicz t-norm are continuous t-norms, whereas drasticproduct is even not left-continuous.

DEFINITION 3. A mapping S : [0, 1]2 → [0, 1] is a triangular conorm (t-conorm forshort) if it satisfies: S(x, 0) = x (boundary condition), y ≤ z implies S(x, y) ≤S(x, z) (increasing first and second partial mappings), S(x, y) = S(y, x) (commu-tativity), and S(x, S(y, z)) = S(S(x, y), z) (associativity).

International Journal of Intelligent Systems DOI 10.1002/int

1282 SHI ET AL.

A t-conorm S can be obtained by a t-norm T and a strong fuzzy negationthrough

(∀(x, y) ∈ [0, 1]2)(S(x, y) = N(T (N(x), N (y))))

Corresponding to the four famous t-norms, the four most famous t-conorms are

• SM(x, y) = max(x, y) (maximum)• SP(x, y) = x + y − xy (probabilistic sum)• SL(x, y) = min(x + y, 1) (Łukasiewicz t-conorm, bounded sum)

• SD(x, y) ={

max(x, y) x = 0 or y = 01 otherwise. (drastic sum)

DEFINITION 4. A mapping I : [0, 1]2 → [0, 1] is an implication if it has decreas-ing first partial mappings and increasing second partial mappings and satisfiesI(0, 0) = I(0, 1) = I(1, 1) = 1, and I(1, 0) = 0.

In the literature devoted to the theory of fuzzy logic, many other poten-tial properties are required for an implication, among which the most importantones are

• left neutrality of truth: (∀x ∈ [0, 1])(I(1, x) = x);• exchange principle: (∀(x, y, z) ∈ [0, 1]3)(I(x,I(y, z)) = I(y,I(x, z)));• ordering principle: (∀(x, y) ∈ [0, 1]2)(I(x, y) = 1 ⇔ x ≤ y);• second place right-continuity.

A conjunction C and an implication I are called adjoint if

(∀(x, y, z) ∈ [0, 1]3)(C(x, z) ≤ y ⇔ z ≤ I(x, y)) (1)

We define an implication I via a conjunction C by

(∀(x, y) ∈ [0, 1]2)(I(x, y) = sup{t ∈ [0, 1]|C(x, t) ≤ y}) (2)

And we define a conjunction C via an implication I by

(∀(x, y) ∈ [0, 1]2)(C(x, y) = inf{t ∈ [0, 1]|I(x, t) ≥ y}) (3)

THEOREM 1. (Ref. 4, Proposition 5.4.2). Let C be a conjunction, and I be animplication defined by (2). Then C and I are adjoint if and only if C is left-continuousw.r.t. the second variable.



According to the proof of Theorem 1, we have the following corollary:

COROLLARY 1. Let I be an implication and C be a conjunction defined by (3). ThenI and C are adjoint if and only if I is right-continuous w.r.t. the second variable.

From Theorem 1 and Corollary 1, we obtain the following corollaries:

COROLLARY 2. (Ref. 4, Corollary 5.4.1). Let C be a conjunction. If C is left-continuous w.r.t. second variable, then formula (2) becomes

(∀(x, y) ∈ [0, 1]2)(I(x, y) = max{t ∈ [0, 1]|C(x, t) ≤ y}) (4)

COROLLARY 3. Let I be an implication. If I is right-continuous w.r.t. the secondvariable, then formula (3) becomes

(∀(x, y) ∈ [0, 1]2)(C(x, y) = min{t ∈ [0, 1]|I(x, t) ≥ y}) (5)

We will see later that adjointness of a conjunction and an implication plays animportant role in defining fuzzy dilation and fuzzy erosion which can form a fuzzyadjunction in F (Rn). There are three important ways to construct implication func-tions from classical logic.5

Let S be a t-conorm, N be a strong fuzzy negation, and T be a t-norm. Thenan S-implication function is defined by

(∀(x, y) ∈ [0, 1]2)(I(x, y) = S(N(x), y)). (6)

Below are the four S-implications constructed by the four aforementioned t-conorms,and the strong fuzzy negation N(x) = 1 − x (Ref. 5):

• Ib(x, y) = max(1 − x, y) (Kleene–Dienes)• Ir (x, y) = 1 − x + xy (Reichenbach)• IL(x, y) = min(1, 1 − x + y) (Łukasiewicz)

• ILS(x, y) =⎧⎨⎩

y x = 11 − x y = 01 else

An R-implication function is defined by

(∀(x, y) ∈ [0, 1]2)(I(x, y) = sup{t ∈ [0, 1]|T (x, t) ≤ y}) (7)


1284 SHI ET AL.

Below are the three R-implications constructed by the three aforementioned contin-uous t-norms5:

• Ig(x, y) ={

1 x ≤ yy else (Godel)

• I�(x, y) ={

1 x ≤ yy/x else (Goguen)

• IL(x, y) = min(1, 1 − x + y) (Łukasiewicz)

A QL-implication is defined by

(∀(x, y) ∈ [0, 1]2)(I(x, y) = S(N(x), T (x, y))) (8)

Below are the four QL-implications constructed by the four aforementioned t-norms,t-conorms, and the strong fuzzy negation N(x) = 1 − x,5:

• Im(x, y) = max(1 − x, min(x, y)) (early Zadeh)• Ip(x, y) = 1 − x + x2y (Klir and Yuan 1)• IL(x, y) = min(1, 1 − x + y) (Łukasiewicz)

• Iq (x, y) =⎧⎨⎩

y x = 11 − x x = 1, y = 11 x = 1, y = 1

(Klir and Yuan 2)

Observe that all S-implication functions and R-implication functions are implica-tions. But only a group of QL-implication functions are implications because theydo not always have decreasing first partial mappings. QL-implication functions thatare implications can be found in Refs. 6–9. Besides these three classes of implicationfunctions, we mention here some other cases.

In Ref. 10, the authors defined a new class of implications obtained via agenerator.

DEFINITION 5. A generator f is a continuous [0, 1] → [0, +∞[ mapping which isstrictly decreasing with f (1) = 0. Moreover the pseudo inverse of f , f (−1) is definedas

f (−1)(x) ={f −1(x), if x ≤ f (0)0, otherwise

DEFINITION 6. A f -generated implication If is defined as

(∀(x, y) ∈ [0, 1]2)(If (x, y) = f (−1)(xf (y)))

Observe that if the generator f is defined as f (x) = − log x, then thef -generated implication is the widely known Yager implication IY (Ref. 10):(∀(x, y) ∈ [0, 1]2)(IY (x, y) = yx). In Ref. 11, three kinds of implications arementioned:



DEFINITION 7. The Zadeh implication is defined by

(∀(x, y) ∈ [0, 1]2)(IZ(x, y) ={

1, if x ≤ y

0, if x > y)

DEFINITION 8. A λ-function is a [0, 1] → [0, 1] mapping λ, which is decreasing withλ(0) = 1, and λ(1) = 0 and satisfies the equation λ(x) = 0 has a unique solution;(∀α ∈ [0.5, 1])(λ(x) = α has a unique solution); (∀x ∈ [0, 1])(λ(x) + λ(1 − x) ≥1).11

From a λ-function, we can define a class of implications.

DEFINITION 9. A generalized Łukasiewicz implication is defined by

(∀(x, y) ∈ [0, 1]2)(Iλ(x, y) = min(λ(x) + λ(1 − y), 1))

Examples of λ-functions with parameter n are given below:

(i) λn(x) = 1 − xn with n ≥ 1. Note that if n = 1 in this case, then λn(x) = 1 −x, for all x ∈ [0, 1], which is the standard strong fuzzy negation. Moreover,the corresponding implication via Definition 9 is the famous Lukasiewiczimplication IL.

(ii) λn(x) = 1−x1+nx

with n ∈ ] − 1, 0].

2.2. Fuzzy Morphological Operations

We define binary morphological operations according to Ref. 12.

DEFINITION 10. Let A, B ⊆ Rn. The binary dilation D(A, B) is defined by D(A, B) =

{y ∈ Rn|Ty(B) ∩ A = ∅}, and the binary erosion E(A, B) is defined by E(A, B) =

{y ∈ Rn|Ty(B) ⊆ A}, where Ty(B) = {x ∈ R

n|x − y ∈ B}.

Define for all x ∈ Rn, −B(x) = B(−x). A binary closing C(A, B) is de-

fined by C(A, B) = E(D(A, B), −B), and a binary opening O(A, B) is definedby O(A, B) = D(E(A, B), −B), where D and E are binary dilation and binaryerosion, respectively.

When extending binary morphology to gray-scale morphology, fuzzy sets in Rn

are proper representations of gray-scale images. Using a conjunction to express theintersection of two fuzzy sets and an implication to express the fuzzy set inclusion,we can fuzzify binary dilation and binary erosion into fuzzy dilation and fuzzyerosion.


1286 SHI ET AL.

DEFINITION 11.3 Let A, B ∈ F (Rn), and let C be a conjunction and I be animplication. The fuzzy dilation DC(A, B) is defined by

(∀y ∈ Rn)(DC(A, B)(y) = sup

x∈Rn

C(B(x − y), A(x)))

and the fuzzy erosion EI(A, B) is defined by

(∀y ∈ Rn)(EI(A, B)(y) = inf

x∈RnI(B(x − y), A(x)))

Observe that according to Refs. 1 and 3, several fuzzy approaches to gray-scalemorphology are embedded in the approach stated in Definition 11. For example,the approach of fuzzy set inclusion of Zadeh uses the Zadeh implication to definea fuzzy erosion via Definition 11. The approach of fuzzy set inclusion of Sinhaand Dougherty uses the generalized Łukasiewicz implication to define a fuzzyerosion via Definition 11. However, both of the aforementioned approaches useconjunctions generated by (∀(x, y) ∈ [0, 1]2)(C(x, y) = N(I(x, N(y)))), where N

is a strong fuzzy negation and I = IZ or I = Iλ to define the fuzzy dilation. In thenext section, we will see that it is more proper to define the conjunction C to beadjoint with the implication I .

Similar to the case in binary morphology, a fuzzy closing C(A, B) is definedby

CC,I(A, B) = EI(DC(A, B), −B)

and an opening O(A, B) is defined by

OC,I(A, B) = DC(EI(A, B), −B),

where D and E are binary dilation and binary erosion respectively.

3. FUZZY I-ADJUNCTIONS IN F (Rn)

3.1. Classical Adjunctions and Fuzzy Adjunctions

In this paper, we discuss adjunctions in a complete lattice. Recall that a lattice(L, �) is a poset such that any two elements L1, L2 ∈ L have a supremum andan infimum. A complete lattice (L, �) is a lattice such that any subset of L has asupremum and an infimum in L. First we define the classical adjunctions.

DEFINITION 12.2 Let δ and ε be two operations on a complete lattice (L, �). The pair(δ, ε) is an adjunction inL if and only if (∀L1, L2 ∈ L)(δ(L1) � L2 ⇔ L1 � ε(L2)).

Define the ordering A ⊆ B ⇔ (∀x ∈ Rn)(A(x) ≤ B(x)), resulting in the com-

plete lattice (F (Rn), ⊆). For any A, B ∈ F (Rn), let δ(A) = DC(A, B), which isa fuzzy dilation on A by means of B, and ε(A) = EI(A, −B), which is a fuzzy



erosion on A by means of −B. If (δ, ε) is a properly adjunction in F (Rn), then thefuzzy dilation DC , the erosion EI , and the fuzzy closing and fuzzy opening definedby them will fulfill the properties mentioned below:

(1) ε(⋂

j∈J Aj ) = ⋂j∈J ε(Aj )

(2) δ(⋃

j∈J Aj ) = ⋃j∈J δ(Aj )

(3) A � εδ(A)(4) δε(A) � A

(5) εδε(A) = ε(A)(6) δεδ(A) = δ(A)(7) ε(A) = ⋃{B ∈ L|δ(B) � A}, ⋃

modeled by sup(8) δ(A) = ⋂{B ∈ L|ε(B) � A}, ⋂

modeled by inf

Properties (1) and (2) represent that ε is an algebraic erosion on L and δ is analgebraic dilation on L, respectively. Properties (3) and (4) represent that εδ isextensive and δε is restrictive, respectively. Properties (5) and (6) represent thatboth εδ and δε are idempotent. Properties (7) and (8) represent that one operatorfrom a given adjunction can be generated by the other one.1

Nachtegael et al.1 defined a fuzzy adjunction by an implicationI , which extendsthe properties of classical adjunctions. The implication I is proved necessary tofulfill the ordering principle.

DEFINITION 13. Let I be an implication that satisfies the ordering principle. More-over, let δ and ε be two F (Rn) → F (Rn) mappings. Then the pair (δ, ε) is a fuzzyI-adjunction in F (Rn) if and only if

(∀(A1, A2) ∈ F (Rn) × F (Rn)( infx∈R

nI(δ(A1)(x), A2(x))

= infx∈R

nI(A1(x), ε(A2)(x)))

Let C be a conjunction, I and I ′be two implications. Next theorem and

propositions will state when (δ, ε) is a fuzzy I-adjunction, where δ(A) = DC(A, B)and ε(A) = EI ′ (A, −B).

THEOREM 2. Let C be a conjunction and I be an implication. Moreover, let δ(A) =DC(A, B) and ε(A) = EI(A, −B). Then (δ, ε) is a fuzzy IZ-adjunction if and onlyif C and I are adjoint, where IZ is the Zadeh implication.

Proof. According to Definition 11, We have

(∀y ∈ Rn)(δ(A1)(y) = sup

x∈Rn

C(B(x − y), A1(x)))


1288 SHI ET AL.

and

(∀y ∈ Rn)(ε(A2)(y) = inf

x∈RnI(B(y − x), A2(x)))

According to Definition 7 and 13, (δ, ε) is a fuzzy Iz-adjunction, which means that

(∀x ∈ Rn)(δ(A1(x)) ≤ A2(x)) ⇔ (∀x ∈ R

n)(A1(x) ≤ ε(A2(x))).

Because (δ, ε) is a fuzzy IZ-adjunction, we have

(∀y ∈ Rn)( sup

x∈Rn

C(B(x − y), A1(x)) ≤ A2(y))

⇔ (∀y ∈ Rn)(A1(y) ≤ inf

x∈RnI (B(y − x), A2(x))), i.e.,

(∀(x, y) ∈ Rn × R

n)(C(B(x − y), A1(x)) ≤ A2(y))

⇔ (∀(x, y) ∈ Rn × R

n)(A1(y) ≤ I(B(y − x), A2(x))), i.e.,

(∀(x, y) ∈ Rn × R

n)(C(B(x − y), A1(x)) ≤ A2(y))

⇔ (∀(x, y) ∈ Rn × R

n)(A1(x) ≤ I(B(x − y), A2(y)))

which is equivalent to

(∀(a, b, c) ∈ [0, 1]3)(C(a, b) ≤ c ⇔ b ≤ I(a, c))

which means C and I are adjoint. �

LEMMA 1. Let f be a [0, 1] → [0, 1] mapping and (ai)i∈I an arbitrary familyindexed by I in [0, 1]. Then the following statements hold:

• if f is left-continuous and decreasing then infi∈I f (ai) = f (supi∈I ai);• if f is left-continuous and increasing then supi∈I f (ai) = f (supi∈I ai);• if f is right-continuous and decreasing then supi∈I f (ai) = f (infi∈I ai);• if f is right-continuous and increasing then infi∈I f (ai) = f (infi∈I ai).

PROPOSITION 1. Let C be an associative conjunction being left-continuous w.r.t.the second variable and I be an implication which is left-continuous w.r.t. the firstvariable and right-continuous w.r.t. the second variable and satisfies the orderingprinciple. Moreover, let δ(A) = DC(A, B) and ε(A) = EI(A, −B). Then (δ, ε) is afuzzy I-adjunction if C and I are adjoint.



Proof.

I(C(a, b), c) = max{t |C(C(a, b), t) ≤ c}

Because C and I are adjoint,

C(a, C(b, t)) ≤ c ⇔ C(b, t) ≤ I(a, c)

So

I(b,I(a, c)) = max{t |C(b, t) ≤ I(a, c)}

= max{t |C(b, t) ≤ max{s|C(a, s) ≤ c}}

Since C is associative, we have

I(b,I(a, c)) = max{t |C(a, C(b, t)) ≤ c}

= max{t |C(C(a, b), t)) ≤ c}

= I(C(a, b), c)

Thus for all A1, A2, B ∈ F (Rn), we get by applying the above formula:

infy∈R

ninf

x∈RnI(C(B(x − y), A1(x)), A2(y))

= infy∈R

ninf

x∈RnI(A1(x),I(B(x − y), A2(y)))

or equivalently,

infy∈R

ninf

x∈RnI(C(B(x − y), A1(x)), A2(y))

= infy∈R

ninf

x∈RnI(A1(y),I(B(y − x), A2(x)))

Since I is left-continuous w.r.t. the first variable and right-continuous w.r.t. thesecond variable, according to Lemma 1, we have

infy∈R

nI( sup

x∈Rn

C(B(x − y), A1(x)), A2(y))

= infy∈R

nI(A1(y), inf

x∈RnI(B(y − x), A2(x)))

⇔ infy∈R

nI(δ(A1)(y), A2(y)) = inf

y∈RnI(A1(y), ε(A2)(y))


1290 SHI ET AL.

Thus according to Definition 13, (δ, ε) is an I-adjunction. �

PROPOSITION 2. Let C be a conjunction, I be an implication being left-continuousw.r.t. the first variable and right-continuous w.r.t. the second variable and which sat-isfies the ordering principle, and I ′

an implication. Moreover, let δ(A) = DC(A, B)and ε(A) = EI ′ (A, −B). If (δ, ε) is a fuzzy I-adjunction, then C and I ′

are adjoint.

Proof. If (δ, ε) is a fuzzy I-adjunction, then for all A1, A2, B ∈ F (Rn),

infy∈R

nI(δ(A1)(y), A2(y)) = inf

y∈RnI(A1(y), ε(A2)(y))

or equivalently

infy∈R

nI( sup

x∈Rn

C(B(x − y), A1(x)), A2(y))

= infy∈R

nI(A1(y), inf

x∈RnI ′

(B(y − x), A2(x)))

Since I is left-continuous w.r.t. the first variable and right-continuous w.r.t. thesecond variable, according to Lemma 1, we have

infy∈R

ninf

x∈RnI(C(B(x − y), A1(x)), A2(y))

= infy∈R

ninf

x∈RnI(A1(y),I ′

(B(y − x), A2(x)))

or equivalently

infy∈R

ninf

x∈RnI(C(B(x − y), A1(x)), A2(y))

= infy∈R

ninf

x∈RnI(A1(x),I ′

(B(x − y), A2(y)))

Thus we have

(∀(a, b, c) ∈ [0, 1]3)(I(C(a, b), c) = I(b,I ′(a, c)))

and hence, taking into account that I satisfies the ordering principle, we get

(∀(a, b, c) ∈ [0, 1]3)(C(a, b) ≤ c ⇔ b ≤ I ′(a, c))

Hence C and I ′are adjoint. �



Thus we can see that in the framework of fuzzy adjunctions, it is very importantfor a conjunction C and an implication I to be adjoint. Let us take a closer look atthis adjointness.

3.2. Adjointness between Conjunctions and Implications

For R-implication functions generated by left-continuous t-norms, the follow-ing theorem and corollary hold.

THEOREM 3 (Ref. 13, Theorem 1.14). A [0, 1]2 → [0, 1] mapping I is anR-implication generated by a left-continuous t-norm via (7) if and only if it hasincreasing second partial mappings, satisfies the exchange principle and the order-ing principle and it is right-continuous w.r.t. the second variable.

COROLLARY 4 (Ref. 14, Corollary 10). A [0, 1]2 → [0, 1] mapping T is a left-continuous t-norm if and only if T can be represented by

T (x, y) = inf{t ∈ [0, 1]|I(x, t) ≥ y} (9)

for some [0, 1]2 → [0, 1] mapping I which has increasing second partial mappings,satisfies the exchange principle and the ordering principle and it is right-continuousw.r.t. the second variable.

We see from Corollary 4 that we cannot always generate a left-continuoust-norm from an implication. For example, we cannot generate a t-norm from anS-implication function, which is not an R-implication function. Using the Reichen-bach implication Ir in (3), we have

inf{t ∈ [0, 1]|Ir (x, t) ≥ y}

= inf{t ∈ [0, 1]|1 − x + xt ≥ y}

= max

(x + y − 1

x, 0

)

which is not a t-norm, because of the lack of commutativity.Nevertheless, the corollary following the next theorem shows that we can

generate conjunctions with implications that are not only R-implication functions.

THEOREM 4. A [0, 1]2 → [0, 1] mapping I is an implication being right-continuousw.r.t. the second variable and satisfies

(∀x ∈ [0, 1])(I(1, x) = 1 ⇔ x = 1) (10)


1292 SHI ET AL.

if and only if I is generated by a conjunction C via (2), where C is left-continuousw.r.t. the second variable and satisfies

(∀x ∈ [0, 1])(C(1, x) = 0 ⇔ x = 0) (11)

Proof. To prove the if-part, assume 0 ≤ x1 < x2 ≤ 1. Then according to Corollary 2for all y ∈ [0, 1], I(x1, y) = max{t |C(x1, t) ≤ y} and I(x2, y) = max{t |C(x2, t) ≤y}. Because C has increasing first partial mappings, then

(∀t ∈ [0, 1])(C(x2, t) ≤ y ⇒ C(x1, t) ≤ y)

Thus max{t |C(x2, t) ≤ y} ∈ {t |C(x1, t) ≤ y}. Therefore

max{t |C(x2, t) ≤ y} ≤ max{t |C(x1, t) ≤ y},

i.e., I(x2, y) ≤ I(x1, y). Hence I has decreasing first partial mappings.Now assume 0 ≤ y1 < y2 ≤ 1. Then for all x ∈ [0, 1], I(x, y1) =

max{t |C(x, t) ≤ y1} and I(x, y2) = max{t |C(x, t) ≤ y2}. We have

(∀t ∈ [0, 1])(C(x, t) ≤ y1 ⇒ C(x, t) ≤ y2)

Thus max{t |C(x, t) ≤ y1} ∈ {t |C(x, t) ≤ y2}. Therefore

max{t |C(x, t) ≤ y1} ≤ max{t |C(x, t) ≤ y2},

i.e., I(x, y1) ≤ I(x, y2). Hence I has increasing second partial mappings.Now we will verify the boundary conditions for I . We have

I(0, 0) = max{t |C(0, t) = 0} = 1

I(0, 1) = max{t |C(0, t) ≤ 1} = 1

I(1, 1) = max{t |C(1, t) ≤ 1} = 1

Moreover, because C satisfies condition (11), we have I(1, 0) = max{t |C(1, t) =0} = 0.

Therefore I is an implication. Because C is left-continuous w.r.t. the secondvariable, according to Theorem 1, C and I are adjoint. Thus we have

C(x, y) ≤ z ⇔ y ≤ I(x, z)

Hence I can be represented by (4). Therefore according to Corollary 1, I is right-continuous w.r.t. the second variable. If I(1, x) = max{t |C(1, t) ≤ x} = 1, thenC(1, 1) ≤ x, which implies that x = 1. If x = 1, then we already know I(1, 1) = 1.



Thus I(1, x) = 1 if and only if x = 1. Hence I is an implication being right-continuous w.r.t. the second variable and satisfies condition (10).

To prove the “only if” part, define a [0, 1]2 → [0, 1] mapping C by (3). As-sume 0 ≤ x1 < x2 ≤ 1. Then for all y ∈ [0, 1], C(x1, y) = min{t |I(x1, t) ≥ y} andC(x2, y) = min{t |I(x2, t) ≥ y}. Because I has decreasing first partial mappings,then

(∀t ∈ [0, 1])(I(x2, t) ≥ y ⇒ I(x1, t) ≥ y)

Thus min{t |I(x2, t) ≥ y} ∈ {t |I(x1, t) ≥ y}. Therefore

min{t |I(x2, t) ≥ y} ≥ min{t |I(x1, t) ≥ y}

i.e., C(x2, y) ≥ C(x1, y). Hence C has increasing first partial mappings.Now assume 0 ≤ y1 < y2 ≤ 1. Then for all x ∈ [0, 1], C(x, y1) =

min{t |I(x, t) ≥ y1} and C(x, y2) = min{t |I(x, t) ≥ y2}. We have

(∀t ∈ [0, 1])(I(x, t) ≥ y2 ⇒ I(x, t) ≥ y1)

Thus min{t |I(x, t) ≥ y2} ∈ {t |I(x, t) ≥ y1}. Therefore

min{t |I(x, t) ≥ y2} ≥ min{t |I(x, t) ≥ y1}

i.e., C(x, y2) ≥ C(x, y1). Hence C has increasing second partial mappings.Now we will consider the boundary conditions of C. We have

C(0, 0) = min{t |I(0, t) ≥ 0} = 0

C(0, 1) = min{t |I(0, t) = 1} = 0

C(1, 0) = min{t |I(1, t) ≥ 0} = 0

Moreover, because I satisfies condition (10), we have C(1, 1) = min{t |I(1, t) =1} = 1.

Therefore C is a conjunction. Because I is left-continuous w.r.t. the secondvariable, according to Corollary 1, C and I are adjoint. Thus we have

C(x, y) ≤ z ⇔ y ≤ I(x, z)

Hence I can be represented by (5). Therefore according to Corollary 1, C is left-continuous w.r.t. the second variable.

If C(1, x) = min{t |I(1, t) ≥ x} = 0, then I(1, 0) ≥ x, which implies that x =0. If x = 0, then we already know C(1, 0) = 0. Thus C(1, x) = 0 if and only ifx = 0. Hence C is a conjunction being left-continuous w.r.t. the second variable andsatisfying condition (11). �


1294 SHI ET AL.

From the proof of Theorem 4, we have the next corollary.

COROLLARY 5. A [0, 1]2 → [0, 1] mapping C is a conjunction being left-continuousw.r.t. the second variable and satisfying condition (11) if and only if C is generatedby an implication I via formula (3), where I is right-continuous w.r.t. the secondvariable and satisfies condition (10).

Below we give several examples of adjoint pairs consisting of right-continuousnon-R-implications and the conjunctions generated by them via formula 2.

Example 1.

(i) Observe that S-implication functions always fulfill condition (10). Con-sider the Kleene–Dienes implication Ib(x, y) = max(1 − x, y), which is anS-implication function generated by the t-conorm SM and the standard fuzzynegation, and the Reichenbach implication Ir (x, y) = 1 − x + xy, which isan S-implication function generated by the t-conorm SP. Ib satisfies

(∀(x, y) ∈ [0, 1]2)(Ib(x, y) = 1 ⇔ x = 0 or y = 1),

and Ir satisfies

(∀(x, y) ∈ [0, 1]2)(Ir (x, y) = 1 ⇔ x = 0 or y = 1).

Thus Ib and Ir are not residuated with any left-continuous t-norm, but theyare right-continuous w.r.t. the second variable. Then

CIb(x, y) = inf{t ∈ [0, 1]|Ib(x, t) ≥ y}

={

0, if x ≤ y

y, if x > y

and

CIr(x, y) = inf{t ∈ [0, 1]|Ir (x, t) ≥ y}

=⎧⎨⎩max

(x + y − 1

x, 0

), if x ≥ 0.

0, if x = 0

(ii) A f-generated implicationIf also fulfills condition (10). MoreoverIf satisfies

(∀(x, y) ∈ [0, 1]2)(If (x, y) = 1 ⇔ x = 0 or y = 1)

Thus If is not residuated to any left-continuous t-norm. Because both f andf (−1) are continuous mappings, If is continuous w.r.t. the second variable.



Then

CIf(x, y) = inf{t ∈ [0, 1]|If (x, t) ≥ y}

=⎧⎨⎩f (−1)

(f (y)

x

), ifx > 0

0, ifx = 0.

(iii) Recall the two aforementioned λ-functions.

(iii,1) If λn(x) = 1 − xn (n ≥ 1), then

Iλn(x, y) = min(λn(x) + λn(1 − y), 1)

= min(1 − xn + 1 − (1 − y)n, 1)

= min(2 − xn − (1 − y)n, 1)

Thus

CIλn(x, y) = inf{t | min(2 − xn − (1 − t)n, 1) ≥ y}

= max(1 − (2 − xn − y)1n , 0)

(iii,2) If λn(x) = 1−x1+nx

(n ∈ ] − 1, 0]), then

Iλn(x, y) = min(λn(x) + λn(1 − y), 1)

= min

(1 − x

1 + nx+ y

1 + n − ny, 1

)

Thus

CIλn(x, y) = inf

{t | min

(1 − x

1 + nx+ t

1 + n(1 − t), 1

)≥ y

}

= max

((1 + n)

(− 1−x1+nx

)1 + n

(y − 1−x

1+nx

) , 0

)

Observe that none of the conjunctions in this example is a semi norm.


1296 SHI ET AL.

4. CONCLUSION REMARKS

In this paper, we extend classical adjunctions to fuzzy adjunctions. We usean implication I to define a fuzzy I-adjunction in F (Rn). And then we study theconditions under which a fuzzy dilation, which is defined by a conjunction C, anda fuzzy erosion, which is defined by an implication I ′

, form a fuzzy I-adjunction,which is important for the fuzzy dilation, the fuzzy erosion, and the fuzzy closing andthe fuzzy opening defined by them to fulfill the required properties of an algebraicdilation, an algebraic erosion, an algebraic closing, and an algebraic opening. Wefind out that the adjointness between the conjunction C and the implication I orthe addjointness between the conjunction C and the implication I ′

play importantroles in such conditions. We then work out Theorem 4 and Corollary 5 on theadjointness between a conjunction and an implication. On the basis of Theorem 4and Corollary 5, we can generate a conjunction from an implication so that they areadjoint. Examples of generating conjunctions from famous existing implications aregiven at the end.

References

1. Nachtegael M, Heijmans H, van der Weken D, Kerre E. Fuzzy adjunctions in mathematicalmorphology. In: Proc. of JCIS 2003 Cary, NC; 2003. pp 202–205.

2. Heijmans H, Morphological image operators. London: Academic Press; 1994.3. Nachtegael M, Kerre E. Connections between binary, gray-scale and fuzzy mathematical

morphologies. Fuzzy Sets Syst 2001;124:73–85.4. Gottwald S. A treatise on many-valued logics. Studies in Logic and Computation. Research

Studies Press: Baldock, UK; 2001.5. Klir G, Yuan B. Fuzzy sets and fuzzy logic, theory and applications. Englewood Cliffs, NJ:

Prentice Hall; 1995.6. Fodor J. On fuzzy implication operators. Fuzzy Sets Syst 1991;42:293–300.7. Mas M, Monserrat M, Torrens J. Ql-implications versus D-implications. Kybernetika

2006;42:351–366.8. Shi Y, Gasse BV, Ruan D, Kerre E. On the first place antitonicity in QL-implications. Fuzzy

Sets Syst 2008;159:2988–3013.9. Trillas E, del Campo C, Cubillo S. When QM-operators are implication functions and

conditional fuzzy relations, Int J Intell Syst 2000;15:647–655.10. Yager R. On some new classes of implication operators and their role in approximate

reasoning. Inform Sci 2004;167:193–216.11. Kerre E, Nachtegael M. Fuzzy techniques in image processing. New York: Physica-Verlag;

2000.12. Serra J, editor. Image analysis and mathematical morphology. (London: MIT Press; 1982.13. Fodor J, Roubens M. Fuzzy preference modelling and multicriteria decision support.

Dordrecht, the Netherlands: Kluwer Academic Publishers; 1994.14. Baczynski M. Residual implications revisited. Notes on the Smets–Magrez theorem. Fuzzy

Sets Syst 2004;145:267–277.


fuzzy adjunctions and fuzzy morphological operations based on implications

Documents