chapter 6 multi-fuzzy logic

CHAPTER 6

Multi-fuzzy Logic

We analyze in this chapter the notion of multi-fuzzy logic on residuated lattices

using multi-fuzzy sets as evaluations of logic formulae. We derive some equations and

inequalities, which are useful in diverse areas like multi dimensional reasoning, pat-

tern recognition, image processing, taste characterization and approximate reasoning.

We study some theorems in basic logic using multi-fuzzy evaluations and show that

multi-fuzzy logic is a tool for approximate reasoning. Also we develop the theory of

multi-fuzzy filters of residuated lattices and some characterizations of multi-fuzzy im-

plicative filters and multi-fuzzy regular filters. It seems that, multi-fuzzy filter theory

is parallel to the filter theory in general residuated lattices. In the last section we

propose theories of lattice implication algebra on multi-fuzzy sets, and implication

relation between propositions with different value domains.

6.1 IntroductionHajek [29] introduced the concepts of Basic Logic and BL-algebras as the logic sys-

tem of fuzzy logic. MV -algebras, Godel algebras and product algebras are the most

6Some results of this chapter are included in the paper Multi-fuzzy logic, which is communicatedfor publication.

113

6.2. MULTI-FUZZY LOGIC 114

known structural extensions of BL-algebras. Hajek [29] proposed the notions of filters

and prime filters in BL-algebras and later Xu [94] introduced the notion of lattice

implication algebras and Wang [90] proved that lattice implication algebras are cate-

gorically equivalent to MV -algebras. Theory of filters plays an important role in the

study of BL-algebras. Various filters correspond to various deduction systems and

in other words, they are the sets of various provable formulae. Turunen [86] intro-

duced the notion of Boolean filters and the notion of implicative filters. He proved

that implicative filters are equivalent to Boolean filters in BL-algebras. Afterwards,

Xu and Qin [96, 97] proposed the notions of positive implicative filters and fuzzy

positive implicative filters in lattice implication algebras. Jun et al. [33, 34, 74] de-

rived some characterizations of fuzzy positive implicative filters of lattice implication

algebras. Liu and Li [42, 43] introduced the notions of fuzzy filters, fuzzy Boolean

filters, fuzzy implicative filters, fuzzy positive implicative filters, fuzzy prime filters

and the cosets of fuzzy filters in BL-algebras. They derived several characterizations

of them and proved that fuzzy filters are useful tools to obtain results on classical

filters of BL-algebras. Also they proved that fuzzy Boolean filters are equivalent to

fuzzy implicative filters and fuzzy Boolean filters are fuzzy positive implicative filters

in BL-algebras. Recently, Zhu and Xu [111] developed the filter theory of general

residuated lattices and extended the above-mentioned types of fuzzy filters to resid-

uated lattices. They proposed the notions of regular filters and fuzzy regular filters

in general residuated lattices, and derived some of their characterizations.

6.2 Multi-fuzzy LogicIn this section we propose the concepts of multi-fuzzy logic and study some basic

properties of basic logic BL with multi-fuzzy sets as its evaluation. Also we study

the algebras of logic in multi-fuzzy sets and propose some relations in multi-fuzzy


logic. Throughout this section the constant propositions 0 and 1 the contradiction

and the tautology respectively. An order relation in∏j∈J

Lj means the product order

(see 1.2.8).

Lemma 6.2.1. If J is an indexing set and {(Lj,∧j,∨j,⊗j,⇒j, 0j, 1j) : j ∈ J} is a

family of residuated lattices, then (∏j∈J

Lj,∧,∨,⊗,⇒, 0, 1) is a residuated lattice with

respect to the operations, for every x = (xj)j∈J and y = (yj)j∈J in∏j∈J

Lj:

x⊗ y = (xj ⊗j yj)j∈J ;

x⇒ y = (xj ⇒j yj)j∈J ;

x ∧ y = (xj ∧j yj)j∈J ;

x ∨ y = (xj ∨j yj)j∈J ,

where 0 = (0j)j∈J , 1 = (1j)j∈J and 0j, 1jxj, yj ∈ Lj for each j ∈ J.

Proof. Follows from the definitions.

Lemma 6.2.2. If J is an indexing set and {(Lj,∧j,∨j,⊗j,⇒j, 0j, 1j) : j ∈ J} is

a family of BL algebra, then (∏j∈J

Lj,∧,∨,⊗,⇒, 0, 1) is a BL algebra with respect to

the operations defined in 6.2.1.

Proof. By Lemma 6.2.1, we have (∏j∈J

Lj,∧,∨,⊗,⇒, 0, 1) is a residuated lattice. We

need to prove the properties of pre-linearity divisibility. For every x = (xj)j∈J and

y = (yj)j∈J in∏j∈J

Lj:

(x⇒ y) ∨ (y ⇒ x) = ((xj ⇒j yj) ∨j (yj ⇒ xj))j∈J = (1j)j∈J = 1.

x⊗ (x⇒ y) = (xj ⊗j (xj ⇒j yj))j∈J = (xj ∧j yj)j∈J = x ∧ y.


Definition 6.2.3. Let X be the set of all formulae (including propositional variables)

in a logic system. A multi-fuzzy set A in X (with a residuated lattice∏j∈J

Lj as its

value domain) is a multi-fuzzy evaluation, if it satisfies the following conditions:

A(0) = 0;

A(φ→ ψ) =A(φ)⇒ A(ψ),∀φ, ψ ∈ X;

A(φ& ψ) =A(φ)⊗ A(ψ), ∀φ, ψ ∈ X.

Theorem 6.2.4. Let A,B,C be multi-fuzzy sets in X with same value domain∏j∈J

Lj.

If A,B,C are evaluations of a BL system, then for every φ, ψ, χ ∈ X :

(1) 0 ≤∧φ∈X

A(φ);

(2)∨ψ∈X

(A(φ)⊗B(ψ)) ≤ A(φ);

(3) (A(φ)⊗B(ψ)) = (B(ψ)⊗ A(φ));

(4) (A(φ)⊗ (B(ψ)⊗ C(χ))) = ((A(φ)⊗B(ψ))⊗ C(χ))).

Proof. (1) to (4) follow from the definition of ⊗ in the product lattice.

Note 6.2.5. Following results are immediately obtained from the definitions of join,

meet, equivalence, negation and 1-tautology mentioned in the subsection 1.6.1.

(1) A(¬φ) = A(φ→ 0) = A(φ)⇒ 0;

(2) A(1) = A(¬0) = A(0)⇒ 0 = 0⇒ 0 = 1;

(3) A(φ ≡ ψ) = A((φ→ ψ) & (ψ → φ)) = (A(φ)⇒ A(ψ))⊗ (A(ψ)⇒ A(φ));

(4) A(φ ∧ ψ) = A(φ& (φ→ ψ)) = A(φ)⊗ (A(φ)⇒ A(ψ)) = A(φ) ∧ A(ψ), since the

part (e) of Definition 1.6.7 ;

(5) A(φ ∨ ψ) = A(((φ→ ψ)→ ψ) ∧ ((ψ → φ)→ φ))

= A(((φ→ ψ)→ ψ)) ∧ A(((ψ → φ)→ φ))


= ((A(φ)⇒ A(ψ))⇒ A(ψ)) ∧ ((A(ψ)⇒ A(φ))⇒ A(φ))

= A(φ) ∨ A(ψ), by the part (5) of Lemma 1.6.8.

Proposition 6.2.6. [29] Each BL-algebra is a distributive lattice.

Theorem 6.2.7. If multi-fuzzy sets A,B,C in X (with same value domain) are

evaluations of a BL system, then for every φ, ψ, χ ∈ X :

(1)∨ψ∈X

(A(φ) ∧B(ψ)) ≤ A(φ);

(2) A(φ) ≤∧ψ∈X

(A(φ) ∨B(ψ));

(3) (A(φ) ∧B(ψ)) = (B(ψ) ∧ A(φ));

(4) (A(φ) ∨B(ψ)) = (B(ψ) ∨ A(φ));

(5) (A(φ) ∧ (B(ψ) ∧ C(χ))) = ((A(φ) ∧B(ψ)) ∧ C(χ)));

(6) (A(φ) ∨ (B(ψ) ∨ C(χ))) = ((A(φ) ∨B(ψ)) ∨ C(χ)));

(7)∧ψ∈X

(A(φ) ∨ (A(φ) ∧B(ψ))) = A(φ) =∨ψ∈X

(A(φ) ∧ (A(φ) ∨B(ψ)));

(8) A(φ) ∨ (B(ψ) ∧ C(χ)) = (A(φ) ∨B(ψ)) ∧ (A(φ) ∨ C(χ));

(9) A(φ) ∧ (B(ψ) ∨ C(χ)) = (A(φ) ∧B(ψ)) ∨ (A(φ) ∧ C(χ)).

Proof. (1) to (7) follow from the definition of lattice, and (8) and (9) follow from the

distributive property of BL algebra (see 6.2.6 ).

Note 6.2.8. If φ→ ψ is a tautology in basic logic BL and if a multi-fuzzy set A is

an evaluation of formulae, then A(φ) ⇒ A(ψ) = 1. Lemma 6.2.2 and Lemma 1.6.8

implies A(φ) ≤ A(ψ).

Theorem 6.2.9. Let A be a multi-fuzzy set in X. If A is an evaluation of a BL

system, then for every φ, ψ, χ ∈ X :

(1) (A(φ)⇒ A(ψ)) ≤∧χ∈X

((A(ψ)⇒ A(χ))⇒ (A(φ)⇒ A(χ)));


(2) ((A(φ)⇒ A(ψ))⇒ A(χ)) ≤ (((A(ψ)⇒ A(φ))⇒ A(χ))⇒ A(χ));

(3) (A(φ)⊗ (A(φ)⇒ A(ψ))) = (A(ψ)⊗ (A(ψ)⇒ A(φ)));

(4) (A(φ)⇒ (A(ψ)⇒ A(χ))) = ((A(φ)⊗ A(ψ))⇒ A(χ)));

(5) (A(φ)⇒ A(ψ)) ∨ (A(ψ)⇒ A(φ)) = 1.

Proof. (1) Since (φ→ ψ)→ ((ψ → χ)→ (φ→ χ)) is a 1-tautology,

A(φ → ψ) ≤ A(((ψ → χ) → (φ → χ)))(see Definition 1.6.1 and Note 6.2.8).

Therefore (A(φ) ⇒ A(ψ)) ≤ ((A(ψ) ⇒ A(χ)) ⇒ (A(φ) ⇒ A(χ))),∀χ ∈ X. Hence

(A(φ)⇒ A(ψ)) ≤∧χ∈X

((A(ψ)⇒ A(χ))⇒ (A(φ)⇒ A(χ))).

Similarly (2) to (4) follow immediately from Definition 1.6.1 and Note 6.2.8, and (5)

follows from 6.2.2.

Proposition 6.2.10. (see [29,78,86,111]). If L is a BL algebra and x, y ∈ L, then:

(1) x ≤ (x′ ⇒ y);

(2) x ≤ (x′)′;

(3) (x⇒ y) ≤ (y′ ⇒ x′);

(4) (x⊗ x′) = 0;

(5) x ≤ (1⊗ x);

(6) (x ∨ y)′ = (x′ ∧ y′);

(7) (x ∧ y)′ = (x′ ∨ y′).


system, then for every φ, ψ ∈ X :

(1) A(φ) ≤∧ψ∈X

(A′(φ)⇒ A(ψ));

(2) A(φ) ≤ (A′(φ))′;


(3) (A(φ)⇒ A(ψ)) ≤ (A′(ψ)⇒ A′(φ));

(4) (A(φ)⊗ A′(φ) = 0;

(5) A(φ) ≤ (1⊗ A(φ));

(6) (A(φ) ∨ A(ψ))′ = (A′(φ) ∧ A′(ψ));

(7) (A(φ) ∧ A(ψ))′ = (A′(φ) ∨ A′(ψ)).

Proof. (1) to (7) are obtained if we replace x by A(φ) and y by A(ψ) in the Proposi-

tion 6.2.10.

Proposition 6.2.12. (see [29]) Basic logic satisfies the following relations:

(1) (φ& (φ→ ψ))→ ψ;

(2) φ→ (ψ → (φ& ψ));

(3) φ→ (ψ → φ);

(4) (φ→ (ψ → χ))→ (ψ → (φ→ χ));

(5) (φ→ ψ)→ ((φ& χ)→ (ψ & χ));

(6) (φ→ ψ) & (χ→ ω)→ ((φ& χ)→ (ψ & ω));

(7) (φ→ ψ)→ (φ→ (φ ∧ ψ));

(8) ((φ→ ψ) ∧ (φ→ χ))→ (φ→ (ψ ∧ χ));

(9) (φ→ ψ)→ ((φ ∨ ψ)→ φ);

(10) ((φ→ χ) ∧ (ψ → χ))→ ((φ ∨ ψ)→ χ);

(11) φ& (ψ ∧ χ) ≡ (φ& ψ) ∧ (φ& χ);

(12) φ& (ψ ∨ χ) ≡ (φ& ψ) ∨ (φ& χ).



system, then for every φ, ψ, χ, ω ∈ X :

(1)∨φ∈X

(A(φ)⊗ (A(φ)⇒ A(ψ))) ≤ A(ψ);

(2) A(φ) ≤∧ψ∈X

(A(ψ)⇒ (A(φ)⊗ A(ψ)));

(3) A(φ) ≤∧ψ∈X

(A(ψ)⇒ A(φ));

(4) (A(φ)⇒ (A(ψ)⇒ A(χ))) ≤ (A(ψ)⇒ (A(φ)⇒ A(χ)));

(5) (A(φ)⇒ A(ψ)) ≤∧χ∈X

((A(φ)⊗ A(χ))⇒ (A(ψ)⊗ A(χ)));

(6) (A(φ)⇒ A(ψ))⊗ (A(χ)⇒ A(ω)) ≤ ((A(φ)⊗ A(χ))⇒ (A(ψ)⊗ A(ω)));

(7) (A(φ)⇒ A(ψ)) ≤ (A(φ)⇒ (A(φ) ∧ A(ψ)));

(8) (A(φ)⇒ A(ψ)) ∧ (A(φ)⇒ A(χ)) ≤ (A(φ)⇒ (A(ψ) ∧ A(χ)));

(9) (A(φ)⇒ A(ψ)) ≤ ((A(φ) ∨ A(ψ))⇒ A(φ));

(10) ((A(φ)⇒ A(χ)) ∧ (A(ψ)⇒ A(χ))) ≤ ((A(φ) ∨ A(ψ))⇒ A(χ));

(11) A(φ)⊗ (A(ψ) ∧ A(χ)) = (A(φ)⊗ A(ψ)) ∧ (A(φ)⊗ A(χ));

(12) A(φ)⊗ (A(ψ) ∨ A(χ)) = (A(φ)⊗ A(ψ)) ∨ (A(φ)⊗ A(χ)).

Proof. (1)-(12) follow from the definitions and the properties of basic logic BL in [29]

(see 6.2.12).

Note 6.2.14. Important generalized inference rules in multi-fuzzy logic are the

following:

(1) Generalized modus ponens, that is,∨φ∈X

(A(φ) ∧ (A(φ)⇒ B(ψ))) = B(ψ),∀ψ ∈ X;

6.3. DEDUCTIVE SYSTEMS OF RESIDUATED LATTICES 121

(2) Generalized modus tollens, that is,∨ψ∈X

(B′(ψ) ∧ (A(φ)⇒ B(ψ))) = A′(φ),∀φ ∈ X;

(3) Generalized hypothetical syllogism, that is,∨ψ∈X

(A(φ)⇒ B(ψ)) ∧ (B(ψ)⇒ C(χ)) = (A(φ)⇒ C(χ)),∀φ, χ ∈ X.

These inference rules are suitable for approximate reasoning.

6.3 Deductive Systems of Residuated LatticesIn this section we introduce the notions of multi-fuzzy filter, multi-fuzzy implicative

filter and multi-fuzzy regular filter, and derive some equivalent conditions for them.

Proposition 6.3.1. [29, 78, 86] Let L be a residuated lattice. For any x, y, z ∈ L,

the following properties hold:

(RL1) (x⊗ y)→ z = x→ (y → z);

(RL2) x→ (y → z) = y → (x→ z);

(RL3) y → z ≤ (x→ y)→ (x→ z);

(RL4) x→ y ≤ (y → z)→ (x→ z);

(RL5) x→ y ≤ y′ → x′;

(RL6) x ≤ y implies z → x ≤ z → y;

(RL7) x ≤ y implies y → z ≤ x→ z and y′ ≤ x′;

(RL8) 1→ x = x, x→ x = 1;

(RL9) 1′ = 0, 0′ = 1;

(RL10) x ≤ y if and only if x→ y = 1;

(RL11) x→ (y ∧ z) = (x→ y) ∧ (x→ z);


(RL12) (x ∨ y)→ z = (x→ z) ∧ (y → z);

(RL13) x→ (y → (x⊗ y)) = 1;

(RL14) x⊗ (y ∨ z) = (x⊗ y) ∨ (x⊗ z);

(RL15) (x ∨ y)′ = x′ ∧ y′;

(RL16) x⊗ y ≤ x ∧ y, x⊗ x′ = 0;

(RL17) y ≤ x→ y, x′ ≤ x→ y;

(RL18) x′ = x′′′, x ≤ x′′;

(RL19) x ∨ x′ = 1 implies x ∧ x′ = 0;

(RL20) x ∨ y ≤ ((x→ y)→ y) ∧ ((y → x)→ x).

6.3.1 Filters of Residuated LatticesDefinition 6.3.2. [36, 37, 42, 111] A non-empty subset F of a residuated lattice L

is called a filter of L, if it satisfies the axioms, for all x, y ∈ L :

(1) x, y ∈ F, implies x⊗ y ∈ F ;

(2) x ∈ F and x ≤ y implies y ∈ F.

Remark 6.3.3. Throughout this subsection we assume that elements of F are

nonzero.

Proposition 6.3.4. [78] Let F be a subset of L containing 1. The following

assertions are equivalent, for all x, y, z ∈ L :

(1) F is a filter of L;

(2) x, x→ y ∈ F implies y ∈ F ;

(3) x→ y, y → z ∈ F implies x→ z ∈ F ;


(4) x→ y, x⊗ z ∈ F implies y ⊗ z ∈ F ;

(5) x, y ∈ F and x ≤ y → z implies z ∈ F.

Definition 6.3.5. [30, 35, 86, 95, 109–111] Let F be a filter of a residuated lattice

L. For all x, y ∈ L :

(1) F is a Boolean filter (B-filter), if x ∨ x′ ∈ F.

(2) F is a G-filter (G-filter), if x2 → y ∈ F implies x→ y ∈ F, where x2 = x⊗ x.

(3) F is a MV filter (MV-filter), if y → x ∈ F implies ((x→ y)→ y)→ x ∈ F.

(4) F is a regular filter (R-filter), if x′′ → x ∈ F, where x′′ = (x′)′.

6.3.2 Fuzzy FilterDefinition 6.3.6. [36,37,42,111] A fuzzy set µ of a residuated lattice L is called a

multi-fuzzy filter of L, if it satisfies the axioms, for all x, y ∈ L :

(1) min{µ(x), µ(y)} ≤ µ(x⊗ y);

(2) x ≤ y implies µ(x) ≤ µ(y).

Remark 6.3.7. Throughout this subsection we assume that µ(0) = 0.

Definition 6.3.8. [30, 31, 35–37, 42, 86, 95, 108–111] Let µ be a fuzzy filter of a

residuated lattice L. For all x, y ∈ L :

(1) µ is a fuzzy Boolean filter (FB-filter), if µ(x ∨ x′) = µ(1);

(2) µ is a fuzzy G-filter (FG-filter), if µ(x→ (x→ y)) ≤ µ(x→ y);

(3) µ is a fuzzy MV filter (FMV-filter), if µ(y → x) ≤ µ(((x→ y)→ y)→ x);

(4) µ is a fuzzy regular filter (FR-filter), if µ(x′′ → x) = µ(1);

(5) µ is a fuzzy implicative filter, if µ(x→ (z′ → y)) ∧ µ(y → z) ≤ µ(x→ z).


6.3.3 Multi-fuzzy FilterMulti-fuzzy filters are the deductive systems of multi-fuzzy logic. In this subsection

we introduce the basic notions of multi-fuzzy filters and propose some results. Some

of the results included in this subsection are similar to the respective results in fuzzy

filters proposed by Liu and Li [42], and Zhu and Xu [111].

Definition 6.3.9. A multi-fuzzy set A of a residuated lattice L is called a multi-fuzzy

filter of L, if it satisfies the axioms, for all x, y ∈ L :

(MFF1) A(x) ∧ A(y) ≤ A(x⊗ y);

(MFF2) x ≤ y implies A(x) ≤ A(y).

Remark 6.3.10. Throughout this subsection we assume that A(0) = 0.

Theorem 6.3.11. Let A be a multi-fuzzy set in a residuated lattice L. Then the

following assertions are equivalent, for all x, y, z ∈ L :

(1) A is a multi-fuzzy filter of L;

(2) x ≤ y → z implies A(x) ∧ A(y) ≤ A(z);

(3) A(x) ≤ A(1) and A(x→ y) ∧ A(x) ≤ A(y).

Proof. (1)⇒ (2). Assume that, A is a multi-fuzzy filter of L.

x ≤ y → z implies x⊗ y ≤ z, by the adjoint property

implies A(x⊗ y) ≤ A(z), by (MFF2)

impliesA(x) ∧ A(y) ≤ A(z), by (MFF1).

(2)⇒ (3). Put y = x and z = 1 in (2), we have A(x) ∧ A(x) ≤ A(1), since

x ≤ x → 1. That is, A(x) ≤ A(1), ∀x ∈ L. Since (x → y) ≤ x → y, ∀x ∈ L and by

the assumption (2) we have A(x→ y) ∧ A(x) ≤ A(y).


(3)⇒ (1). By the assumption (3), we have A(x) ≤ A(1), and if x ≤ y, then

x → y = 1. Hence A(x) = A(x) ∧ A(1) = A(x) ∧ A(x → y) ≤ A(y). That is, x ≤ y

implies A(x) ≤ A(y). The adjoint property of residuated lattices implies,

x→ (y → (x⊗ y)) = (x⊗ y)→ (x⊗ y) = 1. (6.i)

A(x) ∧ A(y) = (A(x) ∧ A(1)) ∧ A(y)

= (A(x) ∧ A(x→ (y → (x⊗ y)))) ∧ A(y), by the equation (6.i)

≤A(y → (x⊗ y)) ∧ A(y), by the assumption (3)

≤A(x⊗ y), by the assumption (3).

Hence A is a multi-fuzzy filter.

Definition 6.3.12. A multi-fuzzy filter A of a residuated lattice L is called a multi-

fuzzy implicative filter, if it satisfies the axiom:

(MFF3) A(x→ (z′ → y)) ∧ A(y → z) ≤ A(x→ z),∀x, y, z ∈ L.

Theorem 6.3.13. Let A be a multi-fuzzy filter of L. Then the following assertions

are equivalent, for all x, y, z ∈ L :

(1) A is a multi-fuzzy implicative filter of L;

(2) A(x→ (z′ → z)) ≤ A(x→ z);

(3) A(x→ (z′ → z)) = A(x→ z);

(4) A(y → (x→ (z′ → z))) ∧ A(y) ≤ A(x→ z).

Proof. (1) ⇒ (2). Assume that, A is a multi-fuzzy implicative filter. In (MFF3)

put y = z, we have A(x → (z′ → z)) ∧ A(z → z) ≤ A(x → z). That is,

A(x→ (z′ → z)) ∧ A(1) ≤ A(x→ z). Hence A(x→ (z′ → z)) ≤ A(x→ z).


(2) ⇒ (3). Suppose that (2) holds. Proposition 6.3.1 (see (RL17) and (RL2))

implies

x→ z ≤ z′ → (x→ z) = x→ (z′ → z). (6.ii)

(6.ii) and (MFF2) imply that A(x → z) ≤ A(x → (z′ → z)). This and the

assumption (2) together imply A(x→ (z′ → z)) = A(x→ z).

(3) ⇒ (4). Suppose that (3) holds. Theorem 6.3.11 (see part 3 of the theorem)

and the assumption (3) together imply

A(y → (x→ (z′ → z))) ∧ A(y) ≤ A(x→ (z′ → z)) = A(x→ z).

(4) ⇒ (1). Suppose that (4) holds for every x, y, z ∈ L. Replace y by y → z in (4),

we have

A((y → z)→ (x→ (z′ → z))) ∧ A(y → z) ≤ A(x→ z). (6.iii)

Using Proposition 6.3.1 (see (RL2)), we have that

(y → z)→ (x→ (z′ → z)) =x→ ((y → z)→ (z′ → z))

=x→ (z′ → ((y → z)→ z)). (6.iv)

y ≤ y ∨ z ≤ ((y → z)→ z) ∧ ((z → y)→ y) ≤ (y → z)→ z. (6.v)

(6.v) and Proposition 6.3.1 (see (RL6)) together imply

z′ → y ≤ z′ → ((y → z)→ z).


Again by Proposition 6.3.1 (see (RL6)), we have

x→ (z′ → y) ≤ x→ (z′ → ((y → z)→ z)). (6.vi)

(6.vi), (MFF2) and (6.iv) together imply

A(x→ (z′ → y))≤A(x→ (z′ → ((y → z)→ z)))

=A((y → z)→ (x→ (z′ → z))). (6.vii)

(6.vii) and (6.iii) together imply

A(x→ (z′ → y)) ∧ A(y → z)≤A((y → z)→ (x→ (z′ → z))) ∧ A(y → z)

≤A(x→ z).

This proves that the condition (1) holds.

Definition 6.3.14. A multi-fuzzy filter A of a residuated lattice L is called a

multi-fuzzy regular filter (MFR-filter), if it satisfies the axiom:

(MFF4) A(x′′ → x) = A(1), for all x ∈ L.

Theorem 6.3.15. Let A be a multi-fuzzy filter of L. Then the following assertions

are equivalent, for all x, y, z ∈ L :

(1) A is an MFR-filter of L;

(2) A(x′ → y′) ≤ A(y → x);

(3) A(x′ → y′) = A(y → x);

(4) A(x′ → y) ≤ A(y′ → x);

(5) A(x′ → y) = A(y′ → x).


Proof. (1) ⇒ (2). Suppose thatA is an MFR-filter. Proposition 6.3.1 (see (RL5), (RL7)

and (RL18)) implies x′ → y′ ≤ y′′ → x′′ ≤ y → x′′ and hence

x′′ → x ≤ (y → x′′)→ (y → x) ≤ (x′ → y′)→ (y → x).

Thus, by (MFF2) we have

A(x′′ → x) ≤ A((x′ → y′)→ (y → x)). (6.viii)

A(x′ → y′) =A(x′ → y′) ∧ A(1).

=A(x′ → y′) ∧ A(x′′ → x), by Definition 6.3.14

≤A(x′ → y′) ∧ A((x′ → y′)→ (y → x)), by (6.viii)

≤A(y → x), ∀x, y ∈ L by Theorem 6.3.11. (6.ix)

(2) ⇒ (3). Suppose that A satisfies the the condition (2). By Proposition 6.3.1

(see (RL5)) and (MFF2), we have

A(y → x) ≤ A(x′ → y′) (6.x)

(6.x) and the assumption (2) together imply A(y → x) = A(x′ → y′).

(3) ⇒ (4). Suppose that (3) holds. Using Proposition 6.3.1 (see (RL18) and

(RL6)), we have

x′ → y ≤ x′ → y′′ = x′ → (y′)′. (6.xi)

6.4. MULTI-FUZZY LOGIC IN LIA 129

The assumption (3) and 6.xi together imply

A(x′ → y) ≤ A(x′ → (y′)′) = A(y′ → x).

(4) ⇒ (5). Suppose that A satisfies the condition (4) for all x, y ∈ L. Replace x

by y and y by x in (4), we have

A(y′ → x) ≤ A(x′ → y),∀x, y ∈ L. (6.xii)

(6.xii) and the assumption (4) together imply A(x′ → y) = A(y′ → x).

(5) ⇒ (1). Suppose that (5) holds. Using Proposition 6.3.1 (see (RL8)) and the

assumption (5), we have

A(1) = A(x′ → x′) = A(x′ → (x′)) = A((x′)′ → x) = A(x′′ → x).

Hence A is an MFR-filter.

6.4 Multi-fuzzy Logic in LIAIn this section we introduce and study the concept of lattice implication algebra on

multi-fuzzy sets and implication relation between propositions with different value

domains.

Definition 6.4.1. [94] A bounded lattice (L,∨,∧, 0, 1) with order-reversing involu-

tion ′ and a binary operation→ is a lattice implication algebra (LIA) if the following

axioms hold for every x, y, z ∈ L :

(1) x→ (y → z) = y → (x→ z);

(2) x→ x = 1;


(3) x→ y = y′ → x′;

(4) x→ y = y → x = 1⇒ x = y;

(5) (x→ y)→ y = (y → x)→ x;

(6) (x ∨ y)→ z = (x→ z) ∧ (y → z);

(7) (x ∧ y)→ z = (x→ z) ∨ (y → z).

Proposition 6.4.2. Let {(Lj,∨j,∧,′j ,→j, 0j, 1j) : j ∈ J} be a family of lattice

implication algebra (LIA), then (∏Lj,t,u,′ ,→, 0, 1) is an LIA with respect to the

following operations. For every x, y ∈∏Lj :

x ∨ y = (xj ∨j yj)j∈J ;

x ∧ y = (xj ∧j yj)j∈J ;

x→ y = (xj →j yj)j∈J ;

(x)′ = (x′j)j∈J .

Proof. Follows immediately from the definition.

Proposition 6.4.3. Let {(Lj,∨j,∧,′j ,→j, 0j, 1j) : j ∈ J} be a family of lattice im-

plication algebra (LIA). For every x, y ∈∏Lj, the operations x∨y, x∧y, (x)′, x→ y

defined as in Proposition 6.4.2 and

x⊕ y = x′ → y, x⊗ y = (x→ y′)′.

Thenx⊕ y = (xj ⊕j yj)j∈J and x⊗ y = (xj ⊗j yj)j∈J ,

wherexj ⊕j yj = x′j →j yj and xj ⊗j yj = (xj →j y

′j)′.


Proof. For every x, y ∈∏Lj,

x⊕ y=x′ → y

= (x′j)j∈J → (yj)j∈J

= (x′j →j yj)j∈J

= (xj ⊗j yj)j∈J .

Similarly

x⊗ y= (x→ y′)′

= ((xj)j∈J → (yj)′j∈J)′

= ((xj →j y′j)′)j∈J

= (xj ⊗j yj)j∈J .

Example 6.4.4. Let X be a nonempty set and {(Lj,∨j,∧,′j ,→j, 0j, 1j) : j ∈ J}

be a family of lattice implication algebra (LIA). For every x ∈ X and A,B ∈∏LXj ;

define

(A tB)(x) = A(x) ∨B(x),

(A uB)(x) = A(x) ∧B(x),

(A→ B)(x) = A(x)⇒ B(x)

and

A′(x) = (A(x))′.

As usual we can verify that (∏LXj ,t,u,′ ,→,0X ,1X) is an LIA. Note that A→ B is


the multi-fuzzy set {(x,A(x) ⇒ B(x)) : x ∈ X}. If A and B are crisp subsets of X,

then A→ B is the set compliment of A \B.

Theorem 6.4.5. [98] Let (L,∨,∧,′ ,→, 0, 1) be a lattice implication algebra, then

(L,∨,∧) is a distributive lattice.

Theorem 6.4.6. [98] Let (L,∨,∧,′ ,→, 0, 1) be a lattice implication algebra, the

lattice operations disjunction ∨, conjunction ∧, implication → and the complement ′

has the following relationships, for all x, y, z ∈ L :

(1) x ∨ y = (x→ y)→ y;

(2) x ∧ y = (x′ ∨ y′)′;

(3) x ≤ y if and only if x→ y = 1;

(4) x→ 0 = x′, 0→ x = 1, 1→ x = x, x→ 1 = 1;

(5) x→ y = 0 if and only if x = 1 and y = 0;

(6) x→ y ≤ (y → z)→ (x→ z);

(7) x′ ∨ y ≤ x→ y.

Theorem 6.4.7. [99] Let (L,∨,∧,′ ,→, 0, 1) be an LIA, for every x, y ∈ L, if x ≤ y,

then the following conditions are equivalent:

(1) y → z ≤ x→ z,∀z ∈ L;

(2) z → x ≤ z → y,∀z ∈ L;

(3) x→ z = x→ (y ∧ z), ∀z ∈ L;

(4) z → y = (z ∨ x)→ y,∀z ∈ L.

Definition 6.4.8. [98] A mapping f : L1 → L2 from lattice implication algebras L1

to L2 is called a lattice implication homomorphism if it satisfies the following axioms,

for every x, y ∈ L1 :


(1) f(x→ y) = f(x)→ f(y);

(2) f(x ∨ y) = f(x) ∨ f(y);

(3) f(x ∧ y) = f(x) ∧ f(y);

(4) f(x′) = f(x)′.

For more details of lattice implication algebra, we refer readers to [94,97,98].

Theorem 6.4.9. Let L and M be lattice implication algebras. If the mapping

h : M → L is a lattice implication homomorphism, then:

(1) h(1M) = 1L;

(2) h(0M) = 0L.

Proof. For every x ∈M,

(1) h(1M) = h(x→M x) = h(x)→L h(x) = 1L.

(2) h(x)→ 0L = (h(x))′ = h(x′) = h(x→ 0) = h(x)→ h(0).

Therefore,

h(0)→ 0L = h(0)→ h(0) = 1L

and implies h(0) ≤ 1L. But 0L ≤ h(x), for every x ∈M and so h(0M) = 0L.

Theorem 6.4.10. Let h :∏Mi →

∏Lj be a lattice implication homomorphism and

x, y ∈∏Mi, then:

(1) h(x⊕M y) = h(x)⊕L h(y);

(2) h(x⊗M y) = h(x)⊗L h(y);

(3) h(1M) = 1L;

(4) h(0M) = 0L,


where 0M , 1M , 0L and 1L are the least element of∏Mi, greatest element of

∏Mi,

least element of∏Li and greatest element of

∏Li respectively.

Proof. For every x, y ∈∏Mi,

(1) h(x⊕M y) = h(x′ →M y) = h(x′)→L h(y) = h(x)′ →L h(y) = h(x)⊕L h(y).

(2) h(x⊗M y) = h((x→M y′)′) = (h(x→M y′))′ = (h(x)→L h(y′))′

= (h(x)→L h(y)′)′ = h(x)⊗L h(y).

(3) Similar to part 1 of Theorem 6.4.9.

(4) Similar to part 2 of Theorem 6.4.9.

6.4.1 Implication Between Different Value DomainsThroughout this subsection M =

∏Mi and L =

∏Lj. The relations →M and →L

represent the implication relations in M and L respectively.

Definition 6.4.11. Let and h :∏Mi →

∏Lj be a lattice implication homomor-

phism. For every x ∈∏Mi and y ∈

∏Lj, an implication relation ’x implies y’ under

the bridge function h denoted by x →h y and defined by h(x) →L y , where →L is

the implication operation in L.

Lemma 6.4.12. Let f : X → Y be a crisp function, A ∈∏MX

i , B ∈∏LYj , 1L be

the greatest element of∏Lj and the mapping h :

∏Mi →

∏Lj be a bridge function

for the multi-fuzzy extension of f . If y = f(x), then A(x)→h B(y) = 1L.

Proof. Since x ∈ f−1(y) and h(A(x)) ≤∨

t∈f−1(y)

h(A(t)),

A(x)→h B(y) = h(A(x))→L B(y) = h(A(x))→L

∨t∈f−1(y)

h(A(t)) = 1L.


If y = f(x), then A(x)→h B(y) = 1L, for every bridge function h and so we write

A(x) ⇒ B(y) = 1L.

Theorem 6.4.13. Let y ∈∏Lj, z ∈

∏Nk and x, x1, x2 ∈

∏Mi. If h1 :

∏Mi →∏

Nk and h2 :∏Lj →

∏Nk are lattice implication homomorphisms, then:

(1) x→h1 (y →h2 z) = y →h2 (x→h1 z);

(2) x→h1 y ≤ h1(h−11 (y′)→M x′);

(3) h1((y →h−11 x)→M x) ≤ (x→h1 y)→L y;

(4) x1 ∨ x2 →h1 y = (x1 →h1 y) ∧ (x2 →h1 y);

(5) x1 ∧ x2 →h1 y = (x1 →h1 y) ∨ (x2 →h1 y).

Proof. 1. For every x ∈∏Mi, y ∈

∏Lj and z ∈

∏Nk,

x→h1 (y →h2 z) =h1(x)→N (h2(y)→N z)

=h2(y)→N (h1(x)→N z)

= y →h2 (x→h1 z).

2. For every x ∈∏Mi and y ∈

∏Lj,

x→h1 y= h1(x)→L y

= y′ →L (h1(x))′

= y′ →L h1(x′)

≤h1(h−11 (y′))→L h1(x′),


since h1(h−11 (y′)) ≤ y′. That is,

x→h1 y ≤ h1(h−11 (y′))→L h1(x

′) = h1(h−11 (y′)→M x′).

Equality holds, if h is surjective.

3. For every x ∈∏Mi and y ∈

∏Lj,

h1((y →h−11 x)→M x) = h1((h

−11 (y)→M x)→M x)

= h1(h−11 (y)→M x)→L h1(x)

= (h1(h−11 (y))→L h1(x))→L h1(x)

≤ (y →L h1(x))→L h1(x)

= (h1(x)→L y)→L y

= (x→h1 y)→L y.

Equality holds, if h is surjective.

4. For every x1, x2 ∈∏Mi and y ∈

∏Lj,

x1 ∨ x2 →h1 y=h1(x1 ∨ x2)→L y

= (h1(x1) ∨ h1(x2))→L y

= (h1(x1)→L y) ∧ (h1(x2)→L y)

= (x1 →h1 y) ∧ (x2 →h1 y).


5. For every x1, x2 ∈∏Mi and y ∈

∏Lj,

x1 ∧ x2 →h1 y=h1(x1 ∧ x2)→L y

= (h1(x1) ∧ h1(x2))→L y

= (h1(x1)→L y) ∨ (h1(x2)→L y)

= (x1 →h1 y) ∨ (x2 →h1 y).

∞

chapter 6 multi-fuzzy logic

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