On Extended Representable Uninorms and Their Extended Fuzzy Implications ( Coimplications )

In this work, by Zadeh’s extension principle, we extend representable uninorms and their fuzzy implications (coimplications) to type-2 fuzzy sets. Emphatically, we investigate in which algebras of fuzzy truth values the extended operations are type-2 uninorms and type-2 fuzzy implications (coimplications), respectively.


Preliminaries
Some concepts and facts will be listed in this section.For the sake of convenience, we use I to denote the unit interval [0, 1].Definition 1.In References [41,42], a binary function U : I 2 → I is called a uninorm if it is commutative, associative, non-decreasing in each place and there exists some element e ∈ I (called neutral element of U) such that U(x, e) = x for all x ∈ I .
Here, h is called an additive generator of U.

Definition 4.
In reference [30], a function J : I 2 → I is called a fuzzy coimplication if it is decreasing in its first variable and is increasing in its second variable and satisfies J(0, 0) = J(1, 1) = 0 and J(0, 1) = 1.

Definition 5.
In references [22,24], fuzzy truth values are mappings of I onto itself.The set of fuzzy truth values is denoted by F .
Example 1. Two special fuzzy truth values are the following: Generally, for any constant e ∈ [0, 1], we define fuzzy truth value e as e(x) = 1, x = e, 0, otherwise.

Definition 6.
In reference [20], a fuzzy truth value f ∈ F is said to be (i) normal if there exists some x 0 ∈ [0, 1] such that f (x 0 ) = 1.The set of all normal fuzzy truth values is denoted by F N . (ii Here, * is called the extended * , or extend operation of * . Example 2. (i) If * is t-norm T M = min or t-conorm S M = max, then we get The forms of ( 1) and ( 2) are rewritten as f g and f g, respectively.
(ii) If * is uninorm U, then we have extended uninorm by The operations and above define two partial orders and on F [20].In particular, f g if and only if f g = f , and f g if and only if f g = g.In general, the two partial orders are not the same and neither implies the other.However, the two partial orders coincide in F CN .

Remark 1.
In reference [20], the following holds.(i) For any fuzzy truth value f , f L is increasing and f R is decreasing.
(ii) A fuzzy truth value f is convex if and only if f = f L ∧ f R .(iii) For any fuzzy truth values f and g, it holds that In reference [20], let T be a t-norm and S be a t-conorm.The following hold for all f , g ∈ F if and only if h is convex: f (y).
Type-1 uninorms and fuzzy implications (coimplications) are defined in the algebra I = (I , ∨, ∧, ≤, 0, 1).We will define type-2 uninorms and fuzzy implications (coimplications) analogously to their respective type-1 counterparts.The underlying set of truth values is generalized from I to a subset of F , and since it may not be a lattice, the two partial orders defined by and are considered instead of ≤.
and it is antitone in the first argument and monotone in the second argument w.r.t. at least one of the partial orders and .
(iii) A function : F 2 −→ F is called a type-2 fuzzy coimplication over A, if it satisfies and it is antitone in the first and monotone in the second argument w.r.t. at least one of the partial orders and .
Remark 2. It is worth pointing out that extended fuzzy implications (coimplications) or uninorms are not necessary type-2 fuzzy implications (coimplications) or uninorms.We will try to find the conditions under which extended fuzzy implications (coimplications) or uninorms are type-2 fuzzy implications (coimplications) or uninorms.

Extended Representable Uninorms
Lemma 1.Let A ⊆ F , U be a type-1 uninorm with neutral element e ∈ (0, 1), and U be its extension.Then U is commutative, associative and has neutral element e.
(⇒) Suppose that (I) = (I I).Let f = e and g(q) = 1, q ≥ e, 0, otherwise.For any z ∈ (0, 1), (I) = U(p∧q,y)=z and It can be proved that In fact, if U(q, t) = z and q ≥ e, then t ≤ z and hence U(q,t)=z, q≥e (h(t)) ≤ t≤z (h(t)).On the contrary, if t ≤ z, there always exists some ) ≥ e such that U(q, t) = z and so From the above, we know that Following this fact, we can get that , U be a type-1 conjunctive representable uninorm with neutral element e ∈ (0, 1) and U be its extension.Then, U is a type-2 uninorm on A with neutral element e if and only if A ⊆ F C .Moreover, where h is an additive generator of U.
Proof.(⇐) Lemma 1 shows that U is associative, commutative and has neutral element e. Suppose From the above proposition, we obtain that ( If z ∈ (0, 1), then x, y ∈ (0, 1) and U(x, y) Similar to the above, we have the following facts for disjunctive representable uninorms.Proposition 3. Let A ⊆ F , U be a type-1 disjunctive representable uninorm with neutral element e ∈ (0, 1) and U be its extension.Then, (( , U be a type-1 disjunctive representable uninorm with neutral element e ∈ (0, 1) and U be its extension.Then, U is a type-2 uninorm on A with neutral element e if and only if A ⊆ F C .Moreover, where h is an additive generator of U. Baczy ński and Jayaram in Reference [32] have proved that I U,N is a type-1 fuzzy implication if and only if U is a disjunctive uninorm.
By the same way, we can define (U,N)-coimplications J U,N from a conjunctive uninorm U and a strong negation N, that is J U,N (x, y) = U(N(x), y), x, y ∈ I .
For a (U,N)-implication I U,N derived from a disjunctive uninorm U and a strong negation N, its extended operation is given by For a (U,N)-coimplication J U,N derived from a disjunctive uninorm U and a strong negation N, its extended counterpart is given by Lemma 2. Let A ⊆ F , I U,N be a (U,N)-implication derived from a disjunctive representable uninorm U and a strong fuzzy negation N, and I U,N be the extended operation of I U,N .For any f , g ∈ A , if f , g ∈ F N , then f I U,N g is normal.
Consequently, we have that Namely, f I U,N g is normal.
Lemma 3. Let A ⊆ F , I U,N be a (U,N)-implication derived from a disjunctive representable uninorm U and a strong fuzzy negation N, and I U,N be the extended operation of I U,N .For any f , Proof.Assume that f , g ∈ F C and 0 < x ≤ z ≤ y < 1.Then, ) and hence Thus, Remark 3. The above proof that I U,N is convex on z ∈ (0, 1) is similar to that of Proposition 3.6 in Ref. [29].However, for the consistency of this proof, we give it again.

Lemma 4.
Let A ⊆ F , I U,N be a (U,N)-implication derived from a disjunctive representable uninorm U and a strong fuzzy negation N, and I U,N be the extended operation of I U,N .Then, for any f , g ∈ F if and only if h is convex on I . Proof. and and Similarly to the proof of Proposition 2, we can prove Theorem 4. Let A ⊆ F , A = (A , 0, 1, , ), I U,N be a (U,N)-implication derived from a disjunctive representable uninorm U and a strong fuzzy negation N, and I U,N be the extended operation of I U,N .If A ⊆ F CN , then I U,N is a type-2 fuzzy implication.In addition, where h is an additive generator of U.
Proof.From Equation ( 4), one can easily obtain that Let f , g ∈ A with f g.Then, f g = f .In the following, we will prove that g I U,N h f I U,N h for any h ∈ A .In fact, according to Lemma 4, it holds that Since f , g, h ∈ A ⊆ F CN , from Lemmas 2 and 3, we obtain that ( f I U,N h), (g I U,N h) ∈ F CN .Again from Proposition 1, we can obtain that Thus, if f g, then (g I U,N h) ( f I U,N h) = (g I U,N h) for any h ∈ A , or g I U,N h f I U,N h, which means that I U,N is decreasing in the first place with respect to the partial order .Remember that the partial orders and coincide in F CN .Then, I U,N is decreasing in the first place with respect to the partial order as well.
Similarly, if f g, then h I U,N f h I U,N g for any h ∈ A , namely, I U,N is increasing in the second place with respect to the partial order , whence I U,N is increasing in the second place with respect to the partial order .
To sum up, I U,N is a type-2 fuzzy implication on A. By simple computation, one can easily obtain (5).
The following are some properties for type-2 fuzzy implications.
Theorem 5. Let A ⊆ F CN , A = (A , 0, 1, , ), I U,N be a (U,N)-implication derived from a disjunctive representable uninorm U and a strong fuzzy negation N, and I U,N be a type-2 fuzzy implication.Then, we have the following properties for I U,N : where U c is a conjunctive uninorm given by U c (x, y) = N U(N(x), N(y)) (namely, U c is a representable uninorm dual with U with respect to N).
Proof.(i) and (v) can be easily obtained. (vi) Just like (U,N)-implications, we can obtain the following facts about (U,N)-coimplications.
Lemma 5. Let A ⊆ F , J U,N be a (U,N)-coimplication derived from a conjunctive representable uninorm U and a strong fuzzy negation N, and J U,N be the extended operation of J U,N .For any f , g ∈ A , if f , g ∈ F N , then f J U,N g is normal.
Lemma 6.Let A ⊆ F , J U,N be a (U,N)-coimplication derived from a conjunctive representable uninorm U and a strong fuzzy negation N, and J U,N be the extended operation of J U,N .For any f , Lemma 7. Let A ⊆ F , J U,N be a (U,N)-coimplication derived from a conjunctive representable uninorm U and a strong fuzzy negation N, and J U,N be the extended operation of I U,N .Then, for any f , g ∈ A if and only if h is convex on I .Theorem 6.Let A ⊆ F , A = (A , 0, 1, , ), J U,N be a (U,N)-coimplication derived from a conjunctive representable uninorm U and a strong fuzzy negation N. If A ⊆ F CN , then J U,N is a type-2 fuzzy coimplication on A. In addition, where h is an additive generator of U.

Theorem 7.
Let A ⊆ F CN , A = (A , 0, 1, , ), J U,N be a (U,N)-coimplication derived from a conjunctive representable uninorm U and a strong fuzzy negation N and J U,N be a type-2 fuzzy coimplication on A.
Then we have the following facts.
, where U d is a disjunctive uninorm given by U d (x, y) = N U(N(x), N(y)) (namely, U d is a representable uninorm dual with U with respect to N). (ii) A function J U : I 2 → I is called an RU-cooperation if there exists a uninorm U such that
Proof.It is easy to prove it.

Lemma 9.
Let A ⊆ F , I U be an RU-implication derived from a representable uninorm U and I U be its extended operation.For any f , g ∈ A , if f , g ∈ F N , then f I U g is normal.
Proof.It is similar to Lemma 2.
Lemma 10.Let A ⊆ F , U be a representable uninorm, I U be a RU-implication derived from U and I U be its extended operation.For any f , g ∈ Proof.It is similar to Lemma 3.

Lemma 11.
Let A ⊆ F , U be a representable uninorm, I U be a RU-implication derived from U and I U be its extended operation.Then, Proof.It is easy to prove it.
For extended RU-coimplications that are derived from representable uninorms, we can similarly obtain the following results.Theorem 10.Let A ⊆ F , A = (A , 0, 1, , ), U be a representable uninorm, J U be a RU-coimplication derived from U and J U be its extended operation.If A ⊆ F CN , then J U is a type-2 fuzzy coimplication on A. In addition, where h is an additive generator of U.
Theorem 11.Let A ⊆ F CN , A = (A , 0, 1, , ), U be a representable uninorm, J U be a RU-coimplication derived from U and J U be a type-2 fuzzy coimplication.Then, the following hold: (i) e J U f = f , where e is the neutral element of uninorm U.

Conclusions
Uninorms and fuzzy implications are important operations in type-1 fuzzy sets.In this work, by Zadeh's extension principle, we extended uninorms and fuzzy implications (coimplications) to type-2 fuzzy sets and defined type-2 uninorms and fuzzy implications (coimplications).We focused on discussing in which algebras of fuzzy truth values extended representable uninorms and its fuzzy implications (coimplications) are type-2 uninorms and fuzzy implications (coimplications), respectively.First, extended representable uninorms were discussed.According to the distributive equation of extended conjunctive representable uninorms over type-2 meet, we had the sufficient and necessary conditions under which extended conjunctive representable uninorms are type-2 uninorms.Similar results were obtained for extended disjunctive representable uninorms.As for extended fuzzy implications, including extended (U,N)-implications and RU-implications, which are derived from representable uninorms, we proved that, in the algebra of convex and normal fuzzy truth values, they are type-2 fuzzy implications.Similarly, we obtained results for extended (U,N)-coimplications and RU-coimplications.
Since Wang and Hu [29] proposed the concept of generated extended fuzzy implications, in future work, we will also study generalized extended uninorms.

9 .
A function I U,N : I 2 → I is called a (U,N)-operation if there exists a uninorm U and a strong negation N such that I U,N (x, y) = U(N(x), y), x, y ∈ I .
) and Their Properties Definition 10. (i) A function I U : I 2 → I is called an RU-operation if there exists a uninorm U such that I U (x, y) = sup{z ∈ [0, 1]|U(x, z) ≤ y}, x, y ∈ I .
The set of all convex fuzzy truth values is denoted by F C .Let F CN denote the set of all convex and normal fuzzy truth values.
According to Zadeh's extension principle, a two-place function * : I 2 → I can be extended to * : F 2 −→ F by the convolution of * with respect to ∧ and ∨.Let f , g ∈ F , then

Definition 8 .
Let A = (A , 0, 1, , ), where A ⊆ F .(i)A function • : F 2 −→ F is called a type-2 uninorm over A , if itis commutative, associative, non-decreasing in each variable with at least one of the partial orders and , and there exists e ∈ F , called the neutral element of •, such that f