Conditional distributivity for uni-nullnorms with continuous and Archimedean underlying t-norms and t-conorms

1 Introduction

The process of merging a given number of data into a representative value is usually carried out by the so-called aggregation functions. There are many fields where such a process is required because aggregation functions is an fundamental tool from mathematics and computer sciences to economics and social sciences. Thus, the interest in aggregation functions has considerably grown in last decades and is supported by some monographs [2, 26].

There are many different classes of aggregation functions depending on the context where they are going to be applied. For instance, t-norms and t-conorms that generalize the logical connectives AND and OR of classical logic, are particularly important in the theory of fuzzy sets and its applications. In the same way, uninorms, generalizations of both t-norms and t-conorms, are useful not only in above commented field but also in many others like decision making, expert systems, fuzzy system modelling, aggregation and so on.

One of the main topics in the study of aggregation functions from the theoretical point of view is directed towards characterizing these aggregation functions with certain properties. One of these properties is conditional distributivity. Its interest comes from what is characterization of all distributivity and conditional distributivity uninorms over a continuous t-conorm, that is also the open problems pointed out in the Linz2000 closing session and recalled by Klement. It is closely related to the so-called pseudo-analysis, where the structure of R as vector space is replaced by the structure of semi-ring on any interval [a, b] ⊆ [- ∞ , + ∞], denoting the corresponding operations as pseudo-addition and pseudo-multiplication. In this context, t-norms and t-conorms, specially recently, uninorms, have been frequently used to model the mentioned pseudo-operations. Moreover, the distributivity property plays a fundamental role in the semi-ring structure and some generalizations, at least the conditional distributivity is essential.

In 2006, Ruiz and Torrent completely characterized conditional distributivity for the most usual known classes of uninorms over a continuous t-conorm [20], the results show that distributivity and conditional distributivity are equivalent for these cases. In 2015, Li and Liu investigated the same topic on uninorms with continuous underlying operators. Meanwhile, Li et al. also characterized all uninorms when the underlying operators are Archimedean. Recently, Liu [16] investigated the conditional distributivity for semi-uninorms over continuous t-conorms and t-norms. The obtained results showed that the left or right conditional distributivity implies that S = max or S is Archimedean for the two classes of semi-uninorms.

On the other hand, as the special case of binary aggregation functions, 2-uninorms and further extended n-uninorms which generalize both nullnorms and uninorms, were introduced by P. Akella and studied by many researchers. In 2017, letting a uninorm and a nullnorm share the same underlying t-conorm (resp. t-norm), Sun et al. [23, 24] introduced concept of a uni-nullnorm (resp. null-uninorm) which is a special case of 2-uninorms. Further, they proved that uni-nullnorms and null-uninorms are a pair of dual operations and finally characterized structures when their underlying operators are Archidemean.

Along this thinking, this paper is focused on and fully characterizes the conditional distributivity for a uni-nullnorm with continuous and Archimedean underlying t-norms and t-conorm over a continuous t-conorm or a uninorm from U_min ∪ U_max with continuous underlying operators.

The article is organized as follows. In Section 2, we recall some basic definitions and facts of uninorms and uni-nullnorms with continuous and Archimedean underlying t-norms and t-conorms. In Section 3, we characterize the conditional distributivity of uni-nullnorm with continuous and Archimedean underlying t-norms and t-conorms over a continuous t-conorm. In Section 4, we do the same topic for a uninorm of the form U_min ∪ U_max with the continuous underlying operators. Section 5 is Conclusions and future work.

2 Preliminaries

In this section, we only recall some basic notations. See [2, 13] and [23] for more details.

A triangular norm (t-norm for short) is a function T : [0, 1] ² → [0, 1] which is commutative, associative, non-decreasing in each variable and has 1 as its neutral element. Dual functions to t-norms are t-conorms. A triangular conorm (t-conorm for short) is a function S : [0, 1] ² → [0, 1] which is commutative, associative, nondecreasing in each variable and has 0 as its neutral element. The duality between t-norms and t-conorms is expressed by the fact that from any t-norm T we can obtain its dual t-conorm S by the equation S (x, y) =1 - T (1 - x, 1 - y) and vice-versa. Therefore, we only give some definitions and results about t-norms.

Definition 2.1. ([13]) A t-norm T is said to be

Archimedean, if for every (x, y) ∈ (0, 1) ², there exists some n ∈ N such that x T ( n ) < y , where x T ( n ) = T ( x , x , ⋯ , x ) ︸ n times ;

Remark 2.2. ([13]) For a continuous t-norm T,

it is Archimedean if and only if T (x, x) < x holds for every x ∈ (0, 1).

Definition 2.3. ([7, 26]) A binary operator Ucolon [0, 1] ² → [0, 1] is called a uninorm if it is commutative, associative, non-decreasing and there exists e ∈ [0, 1] called a neutral element such that U (x, e) = x for all x ∈ [0, 1]. A uninorm with the neutral element e = 1 is a t-norm and a uninorm with the neutral element e = 0 is a t-conorm. The associativity of uninorm U implies that U (0, 1) = U (1, 0) ∈ {0, 1}. U is called a conjunctive (disjunctive) uninorm when U (0, 1) =0 (U (0, 1) =1). A uninorm with the neutral element e is said to be proper if 0 < e < 1. Throughout this paper, we exclusively consider proper uninorm. For any proper uninorm U, we can associate two binary operations T _U , S _U  : [0, 1] ² → [0, 1] defined by T U ( x , y ) = U ( ex , ey ) e and S U ( x , y ) = U ( e + ( 1 - e ) x , e + ( 1 - e ) y ) - e 1 - e . Clearly, T _U and S _U is respectively a t-norm and a t-conorm. In other words, for any e ∈ (0, 1), the uninorm U works as a t-norm in [0, e] ² and as a t-conorm in [e, 1] ², T _U and S _U are respectively called the underlyling t-norm and the underlyling t-conorm. Let us denote the remaining part of the unit square by A (e), i.e, A (e) = [0, e) × (e, 1] ∪ (e, 1] × [0, e) = [0, 1] ² \ ([0, e] ² ∪ [e, 1] ²). On the set A (e), any uninorm U is bounded by the minimum and maximum of its arguments, i.e, for any (x, y) ∈ A (e) it holds that min(x, y) ≤ U (x, y) ≤ max(x, y) . Now we recall several classes of uninorms used later. Theorem 2.4. ([7]) Suppose that U is a uninorm with the neutral element e ∈ (0, 1) and both functions x ↦ U (x, 1) and x ↦ U (x, 0)   (∀ x ∈ [0, 1]) are continuous except at the point x = e. Then U is given by one of the following forms.

If U (0, 1) =0, then

In two previous formulas, the class of all uninorms of the form (1) is denoted by U_min and these of the form (2) by U_max. Remark 2.5. ([11]) The first uninorms considered by Yager and Rybalov [26] are idempotent uninorms U e min and U e max from classes U_min ∪ U_max, and have respectively the following form ( 3 ) U e min ( x , y ) = { max ( x , y ) ( x , y ) ∈ [ e , 1 ] 2 , min ( x , y ) otherwise ,(3) and ( 4 ) U e max ( x , y ) = { min ( x , y ) ( x , y ) ∈ [ 0 , e ] 2 , max ( x , y ) otherwise .(4)Definition 2.6. ([7]) Consider e ∈ (0, 1). A binary operation U : [0, 1] ² → [0, 1] is a representable uninorm if there exists a continuous strictly increasing function h : [0, 1] → [- ∞ , + ∞] with h (0) = -∞, h (e) =0 and h (1) =+ ∞ such that ( 5 ) U ( x , y ) = h - 1 ( h ( x ) + h ( y ) )(5) for all (x, y) ∈ [0, 1] ² ∖ {(0, 1) , (1, 0)} and U (0, 1) = U (1, 0) ∈ {0, 1}. h is usually called an additive generator of U.

Note that for a representable uninorm U, it holds that U (0, x) = U (x, 0) =0 for any x ∈ [0, 1) and U (1, x) = U (x, 1) =1 for any x ∈ (0, 1].

Lemma 2.7. ([8]) Let U be a uninorm with the neutral element e ∈ (0, 1) such that the underlying t-norm T _U and t-conorm S _U are strict. Then U is representable if and only if there exists some (x₀, y₀) ∈ (0, e) × (e, 1) ∪ (e, 1) × (0, e) such that min(x₀, y₀) < U (x₀, y₀) < max(x₀, y₀).

Definition 2.8. ([4]) A nullnorm V is a binary operator Vcolon [0, 1] ² → [0, 1], which is commutative, associative, non-decreasing in each variable and there exists some element k ∈ [0, 1] called the absorbing element satisfying V (k, x) = k for all x ∈ [0, 1] and V (0, x) = x for all x ≤ k and V (1, x) = x for all x ≥ k. Obviously, when k = 0 and 1, then V is respectively a t-norm and a t-conorm. It is easy to check that k = V (1, 0), V (x, 0) = min(x, k) and V (x, 1) = max(x, k) for all x ∈ [0, 1]. Theorem 2.9. ([4]) If the nullnorm V satisfies the condition k = V (1, 0) ∈ (0, 1), then V ( x , y ) = { kS V ( x k , y k ) ( x , y ) ∈ [ 0 , k ] 2 , k + ( 1 - k ) T V ( x - k 1 - k , y - k 1 - k ) ( x , y ) ∈ [ k , 1 ] 2 , k otherwise , where T_V and S_V are respectively a t-norm and a t-conorm.

Definition 2.10. ([2]) Let F be a binary commutative operator on [0, 1]. Then {e₁, e₂}  _a is called the 2-neutral element of F if F (e₁, x) = x for all x ≤ a and F (e₂, x) = x for all x ≥ a, where 0 < a < 1, e₁ ∈ [0, a] and e₂ ∈ [a, 1].

Definition 2.11. ([2]) A commutative, associative, increasing binary operator F having a 2-neutral element {e₁, e₂}  _a is called a 2-uninorm. Definition 2.12. ([23]) A uni-nullnorm F is a 2-uninorm having a 2-neutral element {e, 1}  _a with a as an annihilator over [e, 1]. Obviously, F is a uninorm if a = 1, and a nullnorm if e = 0, and a t-conorm if e = 0 and a = 1, and a t-norm if e = a = 0 or e = a = 1. A uni-nullnorm with the 2-neutral element {e, 1}  _a is proper if 0 < e < a < 1.

Lemma 2.13. ([23]) If F is a proper uni-nullnorm with the 2-neutral element {e, 1} _a, then F (0, a) = F (0, 1) ∈ {0, a}.

For any proper uni-nullnorm F with the 2-neutral element {e, 1}  _a , define binary operators T F ll , S F , T F ur , U F and V _F by T F ll ( x , y ) = F ( ex , ey ) e , x , y ∈ [ 0 , 1 ] , S F ( x , y ) = F ( e + ( a - e ) x , e + ( a - e ) y ) - e a - e , x , y ∈ [ 0 , 1 ] , T F ur ( x , y ) = F ( a + ( 1 - a ) x , a + ( 1 - a ) y ) - a 1 - a , x , y ∈ [ 0 , 1 ] , U F ( x , y ) = F ( ax , ay ) a , x , y ∈ [ 0 , 1 ] , V F ( x , y ) = F ( e + ( 1 - e ) x , e + ( 1 - e ) y ) - e 1 - e , x , y ∈ [ 0 , 1 ] .

Lemma 2.14. ([24]) Let F be a proper uni-nullnorm with the 2-neutral element {e, 1}  _a . Then

T F ll and T F ur are t-norms, S _F is a t-conorm.

Lemma 2.15. ([24]) Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean. Then one of the two following statements holds:

Either F (x, a) = x for all x ∈ (0, e) or F (x, a) = a for all x ∈ (0, e).

Theorem 2.16. ([24]) Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean. Then one of the eight following statements holds:

F (x, y) = ( 6 ) { eT F ll ( x e , y e ) ( x , y ) ∈ [ 0 , e ] 2 , e + ( a - e ) S F ( x - e a - e , y - e a - e ) ( x , y ) ∈ [ e , a ] 2 , a + ( 1 - a ) T F ur ( x - a 1 - a , y - a 1 - a ) ( x , y ) ∈ [ a , 1 ] 2 , a ( x , y ) ∈ [ e , a ] × [ a , 1 ] ∪ [ a , 1 ] × [ e , a ] , min ( x , y ) otherwise .(6)

where U_F is a conjunctive representable uninorm.

Now, we review definition and results of the conditional distributivity equation of two binary operators. Definition 2.17. ([20]) Let F, G:  [0, 1] ² → [0, 1] be two commutative aggregation operators, F is distributive over G, if ( 14 ) F ( x , G ( y , z ) ) = G ( F ( x , y ) , F ( x , z ) ) , for all x , y , z ∈ [ 0 , 1 ] .(14)

Definition 2.18. ([20]) Let F:  [0, 1] ² → [0, 1] be a commutative aggregation operator, G be a uninorm with the neutral element e ∈ [0, 1), F is conditionally distributive over G if ( 15 ) F ( x , G ( y , z ) ) = G ( F ( x , y ) , F ( x , z ) ) , for all x , y , z ∈ [ 0 , 1 ] and G ( y , z ) < 1 .(15)

Theorem 2.19. ([20]) Let U be a representable uninorm with the neutral element e ∈ (0, 1) and S be a continuous t-conorm. The following conditions are equivalent.

U is distributive over S.

Theorem 2.20. ([13]) A continuous t-norm T is conditionally distributive over a continuous t-conorm S if and only if exactly one of the following cases is fulfilled:

S = S _M .

Remark 2.21. ([11]) Because a strict t-norm and a nilpotent t-conorm are isomorphic to product t-norm T _P  (x, y) = xy and Lukasiewicz t-conorm S _L  (x, y) = min {x + y, 1} respectively [13], the previous result is often reduced to the following pair ( 18 ) S ( x , y ) = { c + ( 1 - c ) S L ( x - c 1 - c , y - c 1 - c ) ( x , y ) ∈ [ c , 1 ] 2 , max ( x , y ) otherwise ,(18) and ( 19 ) T ( x , y ) = { cT * * ( x c , y c ) ( x , y ) ∈ [ 0 , c ] 2 , c + ( 1 - c ) T P ( x - c 1 - c , y - c 1 - c ) ( x , y ) ∈ [ c , 1 ] 2 , min ( x , y ) otherwise .(19)

Lemma 2.22. ([19]) Let X≠∅, F : X² → X, Y ⊂ X, e ∈ Y be a neutral element for the operator F on Y ((∀ x ∈ Y) F (e, x) = F (x, e) = x). If the operator F is distributive over some operator G : X² → X that fulfils G (e, e) = e, then G is idempotent on Y.

Lemma 2.23. ([19]) Any increasing function F : [0, 1] ² → [0, 1] is distributive and also conditionally distributive over max and min.

In this section, we characterize all solutions of the conditional distributivity equations consisting of a proper uni-nullnorm F and a continuous t-conorm G.

Lemma 3.1. Let F be a proper uni-nullnorm with the forms Equations (6) or (11) such that T F ll , S F and T F ur are continuous and Archimedean, G be a continuous t-conorm. Then F is conditionally distributive over G if and only if exactly one of the following cases is fulfilled:

G = max.

Proof. (⇒) First, note that F is a proper uni-nullnorm with e ∈ (0, a) and G is a continuous t-conorm, then it holds that G (e, e) ≥ e. Indeed, we can claim G (e, e) = e. If otherwise, then we assume that G (e, e) > e. Since G is continuous, there exists some z′ ∈ (0, e) such that G (z′, z′) = e < 1. Taking e < x < a, y = z = z′ in Equation (15), then we have x = F (x, e) = F (x, G (z′, z′)) = G (F (x, z′) , F (x, z′)) = G (z′, z′) = e . But this is a contradiction. Therefore, it follows that G (e, e) = e . Further, applying Lemma 2.22, then we get G (x, x) = x for all x ≤ a.

Now, let us consider structure of F and G in [a, 1] ². Without loss of generality, assume that there exists c ∈ [a, 1) such that c is an idempotent element of G, then we have that F (x, c) = F (x, G (c, c)) = G (F (x, c) , F (x, c)) for all x ∈ [a, 1], which means that F (x, c) is also an idempotent element of G. Because it follows from continuity of T F ur that {F (x, c) |x ∈ [a, 1]} = [a, c], then we can get that G (x, x) = x for any x ∈ [a, c]. Hence, either G = max and then Claim (i) is proved, or there exists the largest nontrivial idempotent element c ∈ [a, 1) of G such that G (x, y) = (< c, 1, S^* >), where S^* is a continuous and Archimedean t-conorm. As the proof of Theorem 2.20, it is easy to draw the conclusion that c is also an idempotent element of F. Note that T F ur is a continuous and Archimedean t-norm, then we can obtain that c = a. Therefore, it holds from Remark 2.21 that G = (< a, 1, S _L  >) and T F ur = T P and then Claim (ii) is proved.

(⇐) It is enough to prove Claim (ii) because Claim (i) is clearly true. To do this, let G = (< a, 1, S _L  >) and T F ur = T P . Without loss of generality, suppose that y ≤ z. To continue the process, there are the four following cases to consider. Assume that (y, z) ∈ [0, 1] ² ∖ [a, 1] ², then it holds from assumption y ≤ z that G (y, z) = z and y ≤ a, which implies from structure of F that F (x, y) ≤ a and F (x, y) ≤ F (x, z) for all x ∈ [0, 1]. So it holds from structures of F and G that F (x, G (y, z)) = F (x, z) = max(F (x, y) , F (x, z)) = G (F (x, y) , F (x, z)).

Assume that (y, z) ∈ [a, 1] ² and {x ∈ [0, e), then we have G (y, z) ≥ a. To continue the process, there are the two following cases to consider. (1) If F is of the form Equation (6), then it holds from structures of F and G that F (x, G (y, z)) = x = G (x, x) = G (F (x, y) , F (x, z)). (2) If F is of the form Equation (11) and x ∈ (0, e), then it holds from structures of F and G that F (x, G (y, z)) = a = G (a, a) = G (F (x, y) , F (x, z)). Besides, when x = 0, clearly, the value of both sides of Equation (15) is equal to 0. Assume that (y, z) ∈ [a, 1] ² and x ∈ [e, a], then we have G (y, z) ≥ a. Therefore it holds from structures of F and G that F (x, G (y, z)) = a = G (a, a) = G (F (x, y) , F (x, z)). Assume that (y, z) ∈ [a, 1] ² and x ∈ [a, 1], then it holds from Claim (ii) that F (x, G (y, z)) = G (F (x, y) , F (x, z)).□ Lemma 3.2. Let F be a proper uni-nullnorm with the form Equation (7) such that T F ll , S F and T F ur are continuous and Archimedean, G be a continuous t-conorm. Then F is conditionally distributive over G if and only if exactly one of the following cases is fulfilled:

G = max.

Proof. (⇒) Similar to the corresponding proof of Lemma 3.1, so it is omitted. (⇐) It is enough to prove Claim (ii) because Claim (i) is clearly true. To do this, let G = (< a, 1, S _L  >) and T F ur = T P . Without loss of generality, suppose that y ≤ z. To continue the process, there are the three following cases to consider. Assume that (y, z) ∈ [0, 1] ² ∖ [a, 1] ², then it holds from assumption y ≤ z that G (y, z) = z and y ≤ a, which implies from structure of F that F (x, y) ≤ a and F (x, y) ≤ F (x, z) for all x ∈ [0, 1]. Therefore it holds from structures of F and G that F (x, G (y, z)) = F (x, z) = max(F (x, y) , F (x, z)) = G (F (x, y) , F (x, z)). Assume that (y, z) ∈ [a, 1] ² and x ∈ [0, a], then we have G (y, z) ≥ a, F (x, y) = F (x, z) = a. Therefore it holds from structures of F and G that F (x, G (y, z)) = a = G (a, a) = G (F (x, y) , F (x, z)). Assume that (y, z) ∈ [a, 1] ² and x ∈ [a, 1], then it holds from Claim (ii) that F (x, G (y, z)) = G (F (x, y) , F (x, z)). Lemma 3.3. Let F be a proper uni-nullnorm with the forms Equations (8) or (9) or (12) such that T F ll , S F and T F ur are continuous and Archimedean, G be a continuous t-conorm. Then F is conditionally distributive over G if and only if exactly one of the following cases is fulfilled:

G = max.

Proof. (⇒) First, note that F is a proper uni-nullnorm with e ∈ (0, a) and G is a continuous t-conorm, then it holds that G (e, e) ≥ e. Indeed, we can claim G (e, e) = e. If otherwise, then we assume that G (e, e) > e. Since G is continuous, there exists some z′ ∈ (0, e) such that G (z′, z′) = e < 1. Take e < x < a and y = z = z′, then from Equation (15) we get x = F (x, e) = F (x, G (z′, z′)) = G (F (x, z′) , F (x, z′)) = G (x, x) . Further, it follows from monotonicity of G that G (e, e) ≤ G (x, x) = x for all e < x < a, which implies that G ( e , e ) ≤ inf x ∈ ( e , a ) x = e , but this contradicts the assumption G (e, e) > e. Therefore, we have G (e, e) = e. The remaining proof is similar to that of Lemma 3.1, so it be omitted. (⇐) Completely similar to the corresponding proof of Lemma 3.2, so it is omitted.

Lemma 3.4. Let F be a proper uni-nullnorm with the forms Equations (10) or (13) such that T F ll , S F and T F ur are continuous and Archimedean, G be a continuous t-conorm. Then F is conditionally distributive over G if and only if exactly one of the following cases is fulfilled:

G = max.

Proof. (⇒) On one hand, we have clearly from the structure of F that F (a, a) = a. On the other hand, it follows from continuity of G that there exists some z′ ∈ (0, a] such that G (z′, z′) = a. Taking x = a and y = z = z′ in Equation (15), then we have a = F (a, G (z′, z′)) = G (F (a, z′) , F (a, z′)) = G (a, a) . Hence, a is a nontrivial idempotent element of G and then there exist two continuous t-conorms S^* and S^** such that G = (<0, a, S^* > , < a, 1, S^** >) because of continuity of G. Since F is conditionally distributive over G, then we have U _F and T F ur are respectively conditionally distributive over S^* and S^** and they simultaneously hold. Further, we knows from Theorem 2.19 that there exist two possibilities: Either (1) S^* = S _M or, (2) S^* is strict and the additive generator s of S^* satisfying s ( e a ) = 1 , is also a multiplicative generator of U _F . Furthermore, we knows from Theorem 2.20 that there exist two possibilities: Either (A) S^** = S _M or, (B)S^** = S _L and T F ur = T P . Considering combinations of all the above possibilities, we know that all required results are valid. (⇐) As other cases are similar, we only prove Case (iv). To do this, let G = (<0, a, S^* > , < a, 1, S _L  >) and T F ur = T P . Without loss of generality, suppose that y ≤ z. To continue the process, there are only three following cases to consider. Assume that (y, z) ∈ [0, a) × (a, 1), then it follows from y ≤ z that G (y, z) = max(y, z) = z, F (x, y) ≤ a and F (x, y) ≤ F (x, z). Therefore, it follows from structures of F and G that F (x, G (y, z)) = F (x, z) = max(F (x, y) , F (x, z)) = G (F (x, y) , F (x, z)). Assume that (y, z) ∈ [0, a] ², then we have G (y, z) ≤ a. If x ≤ a, then it holds from Theorem 2.19 and (iv) that F is conditionally distributive over G. If x > a, we have from structure of F that F (x, y) = F (x, z) = F (x, G (y, z)) = a, thus it holds that F (x, G (y, z)) = a = G (a, a) = G (F (x, y) , F (x, z)). Assume that (y, z) ∈ [a, 1] ², then we get that G (y, z) ≥ a. If x ≥ a, then it holds from Theorem 2.20 and (iv) that F is conditionally distributive over G. If x < a, we have that F (x, y) = F (x, z) = F (x, G (y, z)) = a, thus it holds that F (x, G (y, z)) = a = G (a, a) = G (F (x, y) , F (x, z)). {Besides, if F is of the form Equation (13) and x = 0, clearly, the value of both sides of Equation (15) is equal to 0. Taking into account the above results, we have the following theorem. Theorem 3.5. Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean, G be a continuous t-conorm. Then the following statements are true.

If F is given by Equations (6), (7), (8), (9), (11) or (12), then F is conditionally distributive over G if and only if exactly one of the following cases is fulfilled.

G = max.

4 The conditional distributivity of a uni-nullnorm over a uninorm

In this section, let us consider the conditional distributivity of a proper uni-nullnorm over a uninorm with neutral element e₁ ∈ (0, 1) and from the class U_min ∪ U_max. To do this, we need the following Lemma.

Lemma 4.1. Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean. G be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators, If F is conditionally distributive over G, then either e₁ = a or e₁ = e.

Proof. On the contrary, then we need consider two possibility. (i) If e₁ < e or e < e₁ < a, then taking x = y = e₁, z = e in Equation (15), we have e₁ = F (e₁, e)  = F (e₁, G (e₁, e)) = G (F (e₁, e₁) , F   (e₁, e)) = G (F (e₁, e₁) , e₁) = F (e₁, e₁) , but this contradicts the fact that T F ll and S _F are continuous and Archimedean.

(ii) If e₁ > a, taking x = z = e₁, then for any y ∈ (e₁, 1), we get from Equation (15) that F (e₁, y) = F (e₁, G (y, e₁)) = G (F (e₁, y) , F (e₁, e₁)) . Due to continuity of T F ur , the previous result can be extend to y = 1 and then we have e₁ = F (e₁, 1) = G (F (e₁, 1) , F (e₁, e₁)) = G (e₁, F (e₁, e₁)) = F (e₁, e₁) , but this contradicts the fact that T F ur is continuous and Archimedean. Thus, we have proven that e₁ = a or e₁ = e.

To continue the process, the following investigation will be divided into two cases: G ∈ U_max and G ∈ U_min. Now, let us study Case: G ∈ U_max.

Lemma 4.2. Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_max be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. If F is conditionally distributive over G, then e₁ = e.

Proof. First, note that Lemma 4.1 assure either e₁ = a or e₁ = e. Now, we can claim that e₁ = e. If otherwise, then we have e₁ = a. Taking x₀, y₀ such that e < x₀ < e₁ = a < y₀ < 1 and putting x = x₀, y = y₀, z = 0 in Equation (15), then we have = F (x₀, y₀) = F (x₀, G (y₀, 0)) = G (F (x₀, y₀) , F (x₀, 0)) = G (a, F (x₀, 0)) = F (x₀, 0) . Besides, it follows from Lemma 2.15 (ii) that F (x₀, 0) ∈ {0, x₀} < a, but this is a contradiction.

Therefore, it holds e₁ = e.

Lemma 4.3. Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_max be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. If F is conditionally distributive over G, then one of the following statements are true.

G = U e 1 max .

Proof. Since it follows from Lemma 4.2 that e₁ = e. Hence, we have G (e, e) = e. Further, applying Lemma 2.22, then we get that G (x, x) = x for all x ≤ a. Next, let us consider structure of F and G in [a, 1] ². Note this case is completely similar to the corresponding proof of Lemma 3.1, then we have either G = U e 1 max or T F ur = T P and G is given by Equation (20).

In fact, from the results in several following lemmas, we know that the conditions in Lemma 4.3 is only necessary but not sufficient.

Lemma 4.4. Let F be a proper uni-nullnorm with the forms Equations (6) or (7) or (11) such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_max be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is not conditionally distributive over G.

Proof. By Theorem 2.16, we know that F (x, y) = min(x, y) for (x, y) ∈ [0, e) × [e, a) ∪ [e, a) × [0, e). On the contrary, if F is conditionally distributive over G, then it follows from Lemma 4.2 that e₁ = e. Taking x, y, z such that y < x < e = e₁ < z < a, then it follows from structure of F thatF (x, y) ≤ min(x, y) = y < x < e₁, F (x, z) = min(x, z) = x and it holds from G ∈ U_max that G (y, z) = max(y, z) = z. Thus the left-hand side of Equation (15) is F (x, G (y, z)) = F (x, z) = x. On the other hand, it follows from Lemma 4.3 that either G = U e 1 max or G is given by Equation (20), which implies that the right-hand side of Equation (15) is G (F (x, y) , F (x, z)) = G (F (x, y) , x) = min(F (x, y) , x) = F (x, y) < x = F (x, G (y, z)) , but this is a contradiction. Therefore, F is not conditionally distributive over G.

Lemma 4.5. Let F be a proper uni-nullnorm with the forms Equations (8) or (12) such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_max be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is not conditionally distributive over G.

Proof. On the contrary, if F is conditionally distributive over G, then it follows from Lemma 4.3 that either G = U e 1 max or T F ur = T P and G has the form in Equation (20). Taking x, y, z such that 0 = y < z < e = e₁ < x < a, then we from structure of F and G that F (x, 0) =0, F (x, z) = x, G (0, z) =0, G (0, x) = x. Thus, the left-hand side of Equation (15) is F (x, G (0, z)) = F (x, 0) =0, while the right-hand side of Equation (15) is G (F (x, 0) , F (x, z)) = G (0, x) = x > F (x, G (0, z)) , but this is a contradiction. Therefore, F is not conditionally distributive over G.

Lemma 4.6. Let F be a proper uni-nullnorm with the form Equation (9) such that T F ll , S F and T F ur are continuous and Archimedean. G ∈ U_max be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is conditionally distributive over G if and only if exactly one of the following cases is fulfilled.

G = U e 1 max .

Proof. (⇒) Clearly, the conclusion is valid.

(⇐) Since Case (i) is very simple, we only check Case (ii) over here. If T F ur = T P and G is given by Equation (20), without loss of generality, suppose further that y ≤ z, then we get from monotonicity of F that F (x, y) ≤ F (x, z) for all x ∈ [0, 1].

Assume that (y, z) ∈ [0, e₁] ², then it holds from structure of G that G (y, z) = y and then we have F (x, G (y, z)) = F (x, y). To continue our progress, there are three cases to consider. If x ≤ e₁, then we get directly from the Lemma 2.23 that Equation (15). If a ≥ x > e₁, we have F (x, y) = F (x, z) = x, thus it follows that F (x, G (y, z)) = F (x, y) = x = G (x, x) = G (F (x, y) , F (x, z)). If x > a, we have F (x, y) = F (x, z) = a, thus it follows that F (x, G (y, z)) = F (x, y) = a = G (a, a) = G (F (x, y) , F (x, z)).

Assume that (y, z) ∈ [0, 1] ² \ ([0, e₁] ² ∪ (a, 1] ²), then it holds from structure of G that G (y, z) = max(y, z) = z and F (x, G (y, z)) = F (x, z). Now, we can claim that F (x, y) ≤ e₁ and F (x, z) ≤ e₁ cannot be simultaneously true. If not, then we have F (x, y) ≤ e₁ and F (x, z) ≤ e₁, which implies from structure of F that (x, y) ∈ [0, e₁] ² and (x, z) ∈ [0, e₁] ². Further, we can get (y, z) ∈ [0, e₁] ², but this is a contradiction. Similarly, we can know F (x, y) > a and F (x, z) > a cannot be simultaneously true. Hence, we obtain that (F (x, y) , F (x, z)) ∈ [0, 1] ² \ ([0, e₁] ² ∪ (a, 1] ²), which implies from structure of G that G (F (x, y) , F (x, z)) = max(F (x, y) , F (x, z)) = F (x, z). Thus it holds that F (x, G (y, z)) = F (x, z) = G (F (x, y) , F (x, z)).

Assume that (y, z) ∈ [a, 1] ², then it holds from structure of G that G (y, z) ≥ a. If x ≤ a, then we have from structure of F that F (x, G (y, z)) = a, F (x, y) = a, F (x, z) = a. Thus it follows that F (x, G (y, z)) = a = G (a, a) = G (F (x, y) , F (x, z)). If x ≥ a, then we get directly from Case (ii) that Equation (15).

Lemma 4.7. Let F be a proper uni-nullnorm with the forms Equations (10) or (13) such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_max be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is not conditionally distributive over G.

Proof. On the contrary, if F is conditionally distributive over G, then it follows from Lemma 4.3 that either G = U e 1 max or T F ur = T P and G has the form in Equation (20). For all x, y ≤ a, applying z = e in Equation (15), then we get

( 21 ) F ( x , y ) = F ( x , G ( y , e ) ) = G ( F ( x , y ) , F ( x , e ) ) = G ( F ( x , y ) , x )(21)

Next, let us show that if F is conditionally distributive over G then we have F (x, y) ∈ {x, y} for all (0, e) × (e, a) ∪ (e, a) × (0, e). Without loss of generality, suppose further that 0 < x < e < y < a, then it follows from structure of F that x ≤ F (x, y) ≤ y.

If F (x, y) = e, then e = F (x, y) = G (e, x) = x, but this is a contradiction.

If x ≤ F (x, y) < e < y, according to structure of G, then Equation (21) can be rewritten to F (x, y) = min(F (x, y) , x) = x.

If x < e < F (x, y) ≤ y < a, similar to Equation (21), then we get that F (y, x) = G (F (y, x) , y), and then it follows that F (x, y) = F (y, x) = G (F (y, x) , y), which implies from structure of G that F (x, y) = max(F (y, x) , y) = y.

Thus, we have proven that F (x, y) ∈ {x, y} for (0, e) × (e, a) ∪ (e, a) × (0, e). On the other hand, since U _F is a representable uninorm, it follows from Lemma 2.7 that there exists at least some (x₀, y₀) ∈ (0, e) × (e, 1) ∪ (e, 1) × (0, e) such that min(x₀, y₀) < U (x₀, y₀) < max(x₀, y₀), but this is a contradiction. Therefore, F is not conditionally distributive over G.

Now, let us study Case: G ∈ U_min, that is, the conditional distributivity of a proper uni-nullnorm over a uninorm from class U_min.

Lemma 4.8. Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. If F is conditionally distributive over G, then e₁ = e.

Proof. Obviously, we know that Lemma 4.1 assure either e₁ = a or e₁ = e. Now, we claim that e₁ = e. On the contrary, assume that e₁ = a, then taking x₀, y₀ such that e < y₀ < e₁ = a < x₀ < 1 and putting x = x₀, y = y₀, z = 1 in Equation (15), we have = F (x₀, y₀) = F (x₀, G (y₀, 1)) = G (F (x₀, y₀) , F (x₀, 1)) = G (a, x₀) = x₀ . But this is a contradiction. Therefore, we have e₁ = e.

Lemma 4.9. Let F be a proper uni-nullnorm with F (0, a) = a such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is not conditionally distributive over G.

Proof. On the contrary, if F is conditionally distributive over G, then it follows from Lemma 4.8 that e₁ = e. Taking x = 0, y = e₁, z = a in Equation (15), we have a = F (0, a) = F (0, G (e₁, a)) = G (F (0, e₁) , F (0, a)) = G (0, a) =0, but this is a contradiction.

Therefore, F is not conditionally distributive over G.

Remark 4.10. If F is given by Eqs.(7) or (8) or (9) or (10), then we know that F (0, a) = a in these four forms. Hence, it follows from Lemma 4.9 that F is not conditionally distributive over G.

Lemma 4.11 Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean. G ∈ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. If F is conditionally distributive over G, then one of the following statements are true.

G = U e 1 min .

Proof. Similar to that of Lemma 4.3, it is omitted.

Lemma 4.12 Let F be a proper uni-nullnorm with the form Equation (6) such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is conditionally distributive over G if and only if exactly one of the following cases is fulfilled.

G = U e 1 min .

Proof. (⇒) By Lemma 4.11, the conclusion is valid.

(⇐) Since Case (i) is very simple, we only check Case (ii) over here. If T F ur = T P and G is given by Equation (22), without loss of generality, suppose further that y ≤ z, then we get from monotonicity of F that F (x, y) ≤ F (x, z) for all x ∈ [0, 1]. Assume that (y, z) ∈ [0, 1] ² \ [e₁, 1] ², then it follows from structure of G and assumption y ≤ z that y ≤ e₁, G (y, z) = y and F (x, G (y, z)) = F (x, y). Further, we have from structure of F that F (x, y) ≤ e₁ for all x ∈ [0, 1]. So it holds that F (x, G (y, z)) = F (x, y) = min(F (x, y) , F (x, z)) = G (F (x, y) , F (x, z)).

Assume that (y, z) ∈ [e₁, 1] ² \ [a, 1] ², then it holds from structure of G and assumption y ≤ z that e₁ ≤ y ≤ a, G (y, z) = max(y, z) = z, F (x, G (y, z)) = F (x, z). If x ≤ e₁, then we have F (x, y) = F (x, z) = x, thus it follows that F (x, G (y, z)) = F (x, z) = x = G (x, x) = G (F (x, y) , F (x, z)). If x ≥ e₁, then we have e ≤ F (x, y) ≤ a, thus it follows that F (x, G (y, z)) = F (x, z) = max(F (x, y) , F (x, z)) = G (F (x, y) , F (x, z)).

Assume that (y, z) ∈ [a, 1] ², then it holds from structure of G that G (y, z) ≥ a. To continue our progress, there are three cases to consider. If x ≤ e₁, then we have F (x, y) = F (x, z) = F (x, G (y, z)) = x, thus it follows that F (x, G (y, z)) = x = G (x, x) = G (F (x, y) , F (x, z)). If e₁ ≤ x ≤ a, then we have F (x, y) = F (x, z) = F (x, G (y, z)) = a, thus it follows that F (x, G (y, z)) = a = G (a, a) = G (F (x, y) , F (x, z)). If x > a, we get directly from the Lemma 2.23 that Equation (15).

Lemma 4.13. Let F be a proper uni-nullnorm with the form Equation (11) such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is not conditionally distributive over G.

Proof. On the contrary, if F is conditionally distributive over G, then it follows from Lemma 4.11 that either G = U e 1 min or T F ur = T P and G is given by Equation (22). Taking x, y, z such that 0 < x < e₁ ≤ y < a ≤ z < 1, then we get G (y, z) = z < 1, F (x, y) = x and F (x, z) = a. Hence, the left-hand side of Equation (15) is F (x, G (y, z)) = F (x, z) = a, while the right-hand side of Equation (15) is G (F (x, y) , F (x, z)) = G (x, a) = x < a = F (x, G (y, z)) , but this is a contradiction. Therefore, F is not conditionally distributive over G.

Lemma 4.14. Let F be a proper uni-nullnorm with the form Equation (12) such that T F ll , S F and T F ur are continuous and Archimedean. G ∈ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is not conditionally distributive over G.

Proof. On the contrary, if F is conditionally distributive over G, then it follows from Lemma 4.11 that either G = U e 1 min or T F ur = T P and G is given by Equation (22). Taking x, y, z such that 0 < y < e₁ < x < z < a, then we get G (y, z) = y, F (x, y) = x and a ≥ F (x, z) ≥ max(x, z) = z > x > e₁. Hence, the left-hand side of Equation (15) is F (x, G (y, z)) = F (x, y) = x, while the right-hand side of Equation (15) is G (F (x, y) , F (x, z)) = G (x, F (x, z)) = max(x, F (x, z))  = F (x, z)  ≥  max    (x, z) = z > x = F (x, G (y, z)) , but this is a contradiction. Therefore, F is not conditionally distributive over G.

Lemma 4.15. Let F be a proper uni-nullnorm with the form Equation (13) such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then F is not conditionally distributive over G.

Proof. Similar to that of Lemma 4.7, so it is omitted.

Considering the above results, we have the following theorem.

Theorem 4.16. Let F be a proper uni-nullnorm such that T F ll , S F and T F ur are continuous and Archimedean, G ∈ U_max ∪ U_min be a uninorm with the neutral element e₁ ∈ (0, 1) and the continuous underlying operators. Then the following statements are true.

If G ∈ U_max and F is given by Eqs.(6), (7), (8), (10), (11), (12) or (13). Then F is not conditionally distributive over G.

This paper was focused on and fully characterized the conditional distributivity for a uni-nullnorm with continuous Archimedean underlying t-norms and t-conorm over a continuous t-conorm or a uninorm from U_min ∪ U_max with continuous underlying operators. Note that a null-uninorm is a 2-uninorm having the 2 -neutral element {0, e}  _a with a as an annihilator over [0, e]. Let n be a strong negation and F a proper uni-nullnorm with the 2-neutral element {e, 1}  _a , then the n-dual operator F′ of F, defined by F′ (x, y) = n (F (n (x) , n (y))), is a proper null-uninorm with the 2-neutral element {0, n (e)} _n(a). Therefore, results of this paper can also applied to null-uninorms.

In the forthcoming work, we will consider the characterizations of conditional distributivity for uni-nullnorms with only continuous underlying t-norms and t-conorms over some other binary operators, such as, idempotent uninorms, representable uninorm, semi-t-operators, etc.