Distributivity equations in the class of semi-t-operators

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Fuzzy Sets and Systems ••• (••••) •••–•••

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Distributivity equations in the class of semi-t-operators

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Paweł Dryga´s ∗ , Ewa Rak

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Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Pigonia 1, 35-959 Rzeszów, Poland

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Received 12 September 2014; received in revised form 23 January 2015; accepted 24 January 2015

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Abstract Recently, the distributivity equations have been discussed in families of certain operations (e.g., triangular norms, conorms, uninorms, and nullnorms). In this study, we present the solutions to distributivity between semi-t-operators. Previous results related to the distributivity between nullnorms and between semi-t-operators and semi-nullnorms can be obtained as simple corollaries. © 2015 Published by Elsevier B.V.

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1. Introduction The problem of distributivity was posed many years ago (cf. Aczél [1], pp. 318–319). A new topic of study in this area is mainly concerned with the distributivity between triangular norms and triangular conorms ([10], p. 17). Recently, many studies have addressed the solutions of distributivity equations for aggregation functions [3], fuzzy implications [2], uninorms and nullnorms [16,21], and semi-nullnorms [9], which are generalizations of triangular norms and conorms. In this study, our aim is to obtain algebraic structures with weaker assumptions than nullnorms and t-operators. The characterization of such binary operations is interesting from a theoretical point of view, but also in terms of their applications because they have proved useful in several fields, such as fuzzy logic frameworks [12], expert systems [15], neural networks [15], and fuzzy quantifiers [12]. This research is also the complement of the results of our previous study [9]. First, we introduce weak algebraic structures (Section 2). We then recall the distributivity equations (Section 3). Next, we characterize the solutions to distributivity equations from described families (Section 4). Finally, our results are applied to semi-nullnorms and nullnorms, which can be compared with the results reported by [16], [6], and [9] (Section 5). 2. Associative, monotonic binary operations We start by giving some basic definitions and facts.

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* Corresponding author.

E-mail addresses: [email protected] (P. Dryga´s), [email protected] (E. Rak). http://dx.doi.org/10.1016/j.fss.2015.01.015 0165-0114/© 2015 Published by Elsevier B.V.

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Fig. 1. Structure of a nullnorm.

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Definition 2.1. (See [11].) A triangular semi-norm T is an increasing, associative operation T : [0, 1]2 → [0, 1] with neutral element 1. A triangular semi-conorm S is an increasing, associative operation S : [0, 1]2 → [0, 1] with neutral element 0. A triangular norm T is a commutative triangular semi-norm. A triangular conorm S is a commutative triangular semi-conorm.

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TM (x, y) = min(x, y), TP (x, y) = x · y,

TL (x, y) = max(x + y − 1, 0), SL (x, y) = min(x + y, 1),   min(x, y) if 1 ∈ {x, y} max(x, y) if 0 ∈ {x, y} , SD (x, y) = . TD (x, y) = 0 otherwise 1 otherwise

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SP (x, y) = x + y − xy,

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Definition 2.3. (See [4].) The operation V : [0, 1]2 → [0, 1] is called a nullnorm if it is commutative, associative, increasing, has a zero element z ∈ [0, 1], and it satisfies V (0, x) = x

for all x ≤ z,

(1)

V (1, x) = x

for all x ≥ z.

(2)

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Theorem 2.4. (See [4].) Let z ∈ (0, 1). A binary operation V is a nullnorm with zero element z if and only if a triangular norm T and a triangular conorm S exist such that (see Fig. 1) ⎧ x y if x, y ∈ [0, z] ⎨ zS( z , z ) y−z V (x, y) = z + (1 − z)T ( x−z (3) , ) if x, y ∈ [z, 1] . 1−z 1−z ⎩ z otherwise

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Definition 2.5. The element s ∈ [0, 1] is called an idempotent element of operation G : [0, 1]2 → [0, 1] if G(s, s) = s. The operation G is called idempotent if all the elements from [0, 1] are idempotent.

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By definition, the case where z = 0 leads back to triangular norms, whereas the case where z = 1 leads back to triangular conorms (cf. [11]). The next theorem shows that in other cases nullnorm is built from a triangular norm, a triangular conorm, and the zero element.

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SM (x, y) = max(x, y),

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Example 2.2. (See [11].) Well-known triangular norms and triangular conorms are:

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Theorem 2.6. (Cf. [7].) The operation V : [0, 1]2 → [0, 1] is an idempotent nullnorm with zero element z if and only if it is given by ⎧ ⎨ max(x, y) if x, y ∈ [0, z] V (x, y) = min(x, y) if x, y ∈ [z, 1] . (4) ⎩ z otherwise

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If we omit assumptions (1) and (2) from the definition of a nullnorm, it cannot be shown that a commutative, associative, increasing binary operator V with zero element z = 0 or z = 1 behaves as a triangular norm and triangular conorm (see [7]). In Definition 2.3, the existence of the zero element z follows from (1) and (2).

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Lemma 2.7. (Cf. [7].) Let V be an increasing, binary operation and z ∈ [0, 1] exists such that

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for all x ≤ z,

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V (1, x) = V (x, 1) = x

for all x ≥ z,

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then V (x, y) = z for (x, y) ∈ [0, z] × [z, 1] ∪ [z, 1] × [0, z], V |[0,z] is an increasing, binary operation with neutral element 0 and zero element z. V |[z,1] is an increasing, binary operation with neutral element 1 and zero element z. Moreover V is associative (commutative, idempotent) if and only if V |[0,z] and V |[z,1] are associative (commutative, idempotent).

and the functions F0 and F1 are continuous, where F0 (x) = F (0, x), F1 (x) = F (1, x).

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F (1, 1) = 1,

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In this definition, the existence of the partial neutral elements (conditions (1) and (2)) follows from the continuity of the operation on the boundary of the unit square ((7) and (8)). If we omit the commutativity condition from the definition of a nullnorm, then we obtain the operation given by (3), where the operations T and S are not necessary commutative. Definition 2.9. (See [8].) The operation V and has a zero element z ∈ [0, 1] that satisfies

→ [0, 1] is called a semi-nullnorm if it is associative, increasing,

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V (0, x) = V (x, 0) = x

for all x ≤ z,

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V (1, x) = V (x, 1) = x

for all x ≥ z.

(10)

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Theorem 2.10. (See [8].) Let z ∈ (0, 1). A binary operation V is a semi-nullnorm with zero element z if and only if V is given by (3), where S is a triangular semi-conorm and T is a triangular semi-norm. The semi-nullnorm V is idempotent if and only if it is given by (4) (i.e., V is an idempotent nullnorm).

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This is different in the case of t-operators. Descriptions of the family of this type of operations can be found in [22,13,8].

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Definition 2.11. (See [8].) The operation F : [0, 1]2 → [0, 1] is called a semi-t-operator if it is associative, increasing, and satisfies (7) such that the functions F0 , F1 , F 0 , F 1 are continuous, where F0 (x) = F (0, x), F1 (x) = F (1, x), F 0 (x) = F (x, 0), F 1 (x) = F (x, 1).

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Let Fa,b denote the family of all semi-t-operators such that F (0, 1) = a, F (1, 0) = b.

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: [0, 1]2

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F (0, 0) = 0,

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Definition 2.8. (See [14].) The operation F : [0, 1]2 → [0, 1] is called a t-operator if it is commutative, associative, and increasing such that

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More general families of operations with zero elements were examined in [18]. As we know, the structure of a nullnorm is the same as the structure of a t-operator.

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V (0, x) = V (x, 0) = x

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Theorem 2.12. (See [8].) Let F → [0, 1], F (0, 1) = a, F (1, 0) = b. The operation F ∈ Fa,b if and only if a triangular semi-norm T and triangular semi-conorm S exist such that (see Fig. 2)

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Fig. 2. Structure of the operations F from Theorem 2.12 (left (11), right (12)).

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⎧ if x, y ∈ [0, a], aS( xa , ya ) ⎪ ⎪ ⎪ y−b ⎪ , ) if x, y ∈ [b, 1], ⎨ b + (1 − b)T ( x−b 1−b 1−b F (x, y) = a if x ≤ a ≤ y, ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎩ x otherwise, for a ≤ b and

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F (x, y) =

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(11)

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⎧ bS( xb , yb ) ⎪ ⎪ ⎪ ⎪ y−a ⎪ ⎨ a + (1 − a)T ( x−a 1−a , 1−a )

if x, y ∈ [a, 1],

a ⎪ ⎪ ⎪ ⎪ b ⎪ ⎩ y

if x ≤ a ≤ y, if y ≤ b ≤ x, otherwise,

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if x, y ∈ [0, b],

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(12)

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for b ≤ a.

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Remark 2.13. The class Fz,z is a class of semi-nullnorms with zero element z.

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Let us denote Fz := Fz,z .

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Theorem 2.14. (See [8].) Let F : [0, 1]2 → [0, 1], F (0, 1) = a, F (1, 0) = b. Operation F ∈ Fa,b is idempotent if and only if it is of the form ⎧ max(x, y) if x, y ∈ [0, a], ⎪ ⎪ ⎪ ⎪ min(x, y) if x, y ∈ [b, 1], ⎨ if x ≤ a ≤ y, F (x, y) = a (13) ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ x otherwise, for a ≤ b and

⎧ max(x, y) ⎪ ⎪ ⎪ ⎪ ⎨ min(x, y) F (x, y) = a ⎪ ⎪ ⎪b ⎪ ⎩ y

if x, y ∈ [0, b], if x, y ∈ [a, 1], if x ≤ a ≤ y, if y ≤ b ≤ x, otherwise,

for b ≤ a.

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(14)

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3. Distributivity equations

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Descriptions of the family of this type of operations can be found in [22,13,8].

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Now, we consider the distributivity equations (cf. [1], p. 318).

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Definition 3.1. Let F, G : [0, 1]2 → [0, 1]. The operation F is distributive over G if the left and right distributivity conditions are fulfilled:     ∀x,y,w∈[0,1] F x, G(y, w) = G F (x, y), F (x, w) , (15)     (16) ∀x,y,w∈[0,1] F G(y, w), x = G F (y, x), F (w, x) .

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Proof. Let x ∈ Y ⊂ X, y, w = e ∈ Y ⊂ X. If F is distributive over G, then x = F (x, e) = F (x, G(e, e)) = G(F (x, e), F (x, e)) = G(x, x). The proof is similar in the case where operation F has a left neutral element. 2 Corollary 3.3. (See [5].) If the operation F : [0, 1]2 → [0, 1] with neutral element e ∈ [0, 1] is distributive over operation G : [0, 1]2 → [0, 1] and it satisfies G(e, e) = e, then G is idempotent. Lemma 3.4. (See [19].) Every increasing operation F : [0, 1]2 → [0, 1] is distributive over max and min.

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4. Distributivity of F ∈ Fa,b over G ∈ Fc,d

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Our main consideration concerns the distributivity between the semi-t-operator F ∈ Fa,b and the semi-t-operator G ∈ Fc,d . We distinguish 24 different cases, which depend on the order between the elements a, b of operation F and the elements c, d of operation G. If c = d, then we obtain the cases considered in [9], and if a = b as well, then we obtain the cases considered in [19].

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Lemma 4.1. Let a, b, c, d ∈ [0, 1]. If F ∈ Fa,b is distributive over G ∈ Fc,d , then G is the idempotent semi-t-operator given by (13) or (14).

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Proof. Let a, b, c, d ∈ [0, 1], then G(0, 0) = 0 and G(1, 1) = 1. If a < b, then F has the form (11), where 0 is the right neutral element on the set [0, b] and 1 is the right neutral element on the set [a, 1]. By applying Lemma 3.2 twice, operation G is an idempotent semi-t-operator, which is given by (13) or (14). Similarly, if b < a, then F has the form (12), where 0 is the left neutral element on the set [0, a] and 1 is the left neutral element on the set [b, 1]. By applying Lemma 3.2 twice, operation G is an idempotent semi-t-operator, which is given by (13) or (14). 2

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Now, we start with the case where c < a < b < d, because both the triangular semi-norm and triangular semiconorm are divided into ordinal sums in this case. In most situations where we need to obtain the proofs of the following theorems, we refer to the proof of the theorem for this case.

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Lemma 3.2. (Cf. [19].) Let F : X 2 → X have a right (left) neutral element e in a subset ∅ = Y ⊂ X (i.e., ∀x∈Y F (x, e) = x (F (e, x) = x)). If operation F is distributive over operation G : X 2 → X and it satisfies G(e, e) = e, then G is idempotent in Y .

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Theorem 4.2. Let a, b, c, d ∈ [0, 1], c < a < b < d. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form (see Fig. 3) ⎧ if x, y ∈ [0, c], cS1 ( xc , yc ) ⎪ ⎪ ⎪ y−c x−c ⎪ ⎪ c + (a − c)S2 ( a−c , a−c ) if x, y ∈ [c, a], ⎪ ⎪ ⎪ ⎪ ⎪ max(x, y) if min(x, y) ≤ c ≤ max(x, y) ≤ a, ⎪ ⎪ ⎪ x−b y−b ⎪ ⎨ b + (d − b)T1 ( d−b , d−b ) if x, y ∈ [b, d], F (x, y) = (17) y−d d + (1 − d)T2 ( x−d ⎪ 1−d , 1−d ) if x, y ∈ [d, 1], ⎪ ⎪ ⎪ ⎪ min(x, y) if b ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎩ x otherwise,

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where S1 , S2 are triangular semi-conorms and T1 , T2 are triangular semi-norms.

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Fig. 3. Structure of operations F and G from Theorem 4.2.

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Proof. Let a, b, c, d ∈ [0, 1], c < a < b < d, F ∈ Fa,b be distributive operation G is an idempotent semi-t-operator and it is given by (13). Using (15), (16), and (11) for x ∈ [0, a], we have     c F (x, c) = F x, G(0, 1) = G F (x, 0), F (x, 1) = G(x, a) = x     c F (c, x) = F G(0, 1), x = G F (0, x), F (1, x) = G(x, b) = x

over G ∈ Fc,d . Directly from Lemma 4.1,

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if x ∈ [0, c] , if x ∈ [c, a] if x ∈ [0, c] . if x ∈ [c, a]

Directly from this, the monotonicity of F , and (11), for x ≤ c ≤ y ≤ a, we have y = F (0, y) ≤ F (x, y) ≤ F (c, y) = y. Thus, F (x, y) = max(x, y). Similarly, F (x, y) = max(x, y) if y ≤ c ≤ x ≤ a. Therefore, S is an ordinal sum of triangular semi-conorms S1 and S2 . Similarly, for x ∈ [b, 1], we have     x if x ∈ [b, d] F (x, d) = F x, G(1, 0) = G F (x, 1), F (x, 0) = G(x, b) = , d if x ∈ [d, 1]     x if x ∈ [a, d] F (d, x) = F G(1, 0), x = G F (1, x), F (0, x) = G(x, a) = . d if x ∈ [d, 1] Directly from the above, the monotonicity of F , and (11), for b ≤ x ≤ d ≤ y, we have x = F (x, d) ≤ F (x, y) ≤ F (x, 1) = x. So, F (x, y) = min(x, y). Similarly, F (x, y) = min(x, y) if b ≤ y ≤ d ≤ x. Therefore, T is an ordinal sum of triangular semi-norms T1 and T2 . Conversely, let F be given by (17) and G by (4). Then, F is an operation of the form (11) and by Theorem 2.12, it is in the class Fa,b and similar to G ∈ Fc,d . To prove (15), we have to consider 125 cases. Moreover, directly from Lemma 3.4, we can omit cases with distributivity over max or min. Let us consider the remaining cases, as follows.

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1. If x, y ≤ c < w, then F (x, G(y, w)) = F (x, c) = c = G(cS1 ( xc , yc ), F (x, w)) = G(F (x, y), F (x, w)). 2. If x, w ≤ c < y ≤ a, then F (x, G(y, w)) = F (x, y) = max(x, y) = y = G(y, cS1 ( xc , wc )) = G(max(x, y), cS1 ( xc , wc )) = G(F (x, y), F (x, w)). 3. If x ≤ c < y ≤ a; c < w, then F (x, G(y, w)) = F (x, y) = max(x, y) = y = G(y, F (x, w)) = G(max(x, y), F (x, w)) = G(F (x, y), F (x, w)). 4. If x ≤ a < y ≤ d; w ∈ [0, 1], then F (x, G(y, w)) = F (x, y) = a = G(a, F (x, w)) = G(F (x, y), F (x, w)). 5. If x ≤ a ≤ d < y; w ≤ d, then F (x, G(y, w)) = F (x, d) = a = G(a, F (x, w)) = G(F (x, y), F (x, w)). 6. If x ≤ a ≤ d < y; d < w, then F (x, G(y, w)) = F (x, min(y, w)) = a = G(a, F (x, w)) = G(F (x, y), F (x, w)). 7. If y, w ≤ c; c < x ≤ a, then F (x, G(y, w)) = F (x, max(y, w)) = max(x, max(y, w)) = x = G(x, x) = G(max(x, y), max(x, w)) = G(F (x, y), F (x, w)). 8. If y ≤ c < w; c < x ≤ a, then F (x, G(y, w)) = F (x, c) = x = G(x, F (x, w)) = G(max(x, y), F (x, w)) = G(F (x, y), F (x, w)). 9. If c < x, y ≤ a, then F (x, y) ∈ [c, a] and F (x, G(y, w)) = F (x, y) = G(F (x, y), F (x, w)). 10. If a < x ≤ b, then F (x, G(y, w)) = x = G(x, x) = G(F (x, y), F (x, w)).

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Fig. 4. Structure of operations F and G from Theorem 4.3.

11. If b < x; y ≤ b; w ∈ [0, 1], then G(y, w) ∈ [0, b] and F (x, G(y, w)) = b = G(b, F (x, w)) = G(F (x, y), F (x, w)). 12. If b < x, y ≤ d, then F (x, y) ∈ [b, d] and F (x, G(y, w)) = F (x, y) = G(F (x, y), F (x, w)). 13. If b < x ≤ d; w ≤ d < y, then F (x, G(y, w)) = F (x, d) = x = G(x, F (x, w)) = G(min(x, y), F (x, w)) = G(F (x, y), F (x, w)). 14. If b < x ≤ d; d < y, w, then F (x, G(y, w)) = F (x, min(y, w)) = min(x, min(y, w)) = x = G(x, F (x, w)) = G(min(x, y), F (x, w)) = G(F (x, y), F (x, w)). 15. If b < y ≤ d < x, then F (x, G(y, w)) = F (x, y) = min(x, y) = y = G(y, F (x, w)) = G(min(x, y), F (x, w)) = G(F (x, y), F (x, w)). 16. If w ≤ d; d < x, y, then F (x, y) > d, F (x, w) ≤ d and F (x, G(y, w)) = F (x, d) = d = G(F (x, y), F (x, w)).

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Thus, F and G satisfy (15). We can prove (16) in a similar manner.

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Theorem 4.3. Let a, b, c, d ∈ [0, 1], c < d < a < b. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form (see Fig. 4) ⎧ if x, y ∈ [0, c], cS1 ( xc , yc ) ⎪ ⎪ ⎪ ⎪ x−c y−c ⎪ ⎪ ⎪ c + (d − c)S2 ( d−c , d−c ) if x, y ∈ [c, d], ⎪ ⎪ y−d ⎪ ⎪ d + (a − d)S3 ( x−d ⎪ a−d , a−d ) if x, y ∈ [d, a], ⎪ ⎪ ⎨ max(x, y) if min(x, y) ≤ c ≤ max(x, y) ≤ a, F (x, y) = (18) or min(x, y) ≤ d ≤ max(x, y) ≤ a, ⎪ ⎪ ⎪ y−b ⎪ b + (1 − b)T ( x−b if x, y ∈ [b, 1], ⎪ ⎪ 1−b , 1−b ) ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎩ x otherwise, where S1 , S2 , S3 are triangular semi-conorms and T is a triangular semi-norm. Proof. Let a, b, c, d ∈ [0, 1], c < d < a < b, F ∈ Fa,b be distributive operation G is an idempotent semi-t-operator and it is given by (13). Using (15), (16), and (11), for x ∈ [0, a], we have     c F (x, c) = F x, G(0, 1) = G F (x, 0), F (x, 1) = G(x, a) = x     c F (c, x) = F G(0, 1), x = G F (0, x), F (1, x) = G(x, b) = x     d F (x, d) = F x, G(1, 0) = G F (x, 1), F (x, 0) = G(a, x) = x     d F (d, x) = F G(1, 0), x = G F (1, x), F (0, x) = G(b, x) = x

over G ∈ Fc,d . Directly from Lemma 4.1,

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

if x ∈ [0, c] , if x ∈ [c, a] if x ∈ [0, c] , if x ∈ [c, a] if x ∈ [0, d] , if x ∈ [d, a] if x ∈ [0, d] . if x ∈ [d, a]

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[m3SC+; v1.201; Prn:2/02/2015; 10:52] P.8 (1-16)

P. Dryga´s, E. Rak / Fuzzy Sets and Systems ••• (••••) •••–•••

Directly from this, the monotonicity of F , and (11), for x ≤ c ≤ y ≤ a, we have y = F (0, y) ≤ F (x, y) ≤ F (c, y) = y. So, F (x, y) = max(x, y). Similarly, F (x, y) = max(x, y) if y ≤ c ≤ x ≤ a. Again, from the above, the monotonicity of F , and (11), for x ≤ d ≤ y ≤ a, we have y = F (0, y) ≤ F (x, y) ≤ F (d, y) = y. So, F (x, y) = max(x, y). Similarly, F (x, y) = max(x, y) if y ≤ d ≤ x ≤ a. Therefore, S is an ordinal sum of the triangular semi-conorms S1 , S2 , and S3 . Similar to the proof of Theorem 4.2, we can check that F and G satisfy (15) and (16). 2

6 7

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

where S1 , S2 are triangular semi-conorms and T is a triangular semi-norm. Theorem 4.5. Let a, b, c, d ∈ [0, 1], a < c < b < d. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form ⎧ if x, y ∈ [0, a], aS( xa , ya ) ⎪ ⎪ ⎪ y−b x−b ⎪ ⎪ b + (d − b)T1 ( d−b , d−b ) if x, y ∈ [b, d], ⎪ ⎪ ⎪ ⎪ ⎨ d + (1 − d)T2 ( x−d , y−d ) if x, y ∈ [d, 1], 1−d 1−d F (x, y) = min(x, y) if b ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎩ x otherwise,

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4 5

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9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

where S is a triangular semi-conorm and T1 , T2 are triangular semi-norms.

32

Theorem 4.6. Let a, b, c, d ∈ [0, 1], a < b < c < d. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form ⎧ if x, y ∈ [0, a], aS( xa , ya ) ⎪ ⎪ ⎪ ⎪ x−b y−b ⎪ b + (c − b)T1 ( c−b , c−b ) if x, y ∈ [b, c], ⎪ ⎪ ⎪ ⎪ x−c y−c ⎪ ⎪ ⎪ c + (d − c)T2 ( d−c , d−c ) if x, y ∈ [c, d], ⎪ ⎪ ⎨ d + (1 − d)T3 ( x−d , y−d ) if x, y ∈ [d, 1], 1−d 1−d F (x, y) = ⎪ min(x, y) if b ≤ min(x, y) ≤ c ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ or b ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎩ x otherwise,

34

where S is a triangular semi-conorm and T1 , T2 , T3 are triangular semi-norms.

47

33

35 36 37 38 39 40 41 42 43 44 45 46

48

If a < c < d < b, then any semi-t-operator F ∈ Fa,b is distributive over the idempotent semi-t-operator G ∈ Fc,d .

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3

8

Theorem 4.4. Let a, b, c, d ∈ [0, 1], c < a < d < b. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form ⎧ if x, y ∈ [0, c], cS1 ( xc , yc ) ⎪ ⎪ ⎪ ⎪ x−c y−c ⎪ ⎪ c + (a − c)S2 ( a−c , a−c ) if x, y ∈ [c, a], ⎪ ⎪ ⎪ ⎪ if min(x, y) ≤ c ≤ max(x, y) ≤ a, ⎨ max(x, y) F (x, y) = b + (1 − b)T ( x−b , y−b ) if x, y ∈ [b, 1], ⎪ 1−b 1−b ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ x otherwise,

48 49

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6

We obtain the following theorems in a similar manner.

8 9

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49 50

Theorem 4.7. Let a, b, c, d ∈ [0, 1], a < c < d < b, F ∈ Fa,b . F is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13).

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[m3SC+; v1.201; Prn:2/02/2015; 10:52] P.9 (1-16)

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Fig. 5. Structure of operations F and G from Theorem 4.7.

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Fig. 6. Structure of operations F and G from Theorem 4.8.

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Proof. Let a, b, z ∈ [0, 1], a < z < b, F ∈ Fa,b be distributive over G ∈ Fc,d . Directly from Lemma 4.1, operation G is an idempotent semi-t-operator and it is given by (13). Conversely, let F be given by (11) and G by (13). Then, by Theorem 2.12, F is in the class Fa,b and similarly G ∈ Fc,d . To prove (15), we have to consider 125 cases. Moreover, directly from Lemma 3.4, we can omit cases with distributivity over max or min. Thus, we only need to consider the following cases.

29 30 31 32

2

39 40 41 42 43 44 45 46 47 48 49 50

27 28

30 31 32

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Theorem 4.8. Let a, b, c, d ∈ [0, 1], d < c < a < b. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form (see Fig. 6) ⎧ if x, y ∈ [0, d], dS1 ( dx , dy ) ⎪ ⎪ ⎪ ⎪ y−d x−d ⎪ ⎪ d + (c − d)S2 ( c−d , c−d ) if x, y ∈ [d, c], ⎪ ⎪ ⎪ ⎪ y−c ⎪ c + (a − c)S3 ( x−c if x, y ∈ [c, a], ⎪ a−c , a−c ) ⎪ ⎪ ⎪ ⎨ max(x, y) if min(x, y) ≤ d ≤ max(x, y) ≤ a, F (x, y) = (19) or min(x, y) ≤ c ≤ max(x, y) ≤ a, ⎪ ⎪ ⎪ y−b x−b ⎪ ⎪ b + (1 − b)T ( 1−b , 1−b ) if x, y ∈ [b, 1], ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ x otherwise,

38 39 40 41 42 43 44 45 46 47 48 49 50 51

51 52

26

35

If we change the order between c and d, then we obtain six theorems, similar to Theorems 4.2–4.7.

37 38

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33

Thus, F and G satisfy (15). We can prove (16) in a similar manner. See Fig. 5.

35 36

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1. If x ≤ a < max(y, w), then G(y, w) ≥ a and F (x, G(y, w)) = a = G(F (x, y), F (x, w)). 2. If a < x < b then F (x, G(y, w)) = x = G(x, x) = G(F (x, y), F (x, w)). 3. If min(y, w) < b ≤ x, then G(y, w) ≤ b and F (x, G(y, w)) = b = G(F (x, y), F (x, w)).

33 34

23

where S1 , S2 , S3 are triangular semi-conorms and T is a triangular semi-norm.

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Theorem 4.9. Let a, b, c, d ∈ [0, 1], d < a < c < b. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form

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[m3SC+; v1.201; Prn:2/02/2015; 10:52] P.10 (1-16)

P. Dryga´s, E. Rak / Fuzzy Sets and Systems ••• (••••) •••–•••

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F (x, y) =

⎧ dS1 ( dx , dy ) ⎪ ⎪ ⎪ ⎪ y−d ⎪ d + (a − d)S2 ( x−d ⎪ a−d , a−d ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ max(x, y) y−b b + (1 − b)T ( x−b 1−b , 1−b )

⎪ ⎪ ⎪ ⎪ a ⎪ ⎪ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎩ x

17 18 19 20 21 22 23 24 25 26 27 28 29

if x, y ∈ [d, a], if min(x, y) ≤ d ≤ max(x, y) ≤ a, if x, y ∈ [b, 1],

34

37 38 39 40 41 42 43 44 45 46 47

where S1 , S2 are triangular semi-conorms and T is a triangular semi-norm.

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Theorem 4.10. Let a, b, c, d ∈ [0, 1], d < a < b < c. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form ⎧ if x, y ∈ [0, d], dS1 ( dx , dy ) ⎪ ⎪ ⎪ ⎪ x−d y−d ⎪ ⎪ ⎪ d + (a − d)S2 ( a−d , a−d ) if x, y ∈ [d, a], ⎪ ⎪ ⎪ max(x, y) if min(x, y) ≤ d ≤ max(x, y) ≤ a, ⎪ ⎪ ⎪ ⎪ y−b x−b ⎪ ⎪ ⎨ b + (c − b)T1 ( c−b , c−b ) if x, y ∈ [b, c], y−c F (x, y) = c + (1 − c)T2 ( x−c if x, y ∈ [c, 1], 1−c , 1−c ) ⎪ ⎪ ⎪ ⎪ min(x, y) if b ≤ min(x, y) ≤ c ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ ⎪a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ x otherwise,

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

where S1 , S2 are triangular semi-conorms and T1 , T2 are triangular semi-norms.

31 32

Theorem 4.11. Let a, b, c, d ∈ [0, 1], a < d < c < b, F ∈ Fa,b . F is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14).

33 34 35

Theorem 4.12. Let a, b, c, d ∈ [0, 1], a < d < b < c. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form ⎧ aS( xa , ya ) ⎪ ⎪ ⎪ ⎪ y−b ⎪ b + (c − b)T1 ( x−b ⎪ c−b , c−b ) ⎪ ⎪ ⎪ x−c y−c ⎪ ⎪ ⎨ c + (1 − c)T2 ( 1−c , 1−c ) F (x, y) = min(x, y) ⎪ ⎪ ⎪ ⎪ a ⎪ ⎪ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎩ x

36 37 38

if x, y ∈ [0, a], if x, y ∈ [b, c], if x, y ∈ [c, 1],

39 40 41 42

if b ≤ min(x, y) ≤ c ≤ max(x, y), if x ≤ a ≤ y, if y ≤ b ≤ x,

43

otherwise,

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44 45 46

48

where S is a triangular semi-conorm and T1 , T2 are triangular semi-norms.

50 51

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48 49

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otherwise,

35 36

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if x ≤ a ≤ y, if y ≤ b ≤ x,

30 31

2 3

if x, y ∈ [0, d],

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49 50

Theorem 4.13. Let a, b, c, d ∈ [0, 1], a < b < d < c. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form

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[m3SC+; v1.201; Prn:2/02/2015; 10:52] P.11 (1-16)

P. Dryga´s, E. Rak / Fuzzy Sets and Systems ••• (••••) •••–•••

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Fig. 7. Structure of operations F and G from Theorem 4.14.

⎧ x y if x, y ∈ [0, a], ⎪ ⎪ aS( a , a ) ⎪ ⎪ y−b ⎪ x−b ⎪ b + (d − b)T1 ( d−b , d−b ) if x, y ∈ [b, d], ⎪ ⎪ ⎪ ⎪ y−d ⎪ ⎪ d + (c − d)T2 ( x−d ⎪ c−d , c−d ) if x, y ∈ [d, c], ⎪ ⎪ ⎨ c + (1 − c)T ( x−c , y−c ) if x, y ∈ [c, 1], 3 1−c 1−c F (x, y) = ⎪ min(x, y) if b ≤ min(x, y) ≤ c ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ or b ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ y otherwise, where S is a triangular semi-conorm and T1 , T2 , T3 are triangular semi-norms. Now, if we change the order between a and b, we obtain similar results to Theorems 4.2–4.13.

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

42 43 44 45 46 47 48 49 50 51 52

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Theorem 4.14. Let a, b, c, d ∈ [0, 1], c < d < b < a. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form (see Fig. 7) ⎧ if x, y ∈ [0, c], cS1 ( xc , yc ) ⎪ ⎪ ⎪ ⎪ x−c y−c ⎪ c + (d − c)S2 ( d−c , d−c ) if x, y ∈ [c, d], ⎪ ⎪ ⎪ ⎪ y−d ⎪ ⎪ d + (b − d)S3 ( x−d ⎪ b−d , b−d ) if x, y ∈ [d, b], ⎪ ⎪ ⎨ max(x, y) if min(x, y) ≤ c ≤ max(x, y) ≤ b, F (x, y) = (20) or min(x, y) ≤ d ≤ max(x, y) ≤ b, ⎪ ⎪ ⎪ x−a y−a ⎪ a + (1 − a)T ( 1−a , 1−a ) if x, y ∈ [a, 1], ⎪ ⎪ ⎪ ⎪ ⎪ a if x ≤ b ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ a ≤ x, ⎪ ⎩ y otherwise,

26

where S1 , S2 , S3 are triangular semi-conorms and T is a triangular semi-norm.

39

40 41

10

27 28 29 30 31 32 33 34 35 36 37 38

40

Theorem 4.15. Let a, b, c, d ∈ [0, 1], c < b < d < a. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form ⎧ if x, y ∈ [0, c], cS1 ( xc , yc ) ⎪ ⎪ ⎪ ⎪ y−c x−c ⎪ c + (b − c)S2 ( b−c , b−c ) if x, y ∈ [c, b], ⎪ ⎪ ⎪ ⎪ ⎪ if min(x, y) ≤ c ≤ max(x, y) ≤ b, ⎨ max(x, y) F (x, y) = a + (1 − a)T ( x−a , y−a ) if x, y ∈ [a, 1], ⎪ 1−a 1−a ⎪ ⎪ ⎪ ⎪ a if x ≤ b ≤ y, ⎪ ⎪ ⎪ ⎪b if y ≤ a ≤ x, ⎪ ⎩ y otherwise, where S1 , S2 are triangular semi-conorms and T is a triangular semi-norm.

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P. Dryga´s, E. Rak / Fuzzy Sets and Systems ••• (••••) •••–•••

Theorem 4.16. Let a, b, c, d ∈ [0, 1], c < b < a < d. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form ⎧ if x, y ∈ [0, c], cS ( x , y ) ⎪ ⎪ 1 c c ⎪ y−c x−c ⎪ ⎪ c + (b − c)S2 ( b−c , b−c ) if x, y ∈ [c, b], ⎪ ⎪ ⎪ ⎪ ⎪ max(x, y) if min(x, y) ≤ c ≤ max(x, y) ≤ b, ⎪ ⎪ ⎪ ⎪ x−b y−a ⎪ ⎪ ⎨ a + (d − a)T1 ( d−a , d−a ) if x, y ∈ [a, d], F (x, y) = d + (1 − d)T2 ( x−d , y−d ) if x, y ∈ [d, 1], 1−d 1−d ⎪ ⎪ ⎪ ⎪ min(x, y) if a ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ ⎪ a if x ≤ b ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ a ≤ x, ⎪ ⎪ ⎩ y otherwise, where S1 , S2 are triangular semi-conorms and T1 , T2 are triangular semi-norms. Theorem 4.17. Let a, b, c, d ∈ [0, 1], b < c < d < a, F ∈ Fa,b . F is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13). Theorem 4.18. Let a, b, c, d ∈ [0, 1], b < c < a < d. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form ⎧ if x, y ∈ [0, b], bS( xb , yb ) ⎪ ⎪ ⎪ ⎪ x−a y−a ⎪ ⎪ ⎪ a + (d − a)T1 ( d−a , d−a ) if x, y ∈ [a, d], ⎪ ⎪ x−d y−d ⎪ ⎪ ⎨ d + (1 − d)T2 ( 1−d , 1−d ) if x, y ∈ [d, 1], F (x, y) = min(x, y) if b ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ b ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ a ≤ x, ⎪ ⎪ ⎩ y otherwise, where S is a triangular semi-conorm and T1 , T2 are triangular semi-norms.

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Theorem 4.19. Let a, b, c, d ∈ [0, 1], b < a < c < d. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form ⎧ if x, y ∈ [0, b], aS( xb , yb ) ⎪ ⎪ ⎪ y−a x−a ⎪ ⎪ a + (c − a)T1 ( c−a , c−a ) if x, y ∈ [a, c], ⎪ ⎪ ⎪ ⎪ x−c y−c ⎪ c + (d − c)T2 ( d−c , d−c ) if x, y ∈ [c, d], ⎪ ⎪ ⎪ ⎪ ⎪ x−d y−d ⎪ ⎨ d + (1 − d)T3 ( 1−d , 1−d ) if x, y ∈ [d, 1], F (x, y) = min(x, y) if a ≤ min(x, y) ≤ c ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ or a ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ ⎪ a if x ≤ b ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ a ≤ x, ⎪ ⎪ ⎩ y otherwise,

34

where S is a triangular semi-conorm and T1 , T2 , T3 are triangular semi-norms.

49

50 51

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35 36 37 38 39 40 41 42 43 44 45 46 47 48

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Theorem 4.20. Let a, b, c, d ∈ [0, 1], d < c < b < a. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form (see Fig. 8)

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[m3SC+; v1.201; Prn:2/02/2015; 10:52] P.13 (1-16)

P. Dryga´s, E. Rak / Fuzzy Sets and Systems ••• (••••) •••–•••

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Fig. 8. Structure of operations F and G from Theorem 4.20.

⎧ if x, y ∈ [0, d], dS1 ( dx , dy ) ⎪ ⎪ ⎪ y−d ⎪ x−d ⎪ d + (c − d)S2 ( c−d , c−d ) if x, y ∈ [d, c], ⎪ ⎪ ⎪ ⎪ y−c ⎪ c + (b − c)S3 ( x−c if x, y ∈ [c, b], ⎪ ⎪ b−c , b−c ) ⎪ ⎪ ⎨ max(x, y) if min(x, y) ≤ d ≤ max(x, y) ≤ b, F (x, y) = or min(x, y) ≤ c ≤ max(x, y) ≤ b, ⎪ ⎪ ⎪ x−a y−a ⎪ , ) if x, y ∈ [a, 1], a + (1 − a)T ( ⎪ 1−a 1−a ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎩ y otherwise,

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(21)

where S1 , S2 , S3 are triangular semi-conorms and T is a triangular semi-norm.

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Theorem 4.21. Let a, b, c, d ∈ [0, 1], d < b < c < a. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form ⎧ if x, y ∈ [0, d], dS ( x , y ) ⎪ ⎪ ⎪ 1 d d ⎪ y−d ⎪ x−d ⎪ ⎪ d + (b − d)S2 ( b−d , b−d ) if x, y ∈ [d, b], ⎪ ⎪ ⎪ ⎪ if min(x, y) ≤ d ≤ max(x, y) ≤ b, ⎨ max(x, y) F (x, y) = a + (1 − a)T ( x−a , y−a ) if x, y ∈ [a, 1], ⎪ 1−a 1−a ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ y otherwise,

24

where S1 , S2 are triangular semi-conorms and T is a triangular semi-norm.

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Theorem 4.22. Let a, b, c, d ∈ [0, 1], d < b < a < c. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form ⎧ if x, y ∈ [0, d], dS1 ( dx , dy ) ⎪ ⎪ ⎪ y−d ⎪ x−d ⎪ ⎪ ⎪ d + (b − d)S2 ( b−d , b−d ) if x, y ∈ [d, b], ⎪ ⎪ ⎪ max(x, y) if min(x, y) ≤ d ≤ max(x, y) ≤ b, ⎪ ⎪ ⎪ ⎪ y−a x−a ⎪ ⎨ a + (c − a)T1 ( c−a , c−a ) if x, y ∈ [a, c], F (x, y) = c + (1 − c)T ( x−c , y−c ) if x, y ∈ [c, 1], 2 1−c 1−c ⎪ ⎪ ⎪ ⎪ ⎪ min(x, y) if a ≤ min(x, y) ≤ c ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎩ y otherwise, where S1 , S2 are triangular semi-conorms and T1 , T2 are triangular semi-norms.

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Theorem 4.23. Let a, b, c, d ∈ [0, 1], b < d < c < a, F ∈ Fa,b . F is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14).

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Theorem 4.24. Let a, b, c, d ∈ [0, 1], b < d < a < c. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (14) and F has the following form ⎧ if x, y ∈ [0, b], bS( xb , yb ) ⎪ ⎪ ⎪ ⎪ y−a x−a ⎪ ⎪ ⎪ a + (c − a)T1 ( c−a , c−a ) if x, y ∈ [a, c], ⎪ ⎪ ⎪ x−c y−c ⎪ ⎨ c + (1 − c)T2 ( 1−c , 1−c ) if x, y ∈ [c, 1], F (x, y) = min(x, y) if a ≤ min(x, y) ≤ c ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ y otherwise, where S is a triangular semi-conorm and T1 , T2 are triangular semi-norms. Theorem 4.25. Let a, b, c, d ∈ [0, 1], b < a < d < c. F ∈ Fa,b is distributive over G ∈ Fc,d if and only if G is the idempotent semi-t-operator (13) and F has the following form: ⎧ if x, y ∈ [0, a], aS( xa , ya ) ⎪ ⎪ ⎪ ⎪ y−b x−b ⎪ ⎪ b + (d − b)T1 ( d−b , d−b ) if x, y ∈ [b, d], ⎪ ⎪ ⎪ ⎪ y−d ⎪ ⎪ d + (c − d)T2 ( x−d c−d , c−d ) if x, y ∈ [d, c], ⎪ ⎪ ⎪ ⎪ c + (1 − c)T ( x−c , y−c ) if x, y ∈ [c, 1], ⎨ 3 1−c 1−c F (x, y) = min(x, y) if b ≤ min(x, y) ≤ c ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ ⎪ or b ≤ min(x, y) ≤ d ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ y otherwise, where S is a triangular semi-conorm and T1 , T2 , T3 are triangular semi-norms. 5. Application to nullnorms If we assume that c = d, then we obtain the results described by [9].

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Theorem 5.1. (See [9].) Let a, b, z ∈ [0, 1], a < b < z. F ∈ Fa,b is distributive over G ∈ Fz if and only if G is the idempotent nullnorm (4) and F has the following form (see Fig. 9) ⎧ if x, y ∈ [0, a], aS( xa , ya ) ⎪ ⎪ ⎪ ⎪ y−z x−z ⎪ ⎪ b + (z − b)T1 ( z−b , z−b ) if x, y ∈ [b, z], ⎪ ⎪ ⎪ ⎪ x−z y−z ⎪ ⎨ z + (1 − z)T2 ( 1−z , 1−z ) if x, y ∈ [z, 1], F (x, y) = min(x, y) if b ≤ min(x, y) ≤ z ≤ max(x, y), ⎪ ⎪ ⎪ ⎪ a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ x otherwise,

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where T1 , T2 are triangular semi-norms and S is a triangular semi-conorm.

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Theorem 5.2. (See [9].) Let a, b, z ∈ [0, 1], z < b < a. F ∈ Fa,b is distributive over G ∈ Fz,z if and only if G is the idempotent nullnorm (4) and F has the following form

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Fig. 9. Structure of operations F and G from Theorem 5.1.

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⎧ zS1 ( xz , yz ) ⎪ ⎪ ⎪ ⎪ y−z ⎪ z + (a − z)S2 ( x−z ⎪ a−z , a−z ) ⎪ ⎪ ⎪ ⎪ ⎨ max(x, y) y−b F (x, y) = b + (1 − b)T ( x−b 1−b , 1−b ) ⎪ ⎪ ⎪ ⎪a ⎪ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎩ y

if x, y ∈ [0, z], if x, y ∈ [z, a], if min(x, y) ≤ z ≤ max(x, y) ≤ a, if x, y ∈ [b, 1], if x ≤ a ≤ y, if y ≤ b ≤ x, otherwise,

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Theorem 5.4. (See [9].) Let a, b, z ∈ [0, 1], b < a < z. F ∈ Fa,b is distributive over G ∈ Fz if and only if G is the idempotent t-operator (4) and F has the following form ⎧ if x, y ∈ [0, a], aS( xa , ya ) ⎪ ⎪ ⎪ ⎪ x−z y−z ⎪ ⎪ b + (z − b)T1 ( z−b , z−b ) if x, y ∈ [b, z], ⎪ ⎪ ⎪ ⎪ x−z y−z ⎪ ⎨ z + (1 − z)T2 ( 1−z , 1−z ) if x, y ∈ [z, 1], F (x, y) = min(x, y) if b ≤ min(x, y) ≤ z ≤ max(x, y), ⎪ ⎪ ⎪ ⎪a if x ≤ a ≤ y, ⎪ ⎪ ⎪ ⎪ ⎪ b if y ≤ b ≤ x, ⎪ ⎪ ⎩ y otherwise,

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where T1 , T2 are triangular semi-norms and S is a triangular semi-conorm.

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If we also add an assumption about the commutativity of F and G, then we obtain the results reported by [16].

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Theorem 5.5. (Cf. [16].) Let z1 , z2 ∈ [0, 1] and the operations F and G are nullnorms with z1 ≤ z2 . Then, F is distributive over G if and only if G is idempotent and F has the following form ⎧ if x, y ∈ [0, z1 ] z1 S( zx1 , zy1 ) ⎪ ⎪ ⎪ ⎪ y−z x−z ⎪ 1 1 ⎪ ⎨ (z2 − z1 )T1 ( z2 −z1 , z2 −z1 ) if x, y ∈ [z1 , z2 ] , F (x, y) = (1 − z )T ( x−z2 , y−z2 ) if x, y ∈ [z2 , 1] 2 2 1−z2 1−z2 ⎪ ⎪ ⎪ ⎪ ⎪ min(x, y) if x ∈ [z1 , z2 ], y ∈ [z2 , 1] or x ∈ [z2 , 1], y ∈ [z1 , z2 ] ⎪ ⎩ z1 otherwise where S is a triangular conorm, and T1 and T2 are triangular norms.

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Theorem 5.3. (See [9].) Let a, b, z ∈ [0, 1], a ≤ z ≤ b. F ∈ Fa,b is distributive over G ∈ Fz if and only if G is the idempotent t-operator (4).

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where S1 , S2 are triangular semi-conorms and T is a triangular semi-norm.

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By duality:

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Theorem 5.6. (Cf. [16].) Let z1 , z2 ∈ [0, 1] and the operations F and G are nullnorms with z2 ≤ z1 . Then, F is distributive over G if and only if G is idempotent and F has the following form ⎧ if x, y ∈ [0, z2 ] z2 S1 ( zx2 , zy2 ) ⎪ ⎪ ⎪ ⎪ ⎪ y−z x−z 2 2 ⎪ ⎨ (z1 − z2 )S2 ( z1 −z2 , z1 −z2 ) if x, y ∈ [z2 , z1 ] F (x, y) = max(x, y) if x ∈ [0, z2 ], y ∈ [z2 , z1 ] or x ∈ [z2 , z1 ], y ∈ [0, z2 ] , ⎪ ⎪ ⎪ y−z x−z 1 1 ⎪ if x, y ∈ [z1 , 1] ⎪ ⎪ (1 − z1 )T ( 1−z1 , 1−z1 ) ⎩ otherwise z1 where S1 , S2 are triangular conorms and T is a triangular norm.

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Acknowledgements

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In this study, inspired by nullnorms and t-operators, we examined functional equations for distributivity in the class of semi-t-operators. We provided full characterizations of Eqs. (15) and (16), i.e., for noncommutative operations F ∈ Fa,b and G ∈ Fc,d in all possible cases, depending on the order between a, b, c, d. This contribution completes the overall consideration published previously for nullnorms and t-operators in [9,16]. We omitted the assumption of commutativity, so in our next study we will consider the left and right distributivity equations in the class of semi-t-operators.

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6. Conclusion

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This study was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge No. RPPK.01.03.00-18-001/10. Uncited references [17] [20] References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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