The distributivity of extended semi-t-operators over extended S-uninorms on fuzzy truth values

Inspired by the thought of distributivity between semi-t-operator and S-uninorm, this paper primarily explores the distributivity between extended semi-t-operator and extended S-uninorm on fuzzy truth value. Firstly, Zadeh-extended semi-t-operator and S-uninorm are proposed on fuzzy truth value and some results of extended semi-operator are studied under special fuzzy truth values. Then, it concentrates on the suﬃcient condition about left and right distributivity of extended semi-t-operator over extended S-uninorm under the condition that semi-t-operator is left and right distributive over S-uninorm, respectively. Finally, when parameters satisfy diﬀerent cases, suﬃcient conditions for the distributivity between extended semi-t-operator and extended S-uninorm are given under the condition that semi-t-operator satisﬁes distributivity or conditional distributivity over S-uninorm.


Introduction
As an extension of fuzzy set, type-2 fuzzy set was defined by Zadeh in 1975 [1], which has been explored comprehensively in theory [2][3][4][5][6][7][8] and practical applications [9][10][11][12][13].Unlike the exact membership function in fuzzy sets, type-2 fuzzy set can be represented as a fuzzy set with fuzzy sets as truth values, that is, the fuzzy membership function (called fuzzy truth value) of type-2 fuzzy set consists of all mappings from [0, 1] into [0, 1].Therefore, type-2 fuzzy set has greater inclusiveness for uncertainty and plays an indispensable role in dealing with fuzzy language uncertainty.Meantime, the operations on type-2 fuzzy set are convolutions of operations on [0, 1] [2].On the other end of the spectrum, aggregation theory of real numbers has an imperative effect both in actual applications and theory [14][15][16].Hence, the aggregation function of real numbers is widely generalized to type-2 fuzzy set [17][18][19].More specifically, Gera and Dombi [20] introduced the extended t-norm and t-conorm on fuzzy truth value by pointwise formulas.Further, the extended aggregation function on convex normal fuzzy truth value was proposed by Takáč [17].On the basis of [17], Torres-Blanc et al. [7] extended the aggregation function of fuzzy set to type-2 fuzzy set with the help of Zadeh's extension principle.
In addition, the distributivity between binary functions was introduced based on the viewpoint of functional equation [21].Subsequently, the distributivity between many special aggregation functions was explored [22][23][24][25].Accordingly, a natural and important problem is to consider the distributivity between convolution operations on fuzzy truth values, which can be viewed as a link connecting two extended aggregation functions.The distributivity between extended aggregation functions not only enriches the theory of logical algebraic structure for type-2 fuzzy set [26][27][28][29], but also enhances its application in approximate reasoning and fuzzy control system [30].For example, Walker and Walker [2,31,32], Harding et al. [5] proposed that the extended minimum was distributive over the extended maximum, meantime, the extended maximum was distributive over the extended minimum.Moreover, the extended t-norm (resp.the extended t-conorm) was distributive over maximum (resp.minimum).In 2014, Hu and Kwong [6] researched the distributivity between the extended t-norm (resp.the extended t-conorm) and maximum (resp.minimum).Further, Xie [33] gave the concept of nullnorm and uninorm on fuzzy truth value and explored the condition that extended uninorm was distributive between minimum or maximum.Afterwards, Zhang and Hu [34] characterized the distributivity of convolution operation over a meet-convolution and a join-convolution.In 2020, Wang and Liu [28] presented the distributivity of extended nullnorm over uninorm under the condition that nullnorm and uninorm were conditionally distributive.Next, when t-norm satisfied conditional distributivity over t-conorm, Liu and Wang [27] investigated the distributivity of extended t-norm over t-conorm.Further, Liu and Wang [29] explored the conditions under which the distributivity between the Z-extended overlap functions and grouping functions holds.
As a further extension of the t-norm and t-conorm, Drygaś [35,36] defined semi-t-operator through reducing the commutative law of t-operator.By the idea of annihilator in aggregation function, Mas et al. [37] presented the concept of S-uninorm, which can be seen as a significant generalization of uninorm.
Based on the above definitions, Fang and Hu [38] researched several conditional distributivity equations for S-uninorm, where t-norm, T -uninorm and semi-t-operator are conditionally distributive over S-uninorm.Later on, Wang et al. [39] complemented the work of conditional distributivity and distributivity between semi-t-operator and S-uninorm.Furthermore, when parameters satisfied different relations, the sufficient and necessary conditions that semi-toperator satisfied left (resp.right) distributivity over S-uninorm were studied with the underlying uninorm in U min .The above conclusions lay a solid foundation for exploring the distributive law between extended aggregation functions on fuzzy truth value.On the other hand, based on the condition that nullnorm and uninorm satisfy conditional distributivity, Wang and Liu [28] explored the distributive law between extended nullnorm and uninorm on fuzzy truth value.It is well known that semi-t-operator [35,36] and S-uninorm [37] are generalizations of toperator (nullnorm) and uninorm, respectively.To further study the extended semi-t-operator and extended S-uninorm and make them easy to apply in application, it is necessary to explore the related properties of semi-t-operator and S-uninorm on fuzzy truth value.The aim of this paper is to generalize the distributivity between extended semi-t-operator and extended S-uninorm, which can be considered as the improvement of the algebraic structure of type-2 fuzzy set in theoretical field.Meanwhile, the specific relationship of distributive law between different fuzzy logical operators is shown in Figure 1.
In this paper, we firstly investigate the extended semi-t-operator and S-uninorm by Zadeh's extension principle, and further explore the distributivity between extended semi-t-operator and S-uninorm based on the results The distributivity of extended semi-t-operators over extended S-uninorms that semi-t-operator satisfies conditional distributivity or distributivity over S-uninorm in [38,39].
The reminder of this paper is arranged as follows.Section 2 reviews some basic concepts and properties, which are essential for the sequel.In Section 3, the Zadeh-extended semi-t-operator and S-uninorm are proposed, and then some properties of extended semi-t-operator are studied under special fuzzy truth values.Section 4 explores the left and right distributivity of extended semi-t-operator over extended S-uninorm on fuzzy truth values when semi-t-operator is left and right distributive to S-uninorm.In Section 5, the distributivity between extended semi-t-operator and S-uninorm is investigated under the condition that semi-t-operator is (conditionally) distributive to S-uninorm.Section 6 summarizes the full works.

Preliminaries
This part is composed of three subsections to review basic knowledge regarding semi-t-operator and S-uninorm, distributivity, conditional distributivity and the extended binary operators on fuzzy truth value.

Semi-t-operator and S-uninorm
In this subsection, we mainly introduce several types of common binary operators.According to reference [40], the concepts of semi-t-norm, semi-t-conorm, t-norm and t-conorm can be shown below.
is an increasing and associative binary function with neutral element 1.
is an increasing and associative binary function with neutral element 0.
that has associativity and the left (resp.right) neutral element 0, i.e., S(0 In light of the positive t-conorm, Wang et al. [39] defined the positive property of left semi-t-conorm and right semi- It is worth reminding that uninorm U degenerates into a t-norm T when e = 1 and a t-conorm S when e = 0.If U (0, 1) = 0, then it is denoted as a conjunctive uninorm.Further, Fodor et al. [42] proposed the characterization of uninorm U in the following.
Proposition 1 [42] The uninorm U with neutral element e ∈ (0, 1) can be expressed as follows: where T U is a t-norm, S U is a t-conorm and A(e) meets the condition that Moreover, the underlying t-norm and t-conorm of uninorm U are denoted as T U and S U , respectively.Specially, the family of all uninorms given by Eq. ( 1) is denoted by U min , if A(e) = min(u, v) for any u ∈ [0, e), v ∈ (e, 1] or u ∈ (e, 1], v ∈ [0, e).
Further, Mas et al. [37] presented the equivalent characterization form of S-uninorm.

Proposition 2 [37] A binary function
] is an S-uninorm if and only if there exists λ ∈ [0, 1), t-conorm S and conjunctive uninorm U with neutral element e U ∈ (0, 1) such that A is expressed as: where U and S are called the underlying uninorm and underlying t-conorm, respectively.
All S-uninorms in the form of Eq. ( 2) consist of a family, called U S e,λ with IFC element e.Further, unless otherwise stated, the family of A ∈ U S e,λ with the underlying uninorm in U min is denoted as (U S e,λ ) min .The distributivity of extended semi-t-operators over extended S-uninorms ] is an increasing, associative binary function with F (0, 0) = 0, F (1, 1) = 1, meantime, F (0, u), F (u, 0), F (1, u) and F (u, 1) are continuous for any u ∈ [0, 1].
The family of all semi-t-operators is denoted as F ε, η with F (0, 1) = ε and F (1, 0) = η.Next, the equivalent characterization of F ∈ F ε, η can be given as follows.

Distributive law and conditional distributive law
Next, the distributivity and conditional distributivity equations between two binary functions are listed as follows.
(3) In a similar way, F satisfies conditional distributivity over A from the right if for any u, v, w ∈ [0, 1] and A(v, w) < 1, it holds that F (A(v, w), u) = A(F (v, u), F (w, u)).
(4) If F and A satisfy both Eqs.(3) and (4), then F satisfies the conditional distributivity for A.

The extended operators on fuzzy truth value
Definition 7 [2,43] A mapping h : [0, 1] −→ [0, 1] is called fuzzy truth value.Further, the family of all fuzzy truth values is represented as There are some special fuzzy truth values [2,31], which can be shown as follows.
Further, if a ∈ [0, 1], a(u) = a for any u ∈ [0, 1] , then a is a mapping of fixed value a.
Definition 8 [2,5] Let h ∈ F , h L and h R are denoted as follows: A fuzzy truth value is normal if h(u) = 1 holds and the family of all normal fuzzy truth values is denoted as F N .
In light of Zadeh's extension principle [1], the binary function and S be a t-norm and a t-conorm.The extended t-norm ⊙ T and extended t-conorm ⊙ S can be showed as follows: In particular, if T and S take T M and S M respectively, then ⊓ and ⊔ are used to denote ⊙ T and ⊙ S , i.e., Further, Liu [44] proved that the extended continuous binary functions with the properties of monotonically increasing satisfy the distributive law for the extended t-norms ⊓ (resp.the extended t-conorms ⊔), which are listed as follows.

The extended semi-t-operator and extended S-uninorm on fuzzy truth value
First, the definitions of extended semi-t-operator ⊙ F and extended S-uninorm ⊙ A are constructed below.
Definition 11 Suppose h 1 , h 2 ∈ F , F is an semi-t-operator and A an S-uninorm.The extended semi-t-operator ⊙ F is represented as The extended S-uninorm ⊙ A is denoted as As a special case of [7], Propositions 5 and 6 hold when semi-t-operator F and S-uninorm A are continuous.
Proposition 5 Let F be a continuous semi-t-operator and ⊙ F the extended operator of F .For any h 1 , h 2 ∈ F , it holds that Proposition 5 illustrates that the order of operations between the extended semi-t-operator and the fuzzy truth value h R (resp.h L ) has no effect on the final results.Further, there is a similar relationship between the extended S-uninorm and the fuzzy truth value h R (resp.h L ).

Proposition 6 Let A be a continuous S-uninorm and ⊙ A the extended operator of A. For any h
Next, the properties of extended semi-t-operator ⊙ F on special fuzzy truth value 1 are discussed.
Proposition 7 Let F ∈ Fε, η be a continuous semi-t-operator, ⊙ F the extended operator of F .For any h ∈ F , the following statements can be obtained.
If ε ⩽ η, then and and Proof First, we verify the case of ε ⩽ η, it holds that (1) If w ⩽ ε, then the following two cases are discussed.
, then according to Proposition 3 that F is semi-t-conorm with neutral element 0. Hence, one concludes that that is, w ≥ u.The distributivity of extended semi-t-operators over extended S-uninorms Further, for any u ∈ [0, w], In summary, we can obtain that (2) If w ⩾ η, the following two cases are discussed.
, then according to Proposition 3 that F is semi-t-norm with neutral element 1.Hence, it holds that Further, for any u ∈ [w, 1], In summary, it can be concluded that Next, we consider that (1 ⊙ F h) (w) with the case of ε ⩽ η. ( ( According to Proposition 3 that F is semi-tconorm with neutral element 0. Hence, one concludes that Therefore, we can obtain that ( According to Proposition 3 that F is semi-tnorm with neutral element 1.Hence, one concludes that Therefore, we can obtain that (3) If ε ≤ w ≤ η, then the following five cases are discussed: In summary, the value of v is not affected by w = F (u, v) in cases (1 ′ ), (3 ′ ) and (4 ′ ), then we can obtain that In a similar way, the corresponding conclusions can be verified when η ≤ ε. □ Similar to Proposition 7, the properties of extended semi-t-operator ⊙ F on the fuzzy truth value 1 are obtained.
Proposition 8 Suppose F ∈ Fε, η is a semi-t-operator, ⊙ F the extension of F , then for any h ∈ F , the following statements can be obtained when Proof At first, we verify the case of ε ⩽ η.If w ⩾ ε, then it holds that On the other hand, if w < ε, then it holds that Similarly, the case of (1 ⊙ F h)(w) can be verified as follows.
If w ⩾ η, then it infers that On the other hand, if w < η, then it infers that Similarly, the corresponding conclusions can be verified when η ≤ ε. □ When ε, η take special values 0 or 1, i.e., F 0,1 or F 1,0 , some properties of the extended semi-t-operator ⊙ F can be obtained.
0 be two semi-t-operators, ⊙ F1 and ⊙ F2 be the extended operators of F 1 and F 2 , then for any h 1 , h 2 ∈ F , the following statements can be obtained.
The distributivity of extended semi-t-operators over extended S-uninorms In a similar way, (h h 1 (t) can be proven when □ Some properties about the extended semi-t-operator ⊙ F and extended Suninorm ⊙ A can be given with the help of literature [6].Since the semi-toperator does not satisfy commutativity, the extended semi-t-operator also does not satisfy commutativity.
Proposition 10 Suppose F is a semi-t-operator, A an S-uninorm, then for any h, h ′ , h ′′ ∈ F , the following statements hold. ( (2) Since the S-uninorm A satisfies the commutativity, it holds that (3) Since the semi-t-operator F satisfies the associativity, it follows that Hence, (h The distributivity of extended semi-t-operators over extended S-uninorms

□
This section mainly explores some properties of the extended semi-toperator ⊙ F and the extended S-uninorm ⊙ A .Next, the left (resp.right) distributive law between ⊙ F and ⊙ A will be obtained.

Left and right distributivity of extended semi-t-operator over extended S-uninorm on fuzzy truth value
In what follows, the sufficient conditions of the left distributive law between extended semi-t-operator ⊙ F and extended S-uninorm ⊙ A on fuzzy truth value are given.

4.1
The situation of e < min(ε, η) < max(ε, η) = 1 In 2021, Wang et al. [39] charactered the equivalent forms about the semi-toperator and S-uninorm when the left or right distributive law between them is satisfied.
Proposition 11 [39] Suppose 0 < λ < e < ε < η = 1, F ∈ Fε, η , A ∈ (U S e,λ ) min .If the left distributivity between F and A is satisfied, then A and F have the following forms, respectively: and where A is an idempotent S-uninorm; S is a semi-t-conorm;  Next, on the basis of left distributivity between F and A, it is natural to consider the left distributive law of the extended operator ⊙ F with respect to the extended operator ⊙ A .
Theorem 12 Suppose 0 < λ < e < ε < η = 1, F ∈ Fε, η is continuous, A ∈ (U S e,λ ) min and F satisfies the left distributivity over A. For any h 1 ∈ F C and h 2 , h 3 ∈ F , it holds that Proof First, it follows from Definition 11 that for any ω ∈ [0, 1], The distributivity of extended semi-t-operators over extended S-uninorms Since F is left distributive over A, that is, F (y, A(u, v)) = A(F (y, u), F (y, v)).According to Proposition 11, A and F can be expressed as Eqs.( 5) and ( 6), respectively.Next, for any u, v ∈ [0, 1], we divide into four cases to prove. (1 then there exist four cases below. (i Since F is left distributive over A, it holds that Next, we verify that (h Firstly, we give the statement P. P. For each y ′ ∈ [λ, e), (q, t) Due to A(F (p, q), F (s, t)) = A(F (y ′ , q), F (y ′ , t)) = ω, then p ≤ y ′ ≤ s or s ≤ y ′ ≤ p. Otherwise, y ′ < p ∧ s or y ′ > p ∨ s.Assume that y ′ < p ∧ s, there is a contradiction in ω = A(F (p, q), F (s, t)) Hence, y ′ ≥ p ∧ s.Moreover, it can be proven that y ′ ≤ p ∨ s in a similar way.Therefore, p ≤ y ′ ≤ s or s ≤ y ′ ≤ p.One can conclude that h 1 (y ′ ) ≥ h 1 (p) ∧ h 1 (s) since h 1 is convex, and then In view of the above discussion, P holds.
In light of the statement P, one can obtain that The distributivity of extended semi-t-operators over extended S-uninorms Next, we prove that (h (3) If ω = F (y, A(u, v)) ∈ [e, ε], then there exist five cases as follows. (i , then it can be proven as case In light of Proposition 4, all of the above cases satisfy that (h Next, we prove that (h In light of Proposition 4, all of the above cases satisfy that Next, we verify that (h Firstly, we give the statement Q. = ω.The distributivity of extended semi-t-operators over extended S-uninorms Hence, y ′ ≥ p ∧ s.Moreover, y ′ ≤ p ∨ s can be proven in a similar way.Therefore, p ≤ y ′ ≤ s or s ≤ y ′ ≤ p.One can conclude that h 1 (y ′ ) ≥ h 1 (p) ∧ h 1 (s) since h 1 is convex, and then In view of the above discussion, Q holds.
In light of the statement Q, one can obtain that Finally, Cases (1), ( 2), ( 3) and (4) satisfy that (h ) min .F satisfies left distributivity over A, if A is idempotent and F is constructed as Eq. ( 6) and B can be represented as where u, v ∈ [λ, e) and S ′ is a positive right semi-t-conorm.
Lemma 1 gives the sufficient condition that F satisfies left distributivity over A. Furthermore, combined with Lemma 1 and Theorem 12, a sufficient condition about left distributivity between extended semi-t-operator ⊙ F and extended S-uninorm ⊙ A can be shown.
if A is idempotent and F is constructed as Eq. ( 6) and B can be represented as where u, v ∈ [λ, e) and S ′ is positive right semi-t-conorm.
Proof It can be immediately verified by Theorem 12 and Lemma 1. □ Similarly, when semi-t-operator F satisfies the right distributive law for S-uninorm A, then the extended semi-t-operator ⊙ F satisfies distributive law over extended S-uninorm ⊙ A .
Proposition 14 [39] Suppose 0 < λ < e < η < ε = 1, F ∈ Fε, η , A ∈ (U S e,λ ) min .If the right distributivity between F and A is satisfied, then A is an idempotent S-uninorm and F has the following form: where S is a semi-t-conorm,

have associativity, the property of monotonically increasing and the respective boundary condition
Figure 3 The characterizations of F (left) and A (right) in Proposition 14 [39] Theorem 15 Suppose 0 < λ < e < η < ε = 1, F ∈ Fε, η is continuous, A ∈ (U S e,λ ) min and F satisfies the right distributivity over A. For any h 1 ∈ F C and h 2 , h 3 ∈ F , then it holds that . The distributivity of extended semi-t-operators over extended S-uninorms Proof It can be verified in an analogical method as Theorem 12. □ Lemma 2 [39] Suppose 0 < λ < e < η < ε = 1, F ∈ Fε, η , A ∈ (U S e,λ ) min .F satisfies right distributivity over A, if A is idempotent and F is constructed as Eq. ( 7) and B can be represented as where u, v ∈ [λ, e) and S ′ is a positive left semi-t-conorm.
Furthermore, a sufficient condition of right distributivity between extended semi-t-operator ⊙ F and S-uninorm ⊙ A can be shown below.
holds, if A is idempotent and F is constructed as (7) where B can be represented as: where u, v ∈ [λ, e) and S ′ is positive left semi-t-conorm.
Proof It follows immediately from Lemma 2 and Theorem 15. □
(1) F satisfies the left distributivity over A ⇐⇒ A is idempotent.
(2) F is right distributive over A.
Proof Consider that ε = 0 and η = 1.According to Proposition 3, F (u, v) = u for all u, v ∈ [0, 1].Meanwhile, since F is left distributive over A, we obtain that A is idempotent by Lemma 3, that is, A(t, t) = t for any t ∈ [0, 1].Hence, it holds that To illustrate the left distributivity of ⊙ F to ⊙ A , only h 1 (ω) = A(p,s)=ω h 1 (p) ∧ h 1 (s) needs to be stated.
On the one hand, it can be obtained that On the other hand, if A(p, s) = ω, then p ∧ s ≤ ω ≤ p ∨ s.In fact, assume that p ∧ s > ω, then which is a contradiction.Similarly, it can be shown that ω > p ∨ s does not hold.Hence, we obtain that p∧s ≤ ω ≤ p∨s.Since In summary, (h Further, we verify that the extended semi-t-operator ⊙ F satisfies the right distributive law over extended S-uninorm ⊙ A under the condition ε = 0, η = 1. e,λ ) min .Then the right distributivity between F and A is satisfied and for any h 1 , h 2 , h 3 ∈ F , it concludes that Proof According to Proposition 3, F (u, v) = u for any u, v ∈ [0, 1] when ε = 0 and η = 1.One concludes that

Hence, ((h
Notice that the right distributivity between extended semi-t-operator ⊙ F and extended S-uninorm ⊙ A does not need the condition of f ∈ F C .
(1) F satisfies left distributive law over A.
(2) F satisfies right distributive law over A ⇐⇒ A is idempotent.
e,λ ) min .Then the following statements hold.
Proof It can be verified in an analogical method as Theorems 17 and 18. □
In light of Proposition 20, the following conclusions can be obtained similarly as Theorem 12.The distributivity of extended semi-t-operators over extended S-uninorms Theorem 21 Suppose 0 < λ < e < ε < η = 1, F ∈ Fε, η is continuous, A ∈ (U S e,λ ) min and F satisfies the distributivity over A. For any h 1 ∈ F C , h 2 , h 3 ∈ F , then we have that Proof It can be verified in an analogical method as Theorem 12. □ Further, combined with Lemma 5, the sufficient condition that ⊙ F is distributive over ⊙ A can be given below.
hold, if A is idempotent, F is constucted as (8) and B can be represented as: where u, v ∈ [λ, e) and S ′ is positive semi-t-conorm.
Proof It follows from Lemma 5 that F satisfies the distributive law for A. Further, the distributivity between ⊙ F and ⊙ A can be proved similar to Theorem 12. □ The following conclusions hold when e < η < ε = 1.
According to Proposition 23, we can obtain the following conclusions similarly as Theorem 12.
Theorem 24 Suppose 0 < λ < e < η < ε = 1, F ∈ Fε, η is continuous, A ∈ (U S e,λ ) min and F satisfies the distributivity over A. For any h 1 ∈ F C and h 2 , h 3 ∈ F , it holds that Proof It can be verified in an analogical method as Theorem 15.
□ Further, combined with Lemma 6, the sufficient condition of the distributivity between ⊙ F and ⊙ A can be given below.The distributivity of extended semi-t-operators over extended S-uninorms Proposition 25 Suppose 0 < λ < e < η < ε = 1, F ∈ Fε, η is continuous, A ∈ (U S e,λ ) min .For any h 1 ∈ F C and h 2 , h 3 ∈ F , then we conclude that and hold, if A is idempotent, F is constructed as (9) and B can be represented as: where u, v ∈ [λ, e) and S ′ is positive semi-t-conorm.
(1) F satisfies conditional distributivity for A from the left ⇐⇒ F satisfies left distributivity for A when e < ε < η = 1.(2) F satisfies conditional distributivity for A from the right ⇐⇒ F satisfies right distributivity for A when e < η < ε = 1.
Next, based on the condition that F has conditional distributivity over A form left (resp.right), the left (resp.right) distributive law between ⊙ F and ⊙ A is given in the following theorems.
Theorem 26 Suppose 0 < λ < e < ε < η = 1, F ∈ Fε, η is continuous, A ∈ (U S e,λ ) min and F satisfies the conditional distributivity over A from the left.For any h 1 ∈ F C and h 2 , h 3 ∈ F , it holds that Proof In light of Theorem 12 and Lemma 7, it can be verified immediately.□ Theorem 27 Suppose 0 < λ < e < η < ε = 1, F ∈ Fε, η is continuous, A ∈ (U S e,λ ) min and F satisfies the conditional distributivity over A from the right.For any h 1 ∈ F C and h 2 , h 3 ∈ F , then we have that Proof In light of Theorem 15 and Lemma 7, it can be verified immediately.□
Theorem 28 Suppose 0 = ε ≤ λ < e < η = 1, F ∈ Fε, η , A ∈ (U S e,λ ) min and F satisfies left distributivity (resp.conditionally distributivity or distributivity) for A. For any h 1 ∈ F C and h 2 , h 3 ∈ F , then we can obtain that Proof In light of Theorem 17 and Lemma 8, it can be verified immediately.□ Theorem 29 Suppose 0 = η ≤ λ < e < ε = 1, F ∈ Fε, η , A ∈ (U S e,λ ) min and F satisfies right distributivity (resp.conditional distributivity or distributivity) for A. For any h 1 ∈ F C and h 2 , h 3 ∈ F , then we get that Proof In light of Theorem 19 and Lemma 8, it can be verified immediately.□

Conclusion
This paper mainly researches the distributive law between extended semi-toperator and S-uninorm on fuzzy truth value where the underlying uninorm of S-uninorm is U min , which further generalizes the corresponding results of [38,39] to fuzzy truth value.The core contents of this paper can be summarized as follows: (1) Applying Zadeh's extension principle to give the extended semi-t-operator and extended S-uninorm, and then some results of extended semi-t-operator are shown, such as the operations of extended semi-t-operator and special fuzzy truth values 1, 1, commutativity, associativity and so on.

Figure 1
Figure 1 The relationship of distributive law between different logical operators 1] is an increasing, commutative and associative binary function with neutral element e ∈ [0, 1].