Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications Satisfying the Order Property
Zhudeng Wang^{1, *}, Yuan Wang^{2}, Keming Tang^{2}
^{1}School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, People's Republic of China
^{2}College of Information Engineering, Yancheng Teachers University, Yancheng, People's Republic of China
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To cite this article:
Zhudeng Wang, Yuan Wang, Keming Tang. Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications Satisfying the Order Property. Applied and Computational Mathematics. Vol. 6, No. 1, 2017, pp. 45-53. doi: 10.11648/j.acm.20170601.13
Received: January 8, 2017; Accepted: January 19, 2017; Published: February 23, 2017
Abstract: We firstly give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation. Then, we lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation. Finally, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary ˅-distributive left (right) semi-uninorms and lower approximation right arbitrary ˄-distributive implications which satisfy the order property.
Keywords: Fuzzy Logic, Fuzzy Connective, Left (Right) Semi-Uninorm, Implication, Strict Left (Right)-Conjunctive
1. Introduction
In fuzzy logic systems (see [1-2]), connectives "and", "or" and "not" are usually modeled by t-norms, t-conorms, and strong negations on (see [3]), respectively. Based on these logical operators on , the three fundamental classes of fuzzy implications on , i.e., R-, S-, and QL-implications on , were defined and extensively studied. But, as was pointed out by Fodor and Keresztfalvi [4], sometimes there is no need of the commutativity or associativity for the connectives "and" and "or". Thus, many authors investigated implications based on some other operators like weak t-norms [5], pseudo t-norms [6], pseudo-uninorms [7], left and right uninorms [8], semi-uninorms [9], aggregation operators [10] and so on.
Uninorms, introduced by Yager and Rybalov [11], and studied by Fodor et al. [12], are special aggregation operators that have proven useful in many fields like fuzzy logic, expert systems, neural networks, aggregation, and fuzzy system modeling. This kind of operation is an important generalization of both t-norms and t-conorms and a special combination of t-norms and t-conorms. But, there are real-life situations when truth functions cannot be associative or commutative. By throwing away the commutativity from the axioms of uninorms, Mas et al. introduced the concepts of left and right uninorms on in [13] and later in a finite chain in [14], and Wang and Fang [8, 15] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [9] introduced the concept of semi-uninorms, and Su et al. [16] discussed the notions of left and right semi-uninorms, on a complete lattice. On the other hand, it is well known that a uninorm (semi-uninorm) can be conjunctive or disjunctive whenever or 1, respectively. This fact allows us to use uninorms in defining fuzzy implications.
Constructing fuzzy connectives is an interesting topic. Recently, Jenei and Montagna [17] introduced several new types of constructions of left-continuous t-norms, Wang [18] laid bare the formulas for calculating the smallest pseudo-t-norm that is stronger than a binary operation and the largest implication that is weaker than a binary operation, Su et al. [16] studied the constructions of left and right semi-uninorms on a complete lattice, Su and Wang [19] investigated the constructions of implications and coimplications on a complete lattice. and Wang et al. [20-22] studied the relations among implications, coimplications and left (right) semi-uninorms, on a complete lattice. Moreover, Wang et al. [23-24] investigated the constructions of conjunctive left (right) semi-uninorms, disjunctive left (right) semi-uninorms, strict left (right)-disjunctive left (right) semi-uninorm, implications and coimplications satisfying the neutrality principle.
This paper is a continuation of [16, 19, 23-24]. Motivated by these works in [16, 19, 23-24], we will further focus on this issue and investigate constructions of the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms and the upper and lower approximation implications which satisfy the order property.
This paper is organized as follows. In Section 2, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation. In Section 3, we lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation. In Section 4, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary -distributive left (right) semi-uninorms and lower approximation right arbitrary -distributive implications which satisfy the order property, and find out some conditions such that the lower approximation strict left (right)-conjunctive left (right) semi-uninorm of a binary operation and upper approximation implication, which satisfies the order property, of left (right) residuum of the binary operation satisfy the generalized modus ponens rule.
The knowledge about lattices required in this paper can be found in [25].
Throughout this paper, unless otherwise stated, always represents any given complete lattice with maximal element 1 and minimal element 0; stands for any index set.
2. Strict Conjunctive Left and Right Semi-Uninorms
In this section, we firstly recall some necessary concepts about the strict conjunctive left (right) semi-uninorms on a complete lattice.
Definition 2.1 (Su et al. [16]). A binary operation on is called a left (right) semi-uninorm if it satisfies the following two conditions:
(U1) there exists a left (right) neutral element, i.e., an element () satisfying ( for all ,
(U2) is non-decreasing in each variable.
Clearly, and hold for any left (right) semi-uninorm on .
If a left (right) semi-uninorm is associative, then is the left (right) uninorm [8, 15] on .
If a left (right) semi-uninorm with the left (right) neutral element () has a right (left) neutral element (), then . Let . Here, is the semi-uninorm [9].
For any left (right) semi-uninorm on , is said to be left-conjunctive and right-conjunctive if and , respectively. is said to be conjunctive if both and since it satisfies the classical boundary conditions of AND.
is said to be strict left-conjunctive and strict right- conjunctive if is conjunctive and for any and , respectively.
Definition 2.2 (Wang and Fang [8]). A binary operation on is called left (right) arbitrary -distributive if
(1)
left (right) arbitrary -distributive if
(2)
If a binary operation is left arbitrary -distributive (- distributive) and also right arbitrary -distributive (-distributive), then is said to be arbitrary -distributive (-distributive).
For the sake of convenience, we introduce the following symbols:
: the set of all strict left-conjunctive left semi-uninorms with the left neutral element on ;
: the set of all strict right-conjunctive right semi-uninorms with the right neutral element on ;
: the set of all strict left-conjunctive left arbitrary -distributive left semi-uninorms with the left neutral element on ;
: the set of all strict right-conjunctive right arbitrary -distributive right semi-uninorms with the right neutral element on .
Below, we illustrate these notions by means of two examples.
Example 2.1. Let ,
where and are elements of . By Example 2 and Theorem 8 in [20], we see that and are two join-semilattices with the greatest element . When and , it is straightforward to verify that is the smallest element of .
Moreover, assume that . is left arbitrary -distributive and the smallest element of .
Example 2.2. Let ,
where and are elements of . By Example 3 and Theorem 8 in [20], we see that and are two join-semilattices with the greatest element .
Similarly, When and , is the smallest element of . Moreover, if, then is the smallest element of .
Constructing aggregation operators is an interesting work. Recently, Jenei and Montagna [17] introduced several new types of constructions of left-continuous t-norms, Su et al. [16] studied the constructions of left and right semi-uninorms on a complete lattice, and Wang et al. [23-24] investigated the constructions of conjunctive left (right) semi-uninorms and disjunctive left (right) semi-uninorms on a complete lattice. Now, we continue this work and give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.
It is easy to verify that for any nonempty subset of . If and , then is a complete lattice with the smallest element and greatest element by Example 2.1. Thus, for a binary operation on , if there exists such that , then
(3)
is the smallest strict left-conjunctive left semi-uninorm that is stronger than on , we call it the upper approximation strict left-conjunctive left semi-uninorm of and write as ; if there exists such that , then
(4)
is the largest strict left-conjunctive left semi-uninorm that is weaker than on , we call it the lower approximation strict left-conjunctive left semi-uninorm of and write as .
Similarly, we introduce the following symbols:
the upper approximation strict right-conjunctive right semi-uninorm of ;
: the lower approximation strict right-conjunctive right semi-uninorm of ;
: the upper approximation strict left-conjunctive left arbitrary -distributive left semi-uninorm of ;
: the lower approximation strict left-conjunctive left arbitrary -distributive left semi-uninorm of ;
: the upper approximation strict right-conjunctive right arbitrary -distributive right semi-uninorm of ;
: the lower approximation strict right-conjunctive right arbitrary -distributive right semi-uninorm of .
Definition 2.3 (Su et al. [16]). Let be a binary operation on . Define the upper approximation aggregator and the lower approximation aggregator of as follows:
(5)
(6)
Theorem 2.1 (Su et al. [16]). Let . Then the following statements hold:
(7)
and
(8)
and are non-decreasing in its each variable.
If is non-decreasing in its each variable, then
(9)
Theorem 2.2. Let .
(1). If is left (right) arbitrary -distributive, then is left (right) arbitrary -distributive.
(2). If is left (right) arbitrary -distributive, then is left (right) arbitrary -distributive.
Proof. We only prove that statement (1) holds.
If is left arbitrary -distributive, then is non-decreasing in its first variable,
(10)
(11)
i.e., is left arbitrary -distributive.
Similarly, we can show that is right arbitrary -distributive when is right arbitrary -distributive.
The theorem is proved.
Below, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.
Theorem 2.3. Suppose that , and .
(1). If , then ;
if , then .
(2). If , and is left arbitrary -distributive, then
(12)
Moreover, if is non-decreasing in its second variable, then .
Proof. Assume that and . Then and are, respectively, the smallest and greatest elements of by Example 2.1.
(1) Let . If , then , . Thus,
(13)
It implies that and for any . If , then and so , i.e., is strict left-conjunctive. By Theorem 2.1(3) and the monotonicity of , we can see that is non-decreasing in its each variable. So, . If and , then and . Therefore,
(14)
Let . If , then
and . (15)
Thus, and for any and is strict left-conjunctive. By Theorem 2.1(3) and the monotonicity of , we know that is non-decreasing in its each variable. So, . If and , then and . Therefore,
(16)
(2) When , and are, respectively, the smallest and greatest elements of by Example 2.1. Let . If , then by statement (1). Noting that is left arbitrary -distributive, we can see that is also left arbitrary -distributive by Theorem 2.2(1). Thus, is left arbitrary -distributive and . By the proof of statement (1), we have that .
Moreover, if is non-decreasing in its second variable, then by Theorem 2.1(4) and so
(17)
The theorem is proved.
Similarly, for calculating the upper and lower approximation strict right-conjunctive right semi-uninorms of a binary operation, we have the following theorem.
Theorem 2.4. Suppose that , and .
(1). If , then ; if , then .
(2). If , and is right arbitrary -distributive, then
(18)
Moreover, if is non-decreasing in its first variable, then .
3. Implications Satisfying the Order Property
Recently, Su and Wang [19] have studied the constructions of implications and coimplications and Wang et al. [23-24] further investigated the constructions of implications and coimplications satisfying the neutrality principle on a complete lattice. This section is a continuation of [19,23,24]. We will study the constructions of the upper and lower approximation implications which satisfy the order property.
Definition 3.1 (Baczynski and Jayaram [26], Bustince et al. [27], De Baets and Fodor [28], Fodor and Roubens [1]). An implication on is a hybrid monotonous (with decreasing first and increasing second partial mappings) binary operation that satisfies the corner conditions and .
An implication is said to satisfy the order property with respect to (w.r.t. e, for short) when if and only if for any .
Note that for any implication on , due to the monotonicity, the absorption principle holds, i.e., for any .
For the sake of convenience, we introduce the following symbols:
: the set of all implications which satisfy the order property w. r. t. e on ;
: the set of all right arbitrary -distributive implications which satisfy the order property w. r. t. e on .
Clearly, and are all meet-semilattices. Definition 3.2. Let be a binary operation on . Define as follows:
(19)
(20)
Here, and are, respectively, called the left and right residuum of the binary operation .
When is non-decreasing in each variable, it is easy to see that and are all decreasing in the first variable and increasing in the second one by Definition 3.2.
Example 3.1. For some left and right semi-uninorms in Examples 2.1-2.2, a simple computation shows that
where and are elements of . By the virtue of Theorem 8 in [20], we see that is the smallest element of both and .
When and , it is easy to see that is the greatest element of .
Moreover, assume that . is the greatest element of .
Similar conclusions hold for and .
It is easy to verify that if , then
(21)
When and , we see that is also a complete lattice with the smallest element and greatest element by Example 3.1. Thus, for a binary operation on , if there exists such that , then
(22)
is the smallest implication that is stronger than and satisfies the order property w. r. t. on . Here, we call it the upper approximation implication, which satisfies the order property w. r. t. , of and write as . Similarly, if there exists such that , then
(23)
is the largest implication that is weaker than and satisfies the order property w. r. t. on . Here, we call it the lower approximation implication, which satisfies the order property w. r. t. , of and write as .
Likewise, for a binary operation on , we may introduce the following symbols:
: the upper approximation implication, which satisfies the order property w. r. t. , of ;
: the lower approximation implication, which satisfies the order property w. r. t. , of ;
(): the upper approximation right arbitrary -distributive implication, which satisfies the order property w. r. t. (), of ;
(): the lower approximation right arbitrary -distributive implication, which satisfies the order property w.r. t. (), of .
Definition 3.3 (see Su and Wang [19]). Let be a binary operation on . Define the upper approximation implicator and the lower approximation implicator of as follows:
(24)
(25)
Theorem 3.1 (see Su and Wang [19]). Let . Then the following statements hold:
(26)
and
(27)
and are hybrid monotonous.
If is are hybrid monotonous, then .
Theorem 3.2. Let .
(1). If is right arbitrary -distributive, then is also right arbitrary -distributive,
(28)
(29)
(2). If is right arbitrary -distributive, then is also right arbitrary -distributive.
(3). If is left arbitrary -distributive, then,
(30)
(31)
Proof. We only prove that statement (1) holds.
Assume that is a right arbitrary -distributive binary operation on . Clearly, is also right arbitrary -distributive. By Definition 3.3, the monotonicity of and , and the right residual principle, we have that
(32)
(33)
Thus, . Similarly, we have that
(34)
(35)
(36)
If , let , then
(37)
So, for any , i.e., . Moreover, we know that is right arbitrary -distributive and hence
(38)
The theorem is proved.
Below, we give out the formulas for calculating the upper and lower approximation implications which satisfy the order property.
Theorem 3.3. Suppose that , and .
(1) If , then ;
if , then .
(2) If , and is right arbitrary -distributive, then
(39)
Moreover, if is non-decreasing in its first variable, then .
Proof. Assume that and. Then and are, respectively, the smallest and greatest elements of by Example 3.1.
(1) If , let , then and
(40)
Thus, and . If , then ; if , then and so , i.e., satisfies the order property w. r. t. . By Theorem 3.1(3) and the hybrid monotonicity of , we know that is hybrid monotonous. So, . If and , then and . Therefore,
(41)
If , let , then ,
(42)
Thus, we can prove in an analogous way that and .
(2) When , and are, respectively, the smallest and greatest elements of by Example 3.1. Let . If , then by statement (1). Noting that is right arbitrary -distributive, we can see that is also right arbitrary -distributive by Theorem 3.2(2). So, is right arbitrary -distributive, i.e., . By the proof of statement (1), we know that .
Moreover, if is non-decreasing in its first variable, then by Theorem 3.1(4) and so
(43)
The theorem is proved.
Analogous to Theorem 3.3, we have the following theorem.
Theorem 3.4. Suppose that , and .
(1) If , then ;
if , then .
(2) If , and is right arbitrary -distributive, then
(44)
Moreover, if is non-decreasing in its first variable, then .
4. The Relations Between Strict (Right)-Conjunctive Left (Right) Semi-Uninorms and Implications
In this section, we reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary -distributive left (right) semi-uninorms and lower approximation right arbitrary -distributive implications which satisfy the order property.
Theorem 4.1. Suppose that , and .
(1) If , and is left arbitrary -distributive, then
(45)
(2) If , and is right arbitrary -distributive, then
(46)
Proof. We only prove the statement (1) holds.
Assume that and is left arbitrary -distributive. Then it follows from Theorem 4.6 in [8] and Definition 3.2 that and is right arbitrary -distributive. Thus, by Theorem 3.3(2). Moreover, it follows from Theorems 2.2(1) and 2.3(2) and the left residual principle that
(47)
i.e., . By Theorem 3.2(3), we know that . Therefore,
(48)
The theorem is proved.
Finally, we give out some conditions such that the lower approximation strict left (right)-conjunctive left (right) semi-uninorm of a binary operation and upper approximation implication, which satisfies the order property, of left (right) residuum of the binary operation satisfy the GMP rule.
Theorem 4.2. Suppose that , and .
(1) If , and is non-decreasing in its second variable and left arbitrary -distributive and and are comparable (see [25]) when , then and satisfy the GMP rule in the form
(49)
(2) If , and is non-decreasing in its first variable and right arbitrary -distributive and and are comparable (see [25]) when , then and satisfy the GMP rule in the form
(50)
Proof. We only prove the statement (1) holds.
Assume that , is non-decreasing in its second variable and left arbitrary -distributive. Then, , , is non-increasing in its first variable by Definition 3.2 and right arbitrary -distributive by Theorem 4.6 in [8], by Theorem 3.1(4),
(51)
By the virtue of Theorem 2.3(2), we see that
(52)
By Example 3.1 and Theorem 3.3(1), we know that
Thus,
When , noting that and are comparable, we see that
So, when ,
(53)
Therefore, for all , i.e., and satisfy the GMP rule.
The theorem is proved.
5. Conclusions and Future Works
Constructing fuzzy connectives is an interesting topic. Recently, Su et al. [16] studied the constructions of left and right semi-uninorms, and Wang et al. [19-20, 22, 24] investigated the constructions of implications and coimplications on a complete lattice. In this paper, motivated by these works, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation; lay bare the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation; reveal the relationships between the upper approximation strict left (right)-conjunctive left (right) arbitrary -distributive left (right) semi-uninorms and lower approximation right arbitrary -distributive implications which satisfy the order property.
In a forthcoming paper, we will further investigate the constructions of left (right) semi-uninorms and coimplications on a complete lattice.
Acknowledgements
This work is supported by Jiangsu Provincial Natural Science Foundation of China (BK20161313), Science Foundation of Yancheng Teachers University (16YCKLQ006) and the National Natural Science Foundation of China (61379064).
References