Automation, Control and Intelligent Systems
Volume 5, Issue 1, February 2017, Pages: 1-7

Constructions of Implications Satisfying the Order Property on a Complete Lattice

Yuan Wang1, *, Keming Tang1, Zhudeng Wang2

1College of Information Science and Technology, Yancheng Teachers University, Yancheng, People's Republic of China

2School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, People's Republic of China

Email address:

(Yuan Wang)
(Keming Tang)
(Zhudeng Wang)

*Corresponding author

To cite this article:

Yuan Wang, Keming Tang, Zhudeng Wang. Constructions of Implications Satisfying the Order Property on a Complete Lattice. Automation, Control and Intelligent Systems. Vol. 5, No. 1, 2017, pp. 1-7. doi: 10.11648/j.acis.20170501.11

Received: January 7, 2017; Accepted: January 19, 2017; Published: February 23, 2017


Abstract: In this paper, we further investigate the constructions of fuzzy connectives on a complete lattice. We firstly illustrate the concepts of left (right) semi-uninorms and implications satisfying the order property by means of some examples. Then we give out the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation.

Keywords: Fuzzy Logic, Fuzzy Connective, Implication, Order Property


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 (see [4-8]). But, as was pointed out by Fodor and Keresztfalvi [9], 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 [10], pseudo t-norms [11], pseudo-uninorms [12], left and right uninorms [13], semi-uninorms [14], aggregation operators [15] and so on.

Uninorms, introduced by Yager and Rybalov [16] and studied by Fodor et al. [17], 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 [17]. However, 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 [0, 1] in [18] and later in a finite chain in [19], Wang and Fang [13, 20] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [14] introduced the concept of semi-uninorms and Su et al. [21] discussed the notion of left and right semi-uninorms on a complete lattice. On the other hand, it is well known that a uninorm (semi-uninorm, left and right uninorms)  can be conjunctive or disjunctive whenever or 1, respectively. This fact allows to use uninorms in defining fuzzy implications [13-14, 14-23].

Constructing fuzzy connectives is an interesting topic. Recently, Wang [24] 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 and Wang [25] investigated the constructions of implications and coimplications on a complete lattice and Wang et al. [26-28] studied the relations among implications, coimplications and left (right) semi-uninorms on a complete lattice. Moreover, Wang et al. [27, 29-30] investigated the constructions of implications and coimplications satisfying the neutrality principle.

In this paper, based on [24-30], we study the constructions of implications satisfying the order property on a complete lattice. After recalling some necessary definitions and examples about the left (right) semi-uninorms and implications on a complete lattice in Section 2, we give out the formulas for calculating the upper and lower approximation implications, which satisfy the order property, of a binary operation in Section 3.

The knowledge about lattices required in this paper can be found in [31].

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. Left (Right) Semi-Uninorms and Implications

In this section, we recall some necessary definitions and examples about the left (right) semi-uninorms and implications on a complete lattice.

Definition 2.1 (Su et al. [21]). 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.

If a left (right) semi-uninorm  is associative, then is the left (right) uninorm [13] 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 [14].

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 [13]). 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).

Noting that the least upper bound of the empty set is 0 and the greatest lower bound of the empty set is 1, we have

(3)

for any  when  is left (right) arbitrary -distributive,

(4)

for any  when  is left (right) arbitrary -distributive.

For the sake of convenience, we introduce the following symbols:

: the set of all left semi-uninorms with the left neutral element  on ;

: the set of all right semi-uninorms with the right neutral element  on ;

: 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 .

Example 2.1 (Su et al. [21]). Let ,

where  and  are elements of . Then  and  are, respectively, the smallest and greatest elements of . By Example 2 and Theorem 8 in [26], we see that and  are two join-semilattices with the greatest element .

Example 2.2. Let ,

When  and , it is straightforward to verify that  is a strict left-conjunctive left semi-uninorm with the left neutral element . If , then

i.e., . Thus,  is the smallest element of .

Moreover, assume that . For any , if , then there exists  such that ,

(5)

if , then  for any and there exists  such that ,

(6)

(7)

if , then  for any ,

(8)

Therefore,  is left arbitrary -distributive and the smallest element of .

Example 2.3. Let ,

 

where  and  are elements of . By Example 2.6 in [26], we know that  and  are, respectively, the smallest and greatest elements of . By Example 3 and Theorem 8 in [26], 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 .

Definition 2.3 (Fodor and Roubens [1], Baczynski and Jayaram [4], Bustince et al. [6], De Baets and Fodor [22]). 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 .

Implications are extensions of the Boolean implication  ( meaning that  is sufficient for ).

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 on ;

: the set of all right arbitrary -distributive implications on ;

: 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. By Example 2.4 in [25], we know that  is not a join-semilattice.

Definition 2.3. Let  be a binary operation on . Define  as follows:

(9)

(10)

Here,  and  are, respectively, called the left and right residuum of the binary operation .

By virtue of Theorems 4.4 and 4.5 in [13], we know that  and  satisfy the following right residual principle:

(11)

when a binary operation  is right arbitrary -distributive;  and  satisfy the following left residual principle:

(12)

when  is left arbitrary -distributive.

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 2.3.

Example 2.4. For some left and right semi-uninorms in Examples 2.1-2.3, a simple computation shows that

where  and  are elements of . By the virtue of Theorem 8 in [26], 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 . For any , if , then there exists  such that ,

(13)

if , then  for any ,

(14)

if , then  for any  and there exists  such that ,

(15)

(16)

Therefore,  is the greatest element of .

 Similar conclusions hold for  and .

3. Constructing the Implications Satisfying the Order Property

Recently, Su and Wang [25] have studied the constructions of implications and coimplications and Wang et al. [27, 29, 30] further investigated the constructions of implications and coimplications satisfying the neutrality principle on a complete lattice.

This section is a continuation of [25, 27, 29, 30]. We will study the constructions of the upper and lower approximation implications which satisfy the order property.

It is easy to verify that if , then

(17)

When  and , we see that  is also a complete lattice with the smallest element and greatest element  by Example 2.4. Thus, for a binary operation  on , if there exists  such that , then

(18)

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

(19)

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 3.1 (see Su and Wang [25]). Let  be a binary operation on . Define the upper approximation implicator  and the lower approximation implicator  of  as follows:

(20)

(21)

Theorem 3.1 (see Su and Wang [25]). Let . Then the following statements hold:

(22)

 and

(23)

 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,

(24)

(25)

(2). If  is right arbitrary -distributive, then  is also right arbitrary -distributive.

(3). If is left arbitrary -distributive, then,

(26)

(27)

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 3.1, the monotonicity of  and , and the right residual principle, we have that

(28)

(29)

Thus, . Similarly, we have that

(30)

(31)

(32)

If , let , then

(33)

So,  for any , i.e., .

Moreover, we know that  is right arbitrary -distributive and hence

(34)

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

(35)

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 2.4.

(1) If , let , then  and

(36)

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,

(37)

If , let , then ,

(38)

Thus, we can prove in an analogous way that  and .

(2) When , and  are, respectively, the smallest and greatest elements of by Example 2.4. 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 (1). 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

(39)

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

(40)

Moreover, if  is non-decreasing in its first variable, then .

4. Conclusions and Future Works

Constructing fuzzy connectives is an interesting topic. Recently, Wang et al. [24-25, 27, 29, 30] 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 implications, which satisfy the order property, of a binary operation.

In a forthcoming paper, we will investigate the relationships between left (right) semi-uninorms and implications on a complete lattice.

Acknowledgements

This work is supported by Science Foundation of Yancheng Teachers University (16YCKLQ006), the National Natural Science Foundation of China (61379064) and Jiangsu Provincial Natural Science Foundation of China (BK20161313).


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