International Journal of Intelligent Information Systems
Volume 6, Issue 1, February 2017, Pages: 1-6

Constructions of Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms on a Complete Lattice

Yuan Wang1, *, Keming Tang1, Zhudeng Wang2

1College of Information Engineering, 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 Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms on a Complete Lattice. International Journal of Intelligent Information Systems. Vol. 6, No. 1, 2017, pp. 1-6. doi: 10.11648/j.ijiis.20170601.11

Received: January 6, 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 strict left (right)-conjunctive left (right) semi-uninorms by means of some examples. Then we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.

Keywords: Fuzzy Logic, Fuzzy Connective, Left (Right) Semi-Uninorm, Strict Left (Right)-Conjunctive


1. Introduction

Uninorms, introduced by Yager and Rybalov [1], and studied by Fodor et al. [2], are special aggregation operators that have proven useful in many fields like fuzzy logic, expert systems, neural networks, aggregation, and fuzzy system modeling (see [3-6]).

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 (see [2]). 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 [7] and later in a finite chain in [8], and Wang and Fang [9-10] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [11] introduced the concept of semi-uninorms, and Su et al. [12] 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 (see [9, 11, 13-14]).

Constructing fuzzy connectives is an interesting topic. Recently, Jenei and Montagna [15] introduced several new types of constructions of left-continuous t-norms, Wang [16] 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. [12] studied the constructions of left and right semi-uninorms on a complete lattice, Su and Wang [17] investigated the constructions of implications and coimplications on a complete lattice. and Wang et al. [18-20] studied the relations among implications, coimplications and left (right) semi-uninorms, on a complete lattice. Moreover, Wang et al. [21-22] investigated the constructions of conjunctive left (right) semi-uninorms, disjunctive left (right) semi-uninorms, strict left (right)-disjunctive left (right) semi-uninorm, implications satisfying the neutrality principle, coimplications satisfying the neutrality principle, and coimplications satisfying the order property.

This paper is a continuation of [12, 21-22]. Motivated by these works in [12, 21-22], we will further focus on this issue and investigate constructions of the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms on a complete lattice.

This paper is organized as follows. In Section 2, we recall some necessary concepts about the left (right) semi-uninorms on a complete lattice and illustrate these notions by means of some examples. In Section 3, we give out the formulas for calculating the upper and lower approximation strict left (right)-conjunctive left (right) semi-uninorms of a binary operation.

The knowledge about lattices required in this paper can be found in [23].

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

Noting that the commutativity and associativity are not desired for aggregation operators in a number of cases, Liu [11] introduced the concept of semi-uninorms and Su et al. [12] studied the notions of left and right semi-uninorms. Here, we recall some necessary definitions and give some examples of the left and right semi-uninorms on a complete lattice.

Definition 2.1 (Su et al. [12]). 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 [9-10] 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 [11].

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 [9]). 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 .

Below, we illustrate these notions by means of several examples.

Example 2.1 (Su et al. [12]). Let ,

where  and  are elements of . Then  and  are, respectively, the smallest and greatest elements of . By Example 2 and Theorem 8 in [18], 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 [20], we know that  and  are, respectively, the smallest and greatest elements of . By Example 3 and Theorem 8 in [18], 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 .

3. Constructing Strict Conjunctive Left and Right Semi-Uninorms

Constructing aggregation operators is an interesting work. Recently, Jenei and Montagna [15] introduced several new types of constructions of left-continuous t-norms, Su et al. [12] studied the constructions of left and right semi-uninorms on a complete lattice, and Wang et al. [21-22] 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 2.2. Thus, for a binary operation  on , if there exists  such that , then

(9)

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

(10)

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 3.1 (Su et al. [12]). Let  be a binary operation on . Define the upper approximation aggregator  and the lower approximation aggregator  of  as follows:

(11)

(12)

Theorem 3.1 (Su et al. [12]). Let . Then the following statements hold:

(13)

 and

(14)

 and  are non-decreasing in its each variable.

If  is non-decreasing in its each variable, then

(15)

Theorem 3.2. 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,

(16)

  (17)

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 3.3. Suppose that ,  and .

(1).   If , then ;

if , then .

(2).   If ,  and   is left arbitrary -distributive, then . 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 Examples 2.1 and 2.2.

(1) Let . If , then , . Thus,

(18)

It implies that and  for any . If , then  and so , i.e.,  is strict left-conjunctive. By Theorem 3.1(3) and the monotonicity of , we can see that  is non-decreasing in its each variable. So, . If  and , then  and . Therefore,

(19)

Let . If , then,  and . Thus,

 and  for any  and  is strict left-conjunctive. By Theorem 3.1(3) and the monotonicity of , we know that  is non-decreasing in its each variable. So, . If  and , then  and . Therefore,

(20)

(2) When ,  and  are, respectively, the smallest and greatest elements of  by Examples 2.1 and 2.2. Let . If , then  by statement (1). Noting that  is left arbitrary -distributive, we can see that  is also left arbitrary -distributive by Theorem 3.2(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 3.1(4) and so

(21)

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 3.4. Suppose that ,  and .

(1).   If , then ; if , then .

(2).   If ,  and   is right arbitrary -distributive, then . Moreover, if  is non-decreasing in its first variable, then .

4. Conclusions and Future Works

Constructing fuzzy connectives is an interesting topic. Recently, Su et al. [12] studied the constructions of left and right semi-uninorms, and Wang et al. [17-18, 20, 22] 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.

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 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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