Constructions of Strict Left (Right)-Conjunctive Left (Right) Semi-Uninorms on a Complete Lattice
Yuan Wang^{1, *}, Keming Tang^{1}, Zhudeng Wang^{2}
^{1}College of Information Engineering, Yancheng Teachers University, Yancheng, People's Republic of China
^{2}School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, People's Republic of China
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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).
References