Abstract
In this paper, we provide a generalized form of ideals that is k-ideals of semirings with the combination of a bipolar fuzzy set (BFS). The BFS is a generalization of fuzzy set (FS) that deals with uncertain problems in both positive and negative aspects. The main theme of this paper is to present the idea of -bipolar fuzzy k-subsemiring (k-BFSS), -bipolar fuzzy k-ideals (k-BFIs), and -bipolar fuzzy k-bi-ideals (k-BFbIs) in semirings by applying belongingness and quasi-coincidence of the bipolar fuzzy (BF) point. After that, upper and lower parts of k-product of BF subsets of semirings are introduced. Lastly, the notions of k-regular and k-intraregular semirings in terms of -k–BFIs and -k–BFbIs are characterized.
1. Introduction and Motivation
In engineering, decision making theory, management science, and medical science, we may face uncertainty and vagueness in the data. In classical mathematics, all the formulae and methods consist on crisp set that cannot tackle the vague problems. After an extensive struggle, many theories are invented to tackle such problems. In 1965, Zadeh [1] gave the idea of fuzzy set theory to handle such complicated problems. This concept is applied on theory of rings, theory of groups, real analysis, logics, and topological space. In FS, membership degree is limited to , but there was a difficulty to deliberate the irrelevancy of data. To resolve this difficulty, Zhang [2] offered the idea of BFS in which the membership degree is . Since the BFS theory is a development of the FS theory, thus a BFS in semirings is also useful.
Bipolarity is a significant theory which is mostly used in real life. It is noticed that people may have different responses at a time for the same qualities of an item or a plan. Someone may have negative thoughts and the other one may have positive thoughts. For example, is a big amount for a needy person but at the same phase, this amount is negligible for a rich man. BFS is also useful in database query, psychology, image processing, multicriteria decision making, argumentation, and in human reasoning problems. Recently, Riaz et al. [3] have worked on BFS with the combination of picture fuzzy set and discussed its application on pattern recognition. Facial recognition or face detection is an artificial intelligence-based computer technology used to find and identify human faces in digital images; see [4–15] for more examples and results which are relevant to BFS.
The role of semirings as an algebraic tool is very important in theoretical computer science ([16, 17]). Semirings are broadly used in formal languages, automata theory, optimization theory, graph theory, coding theory, and theory of discrete event dynamical systems. In the structural theory, ideals play a central role and are beneficial for many other purposes. In general, ideals of semiring do not coincide with ideals of the ring. Many results in rings apparently are not equivalent to results in semirings using only ideals. Henriksen [18] introduced a most generalized form of ideals in semiring, which is k-ideal, with the characterization that if the semiring R is a ring, then a complex in R is a k-ideal equivalent to a ring ideal. The k-ideal is the most restricted form of an ideal. Every k-ideal is ideal but the converse does not hold. The set of whole numbers is a semiring. Let then I is an ideal of while it is not a k-ideal of because If we take a subset of , then I is a k-ideal of . For more details; see [19].
1.1. Related Works
Zadeh’s fuzzy set is a much innovative, crucial, and useful set due to its significance in multiple research dimensions. The fuzzy set addresses the ill-defined situations by which we are often encountered. From these ill-defined situations, we can evaluate results by using degree of membership of fuzzy set but the bipolar fuzzy set is much better set to manage uncertainty, vagueness, and impreciseness than a fuzzy set. The bipolar fuzzy set is much valuable due to its degree of membership [-1,1].
1.2. Applications of the Proposed Model
Bipolar fuzzy ideals are very suitable in the field of image processing, database query, human reasoning, psychology, argumentation, multi-criteria decision making, etc. There are extensive applications of semirings in several fields, such as formal languages, graph theory, automata theory, generalized fuzzy computation, discrete theory, and many branches of applied mathematics. The combination of bipolar fuzzy ideals with semirings is a very useful tool for providing languages to deal with complicated problems in information science.
1.3. Innovative Contribution
In fuzzy sets, the degree of membership was restricted to [0,1]. In our realistic life, someone may have a negative response and other one may have a positive response at a time for the same characteristic of an object. In this regard, bipolarity is a very useful concept, which is commonly used in our real world problems. In the automata theory, many applications of fuzzy semirings are given by Ahsan et al. [20]. Lee [21, 22] introduced BFIs of BCK\BCI-algebras. Jun and Kavikumar [8] describe the idea of a BF finite state machine. Majuder [23] studied characteristics of bipolar fuzzy -ideals in semigroups. Zhou and Li ([24, 25]) utilized BFS in semirings. Manemaran and Chellappa [26] utilized BFSs in the group theory. Bhakat and Das ([27, 28]) initiated the idea of -fuzzy subgroups and extension of this work is -fuzzy ideals of hemirings introduced by Dudek et al. [29]. Shabir et al. [30] gave the idea of characterizations of semigroups by -fuzzy ideals. In [31], Dawaz studied -fuzzy subnear-rings. Ibrar et al. [32] characterized regular ordered semigroups by using bipolar fuzzy generalized bi ideals. In [33], Shabir et al. worked on –BFSSs and –BFIs of semirings. Recently, Shabir et al. [34] studied BF hyperideals in regular semihypergroups. Here, we have extended the above work for -k–BFSSs and –k–BFIs of semirings. See [35–39] for more details.
1.4. Organization of the Paper
In Section 1, we have familiarized with the physical background of the bipolar fuzzy semiring and its characterization toward regular and intraregular semirings. Section 2 describes the literature review of k-BFSS, k-BFI, and some new important definitions which will be used as the ground work for the major works. In Section 3, the concepts of –k–BFSS and –k-BFI of semirings are discussed. In Section 4, we have worked on upper and lower parts of BFSs. In Section 5, the description of theorems of k-regular and k-intraregular semirings in terms of -–BFIs is given. In Section 6, the comparative study and the last section consist of the conclusions and future plans.
The list of acronyms used here is given in Table 1.
2. Preliminaries
Basic definitions, results, and some back ground material are given. Here, a subset means a nonempty subset.
The k-closure of a subset K of a semiring R is . A subsemiring K of is called k-subsemiring of if and only if (iff) . Thus, an ideal (left, right) K of R is a k-ideal (left, right) if A subsemiring K is called a k-bi-ideal of if and A semiring is known as -regular if for all, there are elements such as . A semiring is known as -intraregular if for all , there are elements such that [40].
Lemma 1 (see [40]). A semiring is k-regular iff , for every right k-ideal and left k-ideal of .
Lemma 2 (see [40]). A semiring is k-regular iff for every k-bi-ideal of .
Lemma 3 (see [40]). A semiring is k-intraregular iff for every left k-ideal and every right k-ideal of
Now, we will discuss some essential concepts related to BFS.
A BF subset of a semiring , where gives the satisfaction degree for somewhat opposite property of and gives the degree of satisfaction of the corresponding property of . While in a fuzzy set, the membership function maps to . The difference between a fuzzy set and bipolar fuzzy set is shown by the following example.
Let be a set of workers of a company. Define a fuzzy set on A with fuzzy property “honesty.” The workers having property “honesty” mapped to [0,1] as shown by a bar graph in Figure 1. While other workers have no membership degree in range [0,1] as they are not honest. In the fuzzy set, we can cover only a positive aspect of any situation. We cannot deal with negative aspects of the situation. To facilitate, we deal such problems with a bipolar fuzzy set. The property “dishonesty” is against “honesty”. The workers are mapped to [-1,0] with property “dishonesty.” In such a way, the bipolar fuzzy set gives information about all elements as shown in Figure 2.
For , if , then does not hold the property of and if , then holds the property of If and , then its opposite property and membership function intersect. A BF subset is defined by for any
Let and be two BF subsets of a semiring . Then, if and for all and if and . The intersection and union of and is defined by and , respectively. Throughout the paper, is semiring unless otherwise identified.
Definition 1. Let and be two BF subsets of . The k-product of is defined as follows:for .
Proposition 1. Assume are BF subsets of . At that time, and ; this implies that .
Proof. Let then
As ,
And ,
So,
And,
As ,
Moreover, ,
So,
Hence, .
Definition 2. If is a k-subset of Then, the bipolar characteristic function of is denoted by and is defined by ,
For all , if , then we have a BF subset defined as and for all
Definition 3. A BF subset of is called k-BFSS of if it satisfies the following:(i) and (ii) and (iii) and (iv)If then and for all
Definition 4. A BF subset of is said to be -BFI (right resp. left) of if it satisfies the following:(i) and (ii) and , (iii)If then and for all
Example 1. A BF subset of a semiring Ⱳ is defined as follows:then is a BFI of Ⱳ but not a -BFI because if we take then and
Example 2. Let a BF subset of a semiring Ⱳ defined as follows:Then, it is easy to check that is a -BFI of Ⱳ.
Definition 5. A BF subset of is said to be bipolar fuzzy -BFbI of if it satisfies the following:(i) and (ii) and (iii) and (iv) and (v)If then and for all
Lemma 4. If is -ideal (left, right) of then is -BFI (left, right) of
Proof. We prove it for right –ideal. For this, we have to verify the below three inequalities, for each element (i) and ;(ii), and ;(iii) and .For the proofs of (i) and (ii), see [32]. Here, we just prove part (iii). For this, we discuss the following cases.
Case 1. Suppose
Then, and
Since is a -ideal of , for , and
Case 2. If
Then, and , and
Case 3. If one of the or does not belong to , say
Then, and , and
So, is a -BFI.
Lemma 5. If is a -bi-ideal of , then is -BFbI of
Proof. Straightforward.
3. Bipolar Fuzzy k-Ideals
Here, ideas of -k–BFSS, -k–BFI, and -k–BFbI of semirings are introduced and their related properties are discussed.
Definition 6 (see [32,33]). Let then is called a BF point in if the BF subset of is given in the form of the following:for all
Definition 7 (see [32, 33]). For a BF subset of R and a BF point , we say that belongs to (quasi-coincident to resp.) the is written as , if it satisfies the following:(i) if and (ii) if and (iii) if or (iv) if and Here, we consider and Consider a BF subset such that and for all and . Then, , , and . It follows that , which implies that , thus a contradiction arises. So, , and hence,
Definition 8. A BF subset of is called an -k–BFSS of if for all and it satisfies the following:(i) and (ii) and (iii)Suppose such as , then and here, ,
Definition 9. A BF subset of is called an -k–BFI (left, resp. right) of , if for and for all if it satisfies the following:(i) and (ii) and (resp. (iii)Suppose such as , then and ; here,
Definition 10. A BF subset of is called an --BFbI of , if for and for every it satisfies the following:(i) and (ii) and (iii) and (iv)Suppose such as , then and ; here,
Theorem 1. If a BF subset of is --BFSS of then it fulfills the following conditions for each .(i) and (ii) and (iii)For and
Proof. For the proofs of part (i) and part (ii), see [33]; here, we just prove part (iii).
For four following cases arise:(i) and (ii) and (iii) and (iv) and We will prove with contrary supposition. Then, assume for , or .
Case 1. and , then or implies Now, again consider , then and then Therefore, , but for , . Similarly, , because or So, , , which is a contradiction.
Case 2. and , then or so for , implies , Now, again consider , then and Also, implies and Then, we get and but for , . Similarly, , because Thus, , , which contradicts our supposition.
Case 3. and , then or So, for , implies , . Moreover, implies and Also, indicates and . Then, we get , , but for , . Similarly, , because . Therefore, , which is a contradiction.
Case 4. and , then or Then for and , and then , .
Also, implies and Moreover, implies and . Now, combining all these, we get , .
However, for , . Similarly, , because or Thus, , which is a contradiction.
Therefore, and for each . The reverse of this theorem does not hold in general as shown by the following example.
Example 3. Let be a semiring under operations defined as below:(see Tables 2 and 3)
Let be a BF subset defined by , and .
Then, for , and hold for all . Now, take , then and but as . Also, implies . So, .
Theorem 2. If a BF subset of is (, )--BFbI of , then it fulfills the following conditions for each (i) and (ii) and ;(iii) and ;(iv)For , and
Proof. Proof follows from Theorem 1.
Theorem 3. If a BF subset of is an -kBFI (left, resp. right) of , then it fulfills the following statements for each (i) and (ii) and