Homomorphisms of Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3

Kousar, Sajida; Saleem, Tahzeeb; Kausar, Nasreen; Pamucar, Dragan; Addis, Gezahagne Mulat

doi:https://doi.org/10.1155/2022/6847138

Computational Intelligence and Neuroscience

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

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Fuzzy Swarm Intelligence and its Advances for Real-time Applications

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Research Article | Open Access

Volume 2022 | Article ID 6847138 | https://doi.org/10.1155/2022/6847138

Homomorphisms of Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3

Sajida Kousar,¹Tahzeeb Saleem,¹Nasreen Kausar,²Dragan Pamucar,³and Gezahagne Mulat Addis⁴

Academic Editor: Ahmed M. Anter

Received03 Feb 2022

Revised21 Mar 2022

Accepted04 Apr 2022

Published28 Apr 2022

Abstract

The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the -fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set to any lattice , and lattice homomorphism from to the interval . In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.

1. Introduction

In the sixteenth century, Gerolamo Cardano laid the notion of probability theory to analyze games of chance. In the early nineteenth century, Pierre Laplace complied with the classical interpretation of probability that was assumed to be the best tool to deal with uncertainties in the experimental data. But there are several situations where uncertainty occurs as a vagueness more than a statistical variation. In 1965, Zadeh [1] presented a new concept of the fuzzy subset to carter the situation where probability fails to answer. The fuzzy subset of a nonempty set as described by Zadeh is based on the formulation of a function from to the closed interval . The function is called a membership function, whereas the images of elements of under this function are called membership grades. For instance, let be a collection of finite groups, be the total number of subgroups in . If is the total number of normal subgroups computed by a student in , then defines a fuzzy membership grade to the normal subgroups in . But there is a chance that if the group order is large and the student is unable to compute all the normal subgroups, then will be greater than the one reported by the student. This leads us toward the concept of nonmembership grades first introduced by Atanassov [2], and the fuzzy set that incorporates membership and non-membership grades is termed an intuitionistic fuzzy set (IFS). Atanassov [3] presented basic models, properties, arithmetic operations, algebraic operators, and relations over the intuitionistic fuzzy set.

Over the years, several other generalizations of fuzzy sets have been introduced depending upon various parameters of uncertainty, vagueness, and imprecision by employing membership, nonmembership, hesitancy, and indeterminacy grades. In all these generalizations, the grades are real numbers ranging between 0 and 1. The interval inherits the natural partial order from the set of real numbers and constitutes a lattice. Partial ordering and fuzzy uncertainties are key features of real-life problems with infinite solutions or no solution at all. So it is quite obvious to think about the replacement of by any suitable lattice. Goguen [4] introduced the concept of -fuzzy subsets of where the interval is replaced by a partially ordered set . Atanassov [5] presented the concept of the lattice-valued intuitionistic fuzzy set (LIFS-1) by using a complete lattice , an involutive order reversing unary operation two functions . Due to the compulsion of the operator , the definition of LIFS-1 is not applicable to a larger collection of lattices. Gerstenkorn and Tepa cev [6] refined the concept introduced by Atanassov. They replaced lattice with complete lattice and unary operator with a linearization function ; and termed their finding lattice-valued intuitionistic fuzzy set of a second type (LIFS-2). Different properties, such as the decomposition theorem and synthesis, were established for these fuzzy sets. However, the choice of a linearization map makes LIFS-2 less capable of dealing with basic set operations. For instance, the union of two LIFS-2s need not be a LIFS-2. Thus, the map was replaced with lattice homomorphism , and the refinement is called a lattice-valued intuitionistic fuzzy set type-3 and abbreviated as LIFS-3.

The term group was first used by Évariste Galois in the 1830s for the set of roots of polynomial equations. However, the modern-day definition of the group was established in 1870. Since then, significant research has been carried out in this area, and now the group is one of the most important algebraic structures providing a basic structure for several mathematical branches including analysis, game theory, coding theory, and algebraic geometry. Groups have strong applications in different scientific fields, especially symmetric groups, which play a vital role in theocratical physics and quantum mechanics. In genetics, the four-codon basis constitutes a group isomorphic to the Klein four-group. Gene mutation can be identified by establishing group homomorphism on copies of the sixty-four codon system. The coset diagram depicting group action has a close link with the crystal structure in chemistry. After Zadeh’s invention, many researchers attempted to use and replace the ordinary set with the fuzzy set in various theocratical and experimental areas.

In 1970, Rosenfeld [7] attempted to combine fuzzy concepts in group theory and termed the findings as a fuzzy subgroup. Rosenfeld investigated fundamental group theocratic properties for the newly established algebra. In later years, algebraists examined the structural properties of fuzzy subgroups. Anthony [8, 9] modified the definition of Rosenfeld by strengthening the condition for images of elements and their inverses. In fuzzy groups, it is observed that level sets and proved that a fuzzy subset of a group is a fuzzy subgroup if and only if all the level sets are subgroups of [10, 11]. In 1982, Liu [12] suggested fuzzy invariant subgroups and fuzzy ideals. Ajmal and Prajapati [13] and Mukherjee et al. [14, 15] connected fuzzy normal subgroups and fuzzy cosets and group-theoretic analogs. Kumar et al. [16] resolved fuzzy normal subgroups and fuzzy quotients. Moreover, Tarnauceanu [17] presented the concept of fuzzy normal subgroups for the class of finite groups. Choudhary et al. [18] and Addis [19] investigated structure-preserving maps and fundamental isomorphism theorems. Malik et al. [20] and Mishref [21] introduced the fuzzy normal series to generalize the concept of nilpotancy and solubility of groups to fuzzy subgroups. Zhan and Zhisong [22] defined the intuitionistic fuzzy subgroup as a generalization of Rosenfeld’s fuzzy subgroup. By starting with a given classical group, they define a intuitionistic fuzzy subgroup using the classical binary operation defined over the given classical group. Li and Gui [23] extended Zhan and Zhisong’s work on intuitionistic fuzzy groups. Tarsuslu et al. [24] generalized the action of a group on a set to intuitionistic fuzzy action. Bal et al. [25] investigated the kernel subgroup of intuitionistic fuzzy subgroups.

The employment of lattice order turns the -fuzzy set into an important generalization of the fuzzy set widely applicable in decision language [4], system analysis [26], and coding theory [27]. Group theory is essential not only for mathematical advancements, but also for other scientific fields such as physics [28–30] and chemistry [31, 32]. Since 1970, several mathematicians have been extensively investigated group structure in fuzzy and generalized fuzzy environments. The importance of LIFS-3 and group on their own motivates to combine these two concepts. The main objective of this article is to introduce the notion of lattice-valued intuitionistic fuzzy subgroup type-3 and analyze its algebraic properties.

2. Preliminaries

This section is introductory in nature and contains all the essential definitions and fundamental properties that are necessary to understand the newly established structure of LIFSG-3.

2.1. L-Fuzzy Subset

To understand a -fuzzy subset [4], first we will define the order relation and lattice. For a nonempty set , a reflexive, antisymmetric, and transitive relation is termed as a partial order on , commonly denoted by the notation “less than or equal,” that is, . The set is termed as a partially ordered set. If is a partially ordered set, then the partial order gives an instinct to the concept of the greatest and smallest element in the set. If are related in such a way that , then we can pronounce it as is greater than or another way round is smaller than .

Now if is any subset of , then a member of is called an upper (lower) bound of if is greater (smaller) than all the elements in . Perhaps has more than one upper (lower) bounds, the smallest (greatest) of these upper (lower) bounds is termed to be supremum or meet (infimum or join) of , denoted by . A set together with partial order in which join and meet exist for every pair of elements is called a lattice, denoted by . A lattice is said to be complete if and exits for every nonempty subset of . A lattice is titled to be distributive if and are distributive over each other.

Definition 1. Let be a nonempty set and be a complete distributive lattice which has least (bottom) and greatest (top) elements say and , respectively. Then, an -fuzzy subset of is narrated as follows:

Definition 2. If is a group and the -fuzzy subset satisfy the following conditions for all ,Then, it is termed as a - fuzzy subgroup [33] of .

2.2. L-Intutionistic Fuzzy Subset

Gerstenkorn and Tepa cev [6] refined the concept introduced by Atanassov. They replaced lattice by complete lattice; unary operator by lattice homomorphism and refinement is abbreviated as (LIFS-3).

Definition 3. A LIFS-3 is the set , with a nonempty set, is a complete lattice with top and bottom elements and , respectively, and are membership and nonmemberships functions. The map is a lattice homomorphism with , ,Such that for every , .

Example 1. Consider and the lattice w.r.t partial orderDefine , and Then, for every , imply that is a LIFS-3 of .

3. Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3

We will define lattice-valued intuitionistic fuzzy subgroups (LIFSG) by using a group , membership , nonmembership , and a lattice homomorphism . The combination of and group will provide us a new refined algebraic structure which we can use effectively in real world problems.

Definition 4. For a group , lattice with top and bottom elements and , respectively, and lattice homomorphism . A LIFS-3 of , that is, formulates a LIFSG-3 of provided that for every ,(1)(2)

Proposition 1. Let be a lattice (with top and bottom elements and , respectively) and be a group. Let be a LIFSG-3 of . For and , the -cut set (or level set) is a subgroup of , called -cut subgroup (or level subgroup).

Proof. Recall the definition of cut sets in IFS where these sets are defined as follows:Let be the identity element in . Then, for any By the property of LIFSG, we have is an upper bound for .
Similarly, , this implies that is an lower bound for . We get that . Let . Then,Similarly, , this implies that . Hence, is a subgroup of G.

Proposition 2. For a group and lattice , let be a LIFS-3 such that for each , the -cut set . Then, is a LIFSG-3 of .

Proof. Suppose for all , . Let . Then, there are two possibilities(i)Case 1: Let . Then, and Suppose and and . Then and . We get that Similarly, suppose and and . Then, and . We get that If , then by definition . Now using the same argument as above it is easy to show that(ii)Case 2: if , then three cases arise:(1), then , this implies that (2). As is a lattice so exist. Suppose then Now, (3). As is a lattice so exist. Suppose then ,Now On the basis of previous discussion we conclude that the elements of sustain the axioms of LIFSG-3. Hence, is a LIFSG-3 of .

Remark 1. For every group and lattice with and as the top and bottom element, the LIFS-3 is a LIFSG-3 of if and only if

Proposition 3. For a group and lattice with and as the top and bottom element, let . For , define and from to asLet be a lattice homomorphism. Then, and . Suppose , then is a LIFSG-3 of .

Proof. Let be a lattice homomorphism defined as and and . For we have two cases:(1)Case Let , (2)Case Let , Now, for , we have three cases:(1), this implies that (2), this implies that Thus, Similarly, .(3), then two cases arises:(a)(b)From previous discussion we get that is a lattice-valued intuitionistic fuzzy subgroup type-3 of .
Now, if Then, cut subgroup

Remark 2. Every subgroup of is a cut subgroup of some suitable LIFSG-3 of .

4. Lattice-Valued Intuitionistic Fuzzy Normal Subgroups Type-3

As we defined the normality of the intuitionistic fuzzy subgroup in the previous chapter, similarly, in this chapter, we construct the lattice-valued intuitionistic fuzzy normal subgroup type-3 (LIFNSG-3). Then, using this definition, we proved some useful results in this section.

Definition 5. For a group , lattice with top element and bottom element and lattice homomorphism . Let be a LIFSG-3. Then, is called LIFNSG-3 if

Proposition 4. For a group , lattice with top element and bottom element and lattice homomorphism . If is a LIFNSG-3 of , then for and

Proof. Suppose is a LIFNSG-3 of . Let and . Then, this implies that
,

Proposition 5. For a group , the set is a LIFNSG-3 of if and only if

Proof. Suppose is a LIFNSG-3 of . Then, , implies that . Similarly we get that . Conversely, assume that . Then, and .

Proposition 6. Let be a LIFSG-3 of . Then is LIFNSG-3 if and only if for all ,

Proof. Suppose is LIFNSG-3 of . Then, implies that . We get that . Similarly, . Conversely, assume that . Then . We get that .As implies . If , then . is constant. In this case,If If , then this implies that . Then, is constant. In this case . If From (35) and (38), we get . Similarly, from (36) and (39), we get . Thus, is LIFNSG-3 of . Hence proved.

Theorem 1. Let be a finite group and be a LIFSG-3 of , such that all -cut subgroups of are normal in . Then, is a LIFNSG-3.

Proof. As is finite this implies that and are finite sets. Suppose with and with . Consider be the complete set of -cut subgroup of . As each , soIt is normal in , it can be expressed as union of and . Due to normality of cut subgroups, and for all and . We get similar result from each cut-subgroup. Hence, and . This implies that is a LIFNSG-3 of .

5. Coset and Homomorphism in Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3

In the first section, we introduced the fundamental concepts of the factor group and group homomorphism. In this section, we will discuss these two important features of classical group theory for LIFSG-3.

Definition 6. For a group , lattice with top and bottom element and . Consider the lattice homomorphism . Let be a LIFSG-3 of . For define two mapsThen,This implies that is a LIFS-3 of . The LIFS-3where is called the lattice-valued intuitionistic fuzzy coset type-3 (LIFC-3) of induced by and .

Proposition 7. If is a LIFNSG-3 of , then for any ,

Proof. Suppose is a LIFNSG-3 of . Then, for Similarly, we proved for . Hence proved.

Theorem 2. Let be a LIFNSG-3 of . Let be the collection of all LIFC-3 of induced by and . Then, is a group under the binary operation and is a LIFSG-3 of , where

Proof. Let such that
Consider implies that . As and If , then , because is a lattice-valued intuitionistic fuzzy normal subgroup type-3. As , implies that . Thus, we get thatSimilarly, we getNow we have, Thus, the binary operation is well defined. The associativity of composition of functions implies that the given binary operation is associative. Considerwhere , this implies that and for any .Similarly, this implies that is the identity element in . Let , where . As is a group so , implies that and is a group.
Consider , where is a LIFSG-3 of . Hence proved.

Definition 7. Let be a LIFNSG-3 of . Then, is called a lattice intuitionistic fuzzy quotient subgroup type-3 of denoted by .

Proposition 8. Let be a group and be a LIFNSG-3 of . The map defined by is epimorphism with

Proof. Let , then this implies that is a group homomorphism. Clearly, is onto,From isomorphism theorem we get that . Hence proved.

6. Group Homomorphism and Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3

Proposition 9. Let be a group and be a lattice-valued intuitionistic fuzzy subgroup type-3 of . Suppose be a group isomorphism. Then, , whereis a LIFSG-3 of .

Proof. Let . Then,Similarly, . Let . Then,Similarly, . Now, is a LIFSG-3 of . Hence proved.
Similar to the above proposition, now we will relate LIFNSG-3 of and .

Proposition 10. Let be a group and be a LIFNSG-3 of . If be a group isomorphism, then is a LIFNSG-3 of .

Proof. Suppose is a LIFNSG-3 of . Let . Then, for Similarly, is a LIFNSG-3 of . Hence proved.
In the following proposition we will induce LIFSG-3 of from the LIFS-3 of .

Proposition 11. Let be two groups and be a group homomorphism. Suppose be a LIFSG-3 of . Then, is a LIFSG-3 of . Where

Proof. Let . Then,Similarly, . Now,Similarly, . is a LIFSG-3 of . Hence proved.

Proposition 12. Let and be two groups, be a group homomorphism andbe a LIFNSG-3 of . Then,is a LIFNSG-3 of .

Proof. Let . Then,Similarly, . is a LIFNSG-3 of . Hence proved.

Theorem 3. Let and be any two groups and be a group isomorphism. Let be a lattice with top element and bottom element and be a lattice homomorphism. Let and be the collections of LIFSG-3 of and , respectively. Then, the mapDefined byis bijective.

Proof. From previous propositions we know that ifThen,Now, ifAs is bijectiveSimilarly,As is bijective, this implies . Thus, we get that, ifthen is injective. Now, ifthen and ,Similarly, . This implies that is bijective. Hence proved.

Theorem 4. Let be a group, and be any two lattices with top elements and , and bottom elements and , and be lattice homomorphisms such that , , and . If be a LIFSG-3 of , then is also a LIFSG-3 of . Whereare defined asand

Proof. Let . Then,Similarly, . Let andThen, and are of the formThus,Hence, we get the required result.

Theorem 5. Let and be any two group and be a group isomorphism. Let and be any two lattice with top elements and , and bottom elements and . Let and be lattice homomorphisms such that , , . Let be a LIFSG-3 of . Then,is a LIFSG-3 of , where

Proof. If is a LIFSG-3 of , then from Theorem 4is a LIFSG-3 of . From Proposition 9is a LIFSG-3 of .

Theorem 6. For a group , lattice with top and bottom elements and , respectively, lattice homomorphism . Let be a LIFNSG-3 of . Then, the set

is normal in and is a LIFNSG-3 of .

Conversely, if and is a LIFNSG-3 of , then is a LIFNSG-3 of , where

are defined as

Proof. Let be a LIFNSG-3 of . Then, and , implies that is a cut subgroup so it is normal in and allow to construct factor group Now, for the map Similarly, for the map Also is a LIFNSG-3 of . Conversely, supposebe a LIFNSG-3 of . Define asThen,Define asThen, is a LIFNSG-3 of .

7. Conclusion

The article is about the study of group theocratic concepts in a lattice-valued intuitionistic fuzzy type-3 environment. The structure is established by introducing lattice homomorphism, membership and nonmembership grades obeying certain laws for the binary operation defined on the group and inverses of elements under that operation. It is concluded that the level sets of LIFSG-3 of a group are exactly the subgroups of , and conversely, any LIFS-3 of whose level sets are subgroups of is LIFSG-3. Lattice-valued intuitionistic fuzzy normal subgroups type-3 and lattice-valued intuitionistic fuzzy factor subgroups type-3 of are governed by normal subgroups and factor groups of . Structure preserving LIFSG-3 maps are also discussed, and it is observed that they can be derived by extending group homomorphism. In the future, the notion foundations laid in this article can be used to find the Abelian subgroups of finite -groups [34], verify Lagrange’s theorem [35, 36], compute annihilator [37], aggregation [38], and fundamental isomorphism theorems [39] for lattice-valued intuitionistic fuzzy subgroups type-3. There are several generalizations of fuzzy sets [40–42] where lattice-valued algebraic structures can be defined by replacing with a suitable lattice . The research findings can be utilized for application in algebra [43, 44] and real-life problems [45] to handle uncertainty and ambiguity more accurately.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally to the preparation of this manuscript.

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Copyright © 2022 Sajida Kousar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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