# Homomorphisms of Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3

Authors: Sajida Kousar; Tahzeeb Saleem; Nasreen Kausar; Dragan Pamucar; Gezahagne Mulat Addis
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and theL-fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L, and lattice homomorphism from L to the interval 0,1. 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.


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## Body

## 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 U as described by Zadeh is based on the formulation of a function μ from U to the closed interval 0,1. The function is called a membership function, whereas the images of elements of U under this function are called membership grades. For instance, let U be a collection of finite groups, px be the total number of subgroups in x∈U. If qx is the total number of normal subgroups computed by a student in x, then μx=qx/px defines a fuzzy membership grade to the normal subgroups in x. But there is a chance that if the group order is large and the student is unable to compute all the normal subgroups, then μx 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 interval0,1 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 0,1 by any suitable lattice. Goguen [4] introduced the concept of L-fuzzy subsets of S where the interval 0,1 is replaced by a partially ordered set L. Atanassov [5] presented the concept of the lattice-valued intuitionistic fuzzy set (LIFS-1) by using a complete lattice L, an involutive order reversing unary operation N:L⟶L two functions μ,ν:S⟶L. Due to the compulsion of the operator N, the definition of LIFS-1 is not applicable to a larger collection of lattices. Gerstenkorn and Tepa vˇ cev i´ [6] refined the concept introduced by Atanassov. They replaced lattice with complete lattice and unary operator N with a linearization function ℓ:L⟶0,1; 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 α:L⟶0,1, 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 G is a fuzzy subgroup if and only if all the level sets are subgroups of G [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 theL-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 aL-fuzzy subset [4], first we will define the order relation and lattice. For a nonempty set P, a reflexive, antisymmetric, and transitive relation is termed as a partial order on P, commonly denoted by the notation “less L than or equal,” that is, ≤. The set P is termed as a partially ordered set. If P 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 a,b∈P are related in such a way that a≤b, then we can pronounce it as b is greater than a or another way round a is smaller than b.Now if∅≠Q is any subset of P, then a member ϱ of P is called an upper (lower) bound of Qif ϱ is greater (smaller) than all the elements in Q. Perhaps Q 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 Q, denoted by ∨∧. A set L together with partial order in which join ∧ and meet ∨ exist for every pair of elements is called a lattice, denoted by L,≤,∨,∧. A lattice L is said to be complete if ∨ and ∧ exits for every nonempty subset of L. A lattice L is titled to be distributive if ∨ and ∧ are distributive over each other.Definition 1.
LetS be a nonempty set and L=L,≤,∨,∧ be a complete distributive lattice which has least (bottom) and greatest (top) elements say B and T, respectively. Then, an L-fuzzy subset ζS´˜ of S is narrated as follows:(1)ζS´˜:S⟶L.Definition 2.
IfS is a group and the L-fuzzy subset satisfy the following conditions for all s1,s2∈S,(2)ζS´˜s1s2≥∧ζS´˜s1,ζS´˜s2,ζS´´˜s−1=ζS´´˜s∀s∈G.Then, it is termed as a L- fuzzy subgroup [33] of S.


### 2.2. L-Intutionistic Fuzzy Subset

Gerstenkorn and Tepavˇ cev i´ [6] refined the concept introduced by Atanassov. They replaced lattice by complete lattice; unary operator N by lattice homomorphism ℘:L⟶0,1 and refinement is abbreviated as (LIFS-3).Definition 3.
A LIFS-3 is the setS,L,ζℑL,ξℑL,℘, with S a nonempty set, L is a complete lattice with top and bottom elements T and B, respectively, ζℑL:X⟶L and ξℑL:S⟶L are membership and nonmemberships functions. The map ℘:L⟶0,1 is a lattice homomorphism with ℘T=1, ℘B=0that is,for allℓ1,ℓ2∈L,(3)℘ℓ1∧ℓ2=℘ℓ1∧℘ℓ2℘ℓ1∨ℓ2=℘ℓ1∨℘ℓ2.
Such that for everys∈S, ℘ζℑLs+℘ξℑLs≤1.Example 1.
ConsiderS=a,b,c,d,e,f and the lattice L=B,r,s,t,τ w.r.t partial order(4)B≤B,B≤r,B≤s,B≤t,B≤ττ≤τ,B≤τ,r≤τ,s≤τ,t≤τ,r≤s.
Define℘:L⟶0,1, ζℑL:S⟶L and ξℑL:S⟶L(5)℘τ=1,℘B=0,℘r=0.2,℘s=0.5,℘t=0.3,ζℑLa=τ,ζℑLb=ζℑLc=s,ζℑLd=ζℑLe=ζℑLf=r,ζℑLa=τ,ζℑLb=ζℑLc=s,ζℑLd=ζℑLe=ζℑLf=r.
Then, for everys∈S, ℘ζℑLs+℘ξℑLs≤1 imply that S,L,ζℑL,ξℑL,℘ is a LIFS-3 of S.


## 2.1. L-Fuzzy Subset

To understand aL-fuzzy subset [4], first we will define the order relation and lattice. For a nonempty set P, a reflexive, antisymmetric, and transitive relation is termed as a partial order on P, commonly denoted by the notation “less L than or equal,” that is, ≤. The set P is termed as a partially ordered set. If P 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 a,b∈P are related in such a way that a≤b, then we can pronounce it as b is greater than a or another way round a is smaller than b.Now if∅≠Q is any subset of P, then a member ϱ of P is called an upper (lower) bound of Qif ϱ is greater (smaller) than all the elements in Q. Perhaps Q 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 Q, denoted by ∨∧. A set L together with partial order in which join ∧ and meet ∨ exist for every pair of elements is called a lattice, denoted by L,≤,∨,∧. A lattice L is said to be complete if ∨ and ∧ exits for every nonempty subset of L. A lattice L is titled to be distributive if ∨ and ∧ are distributive over each other.Definition 1.
LetS be a nonempty set and L=L,≤,∨,∧ be a complete distributive lattice which has least (bottom) and greatest (top) elements say B and T, respectively. Then, an L-fuzzy subset ζS´˜ of S is narrated as follows:(1)ζS´˜:S⟶L.Definition 2.
IfS is a group and the L-fuzzy subset satisfy the following conditions for all s1,s2∈S,(2)ζS´˜s1s2≥∧ζS´˜s1,ζS´˜s2,ζS´´˜s−1=ζS´´˜s∀s∈G.Then, it is termed as a L- fuzzy subgroup [33] of S.


## 2.2. L-Intutionistic Fuzzy Subset

Gerstenkorn and Tepavˇ cev i´ [6] refined the concept introduced by Atanassov. They replaced lattice by complete lattice; unary operator N by lattice homomorphism ℘:L⟶0,1 and refinement is abbreviated as (LIFS-3).Definition 3.
A LIFS-3 is the setS,L,ζℑL,ξℑL,℘, with S a nonempty set, L is a complete lattice with top and bottom elements T and B, respectively, ζℑL:X⟶L and ξℑL:S⟶L are membership and nonmemberships functions. The map ℘:L⟶0,1 is a lattice homomorphism with ℘T=1, ℘B=0that is,for allℓ1,ℓ2∈L,(3)℘ℓ1∧ℓ2=℘ℓ1∧℘ℓ2℘ℓ1∨ℓ2=℘ℓ1∨℘ℓ2.
Such that for everys∈S, ℘ζℑLs+℘ξℑLs≤1.Example 1.
ConsiderS=a,b,c,d,e,f and the lattice L=B,r,s,t,τ w.r.t partial order(4)B≤B,B≤r,B≤s,B≤t,B≤ττ≤τ,B≤τ,r≤τ,s≤τ,t≤τ,r≤s.
Define℘:L⟶0,1, ζℑL:S⟶L and ξℑL:S⟶L(5)℘τ=1,℘B=0,℘r=0.2,℘s=0.5,℘t=0.3,ζℑLa=τ,ζℑLb=ζℑLc=s,ζℑLd=ζℑLe=ζℑLf=r,ζℑLa=τ,ζℑLb=ζℑLc=s,ζℑLd=ζℑLe=ζℑLf=r.
Then, for everys∈S, ℘ζℑLs+℘ξℑLs≤1 imply that S,L,ζℑL,ξℑL,℘ is a LIFS-3 of S.


## 3. Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3

We will define lattice-valued intuitionistic fuzzy subgroups (LIFSG) by using a groupG, membership ζIL:G⟶L, nonmembership ξIL:G⟶L, and a lattice homomorphism ℘:L⟶L. The combination of LIFS−3 and group will provide us a new refined algebraic structure which we can use effectively in real world problems.Definition 4.
For a groupG, lattice L with top and bottom elements T and B, respectively, and lattice homomorphism ℘:L⟶0,1. A LIFS-3 of G, that is, IL=G,L,ζIL,ξIL,℘ formulates a LIFSG-3 of G provided that for every x,y∈G,(1)
ζILxy≥ζILx∧ζILyandξILxy≤ξILx∨ξILy(2)
ζILx=ζILx−1andξILx=ξILx−1Proposition 1.
LetL be a lattice (with top and bottom elements T and B, respectively) and G be a group. Let IL=G,L,ζIL,ξIL,℘ be a LIFSG-3 of G. For t1∈ImζIL and t2∈ImξIL, the t1,t2-cut set (or level set) is a subgroup of G, called t1,t2-cut subgroup (or level subgroup).Proof.
Recall the definition of cut sets in IFS where these sets are defined as follows:(6)ILt1,t2=x∈G:ζILx≥t1andξILx≤t2.
Lete be the identity element in G. Then, for any u∈Ge∈uu−1(7)⇒ζILe=ζILuu−1≥ζILu∧ζILu−1.
By the property of LIFSG, we have(8)ζILe=ζILuu−1≥ζILu∧ζILu,⇒ζILe≥ζILu∀u∈G.⇒ζILe is an upper bound for ImζIL.
Similarly,ξILe≤ξILu∀u∈G, this implies that ξILe is an lower bound for ImξIL. We get that e∈ILt1,t2. Let u,v∈ILt1,t2. Then,(9)ζILu,ζILv≥t1,ξILu,ξILv≤t2,ζILuv−1≥ζILu∧ξILv−1,⇒ζILuv−1≥t1.
Similarly,ξILuv−1≤t2, this implies that uv−1∈ILt1,t2. Hence, ILt1,t2 is a subgroup of G.Proposition 2.
For a groupG and lattice L, let IL=G,L,ζIL,ξIL,℘ be a LIFS-3 such that for each t1,t2∈L, the t1,t2-cut set ILt1,t2≤G. Then, I is a LIFSG-3 of G.Proof.
Suppose for allt1∈ImζIL,t2∈ImξIL, ILt1,t2≤G. Let u,v∈G. Then, there are two possibilities(i)
Case 1: Letu,v∈ILt1,t2. Then, uv∈ILt1,t2 and(10)ζILu≥t1,ζILv≥t1,ξILu≤t2,ξILv≤t2⇒ζILuv≥t1,ξILuv≤t2.SupposeζILu=ℓ1 and ζILv=ℓ2 and ℓ3=ℓ1∧ℓ2. Then ℓ3≤ℓ1andℓ2 and uv∈ILℓ3,t2. We get that(11)ζILuv≥ℓ3=ζILu∧ζILv.Similarly, supposeξILu=ℓ42 and ξILv=ℓ5 and ℓ6=ℓ4∨ℓ5. Then, ℓ6≥ℓ4andℓ5 and uv∈ILt1,ℓ6. We get that(12)ξILuv≤ℓ6=ξILu∨ξILv.Ifu∈ILt1,t2≤G, then by definition u−1∈ILt1,t2. Now using the same argument as above it is easy to show that(13)ζILu−1=ζILu,ξILu−1=ξILu.(ii)
Case 2: ifu∈ILt1,t2,v∈ILt3,t4, then three cases arise:(1)
t1≤t3,t4≤t2, then ILt3,t4≤ILt1,t2, this implies that u,v∈ILt1,t2(2)
t1≤t3,t2≤t4. As L is a lattice so t2∨t4 exist. Suppose t5=t2∨t4 then t2,t4≤t5(14)ILt1,t5=g∈G:ζILg≥t1,ξILg≤t5.Now,ζILv≥t3≥t1,ξILu≤t2≤t5,ξILv≤t4≤t5(15)⇒u,v∈ILt1,t5.(3)
t1≥t3,t4≤t2. As L is a lattice so t1∧t3 exist. Suppose t6=t1∧t3 then t1,t3≥t6,(16)ILt6,t4=g∈G:ζIg≥t6,ξIg≤t4.
NowζILv≥t3≥t6,ζILu≥t1≥t6,ξILv≤t4≤t2(17)⇒u,v∈ILt6,t2.
On the basis of previous discussion we conclude that the elements ofG sustain the axioms of LIFSG-3. Hence, IL is a LIFSG-3 of G.Remark 1.
For every groupG and lattice L with T and B as the top and bottom element, the LIFS-3 IL=G,L,ζIL,ξIL,℘ is a LIFSG-3 of G if and only if(18)ILt1,t2≤G∀t1,t2∈L.Proposition 3.
For a groupG and lattice L with T and B as the top and bottom element, let H≤G. For t∈L, define ζHt and ξHt from G to L as(19)ζHtg=t,ifg∈H,B,ifg∈GH,ξHtg=t,ifg∈H,T,ifg∈GH.
Let℘:L⟶0,1 be a lattice homomorphism. Then, ℘T=1 and ℘B=0. Suppose ℘t=0.2, then I˙L=G,L,ζHt,ξHt,℘ is a LIFSG-3 of G.Proof.
Let℘:L⟶0,1 be a lattice homomorphism defined as ℘T=1 and ℘B=0 and ℘t=0.2. For g∈G we have two cases:(1)
Case1: Let g∈H, ζHtg=t,ξHtg=t(20)⇒℘ζHtg+℘ξHtg=℘t+℘t=0.4<1.(2)
Case2: Let g∈G/H, ζHtg=B,ξHtg=T(21)⇒℘ζHtg+℘ξHtg=℘B+℘T=1.
Now, forg1,g2∈G, we have three cases:(1)
g1,g2∈H, this implies that g1g2∈H(22)ζHtg1=t,ζHtg2=t,ζHtg1g2=t,ξHtg1=t,ξHtg2=t,ξHtg1g2=t.(2)
g1∈H,g2∈G/H, this implies that g1g2∉H(23)⇒g1g2∈G/H.Thus,ζHtg1=t,ζHtg2=B,ζHtg1g2=B(24)⇒ζHtg1∧ζHtg2=t∧B=B,⇒ζHtg1g2=B=ζHtg1∧ζHtg2.Similarly,ξHtg1g2=T=ξHtg1∨ξHtg2.(3)
g1,g2∈G/H, then two cases arises:(a)
g1g2∈G/H(25)ζHtg1=B,ζHtg2,=B,ζHtg1g2,=BξHtg1,=T,ξHtg2=T,ξHtg1g2=T.(b)
g1g2∈H(26)ζHtg1=B,ζHtg2=B,ζHtg1g2=t≥BξHtg1=T,ξHtg2=T,ξHtg1g2=t≤T.
From previous discussion we get that(27)ζHtg1g2≥ζHtg1∧ζHtg2,ξHtg1g2≤ξHtg1∨ξHtg2,ζHtg=ζHtg−1,ξHtg=ξHtg−1.
⇒I˙L is a lattice-valued intuitionistic fuzzy subgroup type-3 of G.
Now, ifs1∈ImζHt,s2∈ImξHt(28)⇒s1∈B,t,s2∈t,T.
Then,t,t− cut subgroup(29)I˙Lt,t=g∈G:ζHtg≥t,ξHtg≤t,=H.Remark 2.
Every subgroupH of G is a cut subgroup of some suitable LIFSG-3 of G.


## 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 groupG, lattice L with top element T and bottom element B and lattice homomorphism ℘:L⟶0,1. Let IL=G,L,ζIL,ξIL,℘ be a LIFSG-3. Then, IL is called LIFNSG-3 G if(30)ζILxyx−1=ζILy,ξILxyx−1=ξILy∀x,y∈G.Proposition 4.
For a groupG, lattice L with top element T and bottom element B and lattice homomorphism ℘:L⟶0,1. If IL=G,L,ζIL,ξIL,℘ is a LIFNSG-3 of G, then for t1∈ImζIL and t2∈ImξIL(31)ILt1,t2⊲¯G.Proof.
SupposeIL is a LIFNSG-3 of G. Let y∈ILt1,t2 and x∈G. Then, ζILy≥t1andξILy≤t2 this implies that
xyx−1∈ILt1,t2,(32)⇒ILt1,t2⊲¯G.Proposition 5.
For a groupG, the set IL=G,L,ζIL,ξIL,℘ is a LIFNSG-3 of G if and only if(33)ζILxy=ζILyx,ξILxy=ξILyx∀x,y∈G.Proof.
SupposeIL is a LIFNSG-3 of G. Then, ζILxyx−1=ζILy∀x,y∈G, implies that ζILxy=ζILyx. Similarly we get that ξILxy=ξILyx. Conversely, assume that ζILxy=ζILyxandξILxy=ξILyx∀x,y∈G. Then, ζILxyx−1=ζILx−1xy=ζILy and ξILxyx−1=ξILx−1xy=ξILy.Proposition 6.
LetIL=G,L,ζIL,ξIL,℘ be a LIFSG-3 of G. Then IL is LIFNSG-3 if and only if for all x,y∈G,(34)ζILx,y≥ζILy,ξILx,y≤ξILy.Proof.
SupposeIL is LIFNSG-3 of G. Then, ζILx,y=ζILxyx−1y−1 implies that ζILx,y≥ζILxyx−1∧ζILy−1. We get that ζILx,y≥ζILy. Similarly, ξILx,y≤ξILy. Conversely, assume that ζILx,y≥ζILyandξILx,y≤ξILy. Then ζILxyx−1=ζILxyx−1y−1y. We get that ζILxyx−1≥ζILx,y∧ζILy.(35)⇒ζILxyx−1≥ζILy.(36)ξILxyx−1≤ξILy.
AsζILy=ζILx−1xyx−1x implies ζILy≥ζILx∧ζILxyx−1∧ζILx. If ζILx∧ζILxyx−1=ζILx, then ζILy≥ζILx∀x,y∈G. ⇒ζIL is constant. In this case,(37)ζILy=ζILxyx−1.
IfζILx∧ζILxyx−1=ζILxyx−1(38)⇒ζILy≥ζILxyx−1.
IfξILx∨ξILxyx−1=ξILx, then this implies that ξILy≤ξILx∀x,y∈G. Then, ξIL is constant. In this case ξILy=ξILxyx−1. If ξILx∨ξILxyx−1=ξILxyx−1(39)⇒ξILy≤ξILxyx−1.
From (35) and (38), we get ζILxyx−1=ζILy. Similarly, from (36) and (39), we get ξILxyx−1=ξILy. Thus, IL is LIFNSG-3 of G. Hence proved.Theorem 1.
LetG be a finite group and IL=G,L,ζIL,ξIL,℘ be a LIFSG-3 of G, such that all t,s-cut subgroups of IL are normal in G. Then, IL is a LIFNSG-3.Proof.
AsG is finite this implies that ImζIL and ImξIL are finite sets. Suppose ImζIL=t1,t2,…,tk with t1<t2,…,<tk and ImξIL=s1,s2,…,sm with s1>s2,…,>sm−1>sm. Consider X=ILti,sj:1≤i≤k,1≤j≤m be the complete set of ti,sj-cut subgroup of IL. As each ILti,sj⊲¯G, so(40)ILti,sjILti−1,sj+1=x∈G:ζILx=ti,ξILx=sj.
It is normal inG, it can be expressed as union of C=xyx−1:x∈G and y∈ILti,sj/ILti−1,sj+1. Due to normality of cut subgroups, ζILxyx−1=ζILy and ξILxyx−1=ξILy for all x∈G and y∈ILti,sj/ILti−1,sj+1. We get similar result from each cut-subgroup. Hence, ζILxyx−1=ζILy and ξILxyx−1=ξILy∀x,y∈G. This implies that IL is a LIFNSG-3 of G.


## 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 groupG, lattice L with top and bottom element T and B. Consider the lattice homomorphism ℘:L⟶0,1. Let IL=G,L,ζIL,ξIL,℘ be a LIFSG-3 of G. For x∈G define two maps(41)ζILx:G⟶LasζILxg=ζILgx−1∀g∈G,ξILx:G⟶LasξILxg=ξILgx−1∀g∈G.
Then,(42)℘ζILxg+℘ξILxg=℘ζILgx−1+℘ξILgx−1≤1.
This implies thatILx=G,L,ζILx,ξILx,℘ is a LIFS-3 of G. The LIFS-3(43)ILx=G,L,ζILx,ξILx,℘,
whereG is called the lattice-valued intuitionistic fuzzy coset type-3 (LIFC-3) of G induced by x and IL.Proposition 7.
IfIL is a LIFNSG-3 of G, then for any x∈G,(44)ζILxxg=ζILxgx=ζILg,ξILxxg=ξILxgx=ξILg∀g∈G.Proof.
SupposeIL is a LIFNSG-3 of G. Then, for xinG(45)ζILxxg=ζILxgx−1=ζILg,ζILxgx=ζILgxx−1=ζILg.
Similarly, we proved forξIL. Hence proved.Theorem 2.
LetIL=G,L,ζIL,ξIL,℘ be a LIFNSG-3 of G. Let G/IL=ILx:x∈G be the collection of all LIFC-3 of G induced by x∈G and IL. Then, G/IL is a group under the binary operation ILxoILy=G,L,ζILxy,ξILxy,℘ and IL¯=G/IL,L,ζIL¯,ξIL¯,℘ is a LIFSG-3 of G/IL, where(46)ζIL¯ILx=ζILx,ξIL¯ILx=ξILx∀ILx∈G/IL.Proof.
LetILx1,ILx2,ILy1,ILy2∈G/IL such that ILx1=ILy1andILx2=ILy2
ConsiderζILx1x2g=ζILgx1x2−1 implies that ζILx1x2g≥ζILgy2−1y1−1∧ζILy1y2x2−1x1−1. As ζILx1=ζILy1 and ζILx2=ζILy2(47)⇒ζILgx1−1=ζILgy1−1,⇒ζILgx2−1=ζILgy2−1∀g∈G.
Ifg=y1y2x2−1, then ζILy1y2x2−1x1−1=ζILy2x2−1, because IL is a lattice-valued intuitionistic fuzzy normal subgroup type-3. As ζILgx2−1=ζILgy2−1, implies that ζILy1y2x2−1x1−1=ζILe. Thus, we get that(48)ζILx1x2g≥ζILy1y2g.
Similarly, we get(49)ξILx1x2g=ξILy1y2g.
Now we have,℘ζILx1x2g+℘ξILx1x2g=℘ζILgx1x2−1+℘ξILgx1x2−1(50)℘ζILx1x2g+℘ξILx1x2g≤1.
Thus, the binary operation is well defined. The associativity of composition of functions implies that the given binary operation is associative. Consider(51)ILe=G,L,ζILe,ξILe,℘,where ζILeg=ζILge−1=ζILgandξILeg=ξILge−1=ξILg∀g∈G, this implies that ILe=IL and for any ILx∈G/IL.(52)ILeoILx=G,L,ζILx,ξILx,℘=ILx.
Similarly,ILxoILe=ILx this implies that ILe is the identity element in G/IL. Let ILx∈G/IL, where x∈G. As G is a group so x−1∈G, implies that ILx−1∈G/IL and(53)ILxoILx−1=G,L,ζILxx−1,ξILxx−1,℘,ILx−1oILx=ILe.⇒G/IL is a group.
ConsiderIL¯=G/IL,L,ζIL¯,ξIL¯,℘, where(54)ζIL¯ILx=ζILx,ξIL¯ILx=ξILx∀ILx∈GIL,ζIL¯ILxoILy=ζIL¯ILxyζIL¯ILxoILy≥ζIL¯ILx∧ζIL¯ILy,ζIL¯ILxoILy=ζIL¯ILxy,ζIL¯ILxoILy≥ζIL¯ILx∧ζIL¯ILy,ξIL¯ILxoILy=ξIL¯ILxy,ξIL¯ILxoILy≤ξIL¯ILx∨ξIL¯ILy,=℘ζILx+℘ξILx≤1℘ζIL¯ILx+℘ξIL¯ILx≤1℘ζIL¯ILx+℘ξIL¯ILx,⇒IL¯ is a LIFSG-3 of G/IL. Hence proved.Definition 7.
LetIL be a LIFNSG-3 of G. Then, IL¯ is called a lattice intuitionistic fuzzy quotient subgroup type-3 of G denoted by LIFQSG−3.Proposition 8.
LetG be a group and IL be a LIFNSG-3 of G. The map φ:G⟶G/IL defined by φx=ILx∀xnG is epimorphism with(55)kerφ=GζILξIL=x∈G:ζILe=ζILx,ξILe=ξILx.Proof.
Letx,y∈G, then φxy=ILxy=ILxoILy=φxoφy this implies that φ is a group homomorphism. Clearly, φ is onto,(56)kerφ=x∈G:ζILe=ζILx,ξILe=ξILx,kerφ=GζILξIL.
From1st isomorphism theorem we get that G/GζILξIL≅G/IL. Hence proved.


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

Proposition 9.
LetG be a group and IL=G,L,ζIL,ξIL,℘ be a lattice-valued intuitionistic fuzzy subgroup type-3 of G. Suppose ϑ:G⟶G´ be a group isomorphism. Then, JL=G´,L,ϑoζIL,ϑoξIL,℘, where(57)ϑoζILg´=ζILϑ−1g´,ϑoξILg´=ξILϑ−1g´,
is a LIFSG-3 ofG´.Proof.
Letg´1,g´2∈G´. Then,(58)ϑoζILg´1g´2=ζILϑ−1g´1g´2,ϑoζILg´1g´2≥ϑoζILg´1∧ϑoζILg´2.
Similarly,ϑoξILg´1g´2≤ϑoξILg´1∨ϑoξILg´2. Let a∈G´. Then,(59)ϑoζILa−1=ζILϑ−1a−1,ϑoζILa−1=ϑoζILa.
Similarly,ϑoξILa−1=ϑoξILa. Now,(60)℘ϑoζILg´+℘ϑoξILg´=℘ζILϑ−1g´+℘ξILϑ−1g´≤1℘ϑoζILg´+℘ϑoξILg´≤1.12.⇒JL is a LIFSG-3 of G´. Hence proved.
Similar to the above proposition, now we will relate LIFNSG-3 ofG and G´.Proposition 10.
LetG be a group and IL=G,L,ζIL,ξIL,℘ be a LIFNSG-3 of G. If ϑ:G⟶G´ be a group isomorphism, then JL=G´,L,ϑoζIL,ϑoξIL,℘ is a LIFNSG-3 of G´.Proof.
SupposeIL is a LIFNSG-3 of G. Let u,v∈G´. Then, for u,v∈G´(61)ϑoζILuvu−1=ζILϑ−1uvu−1=ζILϑ−1v=ϑoζILv.
Similarly,ϑoξILuvu−1=ξILϑ−1v=ϑoξILv⇒JL is a LIFNSG-3 of G´. Hence proved.
In the following proposition we will induce LIFSG-3 ofG from the LIFS-3 of G´.Proposition 11.
LetG,G´ be two groups and φ:G⟶G´ be a group homomorphism. Suppose JL=G´,L,ζJL,ξJL,℘ be a LIFSG-3 of G´. Then, IL=G,L,ζJLoφ,ξJLoφ,℘ is a LIFSG-3 of G. Where ∀g∈G(62)ζJLoφg=ζJLφg,ξJLoφg=ξJLφg.Proof.
Letg1,g2∈G. Then,(63)ζJLoφg1g2=ζJLφg2g2,ζJLoφg1g2≥ζJLoφg2∧ζJLoφg2.
Similarly,ξJLoφg1g2≤ξJLoφg2∨ξJLoφg2. Now,(64)ζJLoφg−1=ζJLφg−1,ζJLoφg−1=ζJLφg,ζJLoφg−1=ζJLoφg.
Similarly,ξJLoφg−1=ξJLoφg.(65)℘ζJLoφg+℘ξJLoφg=℘ζJLφg+℘ξJLφg≤1℘ζJLoφg+℘ξJLoφg≤1.
⇒G,L,ζJLoφ,ξJLoφ,℘ is a LIFSG-3 of G. Hence proved.Proposition 12.
LetG and G´ be two groups, φ:G⟶G´ be a group homomorphism and(66)JL=G´,L,ζJL,ξJL,℘,
be a LIFNSG-3 ofG´. Then,(67)IL=G,L,ζJLoφ,ξJLoφ,℘,
is a LIFNSG-3 ofG.Proof.
Letu,v∈G. Then,(68)ζJLoφuvu−1=ζJLφuvu−1,=ζJLφuφvφu−1,=ζJLφv,=ζJLoφv.
Similarly,ξJLoφuvu−1=ξJLoφv. ⇒IL is a LIFNSG-3 of G. Hence proved.Theorem 3.
LetG and G´ be any two groups and ϑ:G⟶G´ be a group isomorphism. Let L be a lattice with top element T and bottom element B and ℘:L⟶0,1 be a lattice homomorphism. Let IGLIFSG−3 and IG´LIFSG−3 be the collections of LIFSG-3 of G and G´, respectively. Then, the map(69)ϑ^:IGLIFSG−3⟶IG´LIFSG−3
Defined by(70)ϑ^G,L,ζIL,ξIL,℘=G¯´,L,ϑoζIL,ϑoξIL,℘,
is bijective.Proof.
From previous propositions we know that if(71)IL=G,L,ζIL,ξIL,℘∈IGLIFSG−3.
Then,(72)JL=G´,L,ϑoζIL,ϑoξIL,℘∈IG´LIFSG−3.
Now, if(73)ϑoζIL1=ϑoζIL2,⇒ϑoζIL1g´=ϑoζIL2g´∀g´∈G´,⇒ζIL1ϑ−1g´=ζIL2ϑ−1g´∀g´∈G´.
Asϑ is bijective(74)⇒ϑ−1g´:g´∈G´=G±,⇒ζIL1=ζIL2.
Similarly,(75)ϑoξIL1=ϑoξIL2,⇒ξIL1ϑ−1g´=ξIL2ϑ−1g´∀g´∈G´.
Asϑ is bijective, this implies ξIL1=ξIL2. Thus, we get that, if(76)ϑ^G,L,ζIL1,ξIL1,℘=ϑ^G,L,ζIL2,ξIL2,℘,
thenG,L,ζIL1,ξIL1,℘=G,L,ζIL2,ξIL2,℘⇒ϑ^ is injective. Now, if(77)G´,L,ζJL,ξJL,℘∈IG´LIFSG−3,
thenG,L,ζJLoϑ,ξJLoϑ,℘∈IGLIFSG−3 and ∀g´∈G´,(78)ϑoζJLoϑg´=ζJLoϑϑ−1g´=ζJLg´,⇒ϑoζJLoϑ=ζJL.
Similarly,ϑoξJLoϑ=ξJL. This implies that ϑ^ is bijective. Hence proved.Theorem 4.
LetG be a group, L and L´ be any two lattices with top elements T and T´, and bottom elements B and B´, f:L⟶L´ and ℘:L⟶0,1 be lattice homomorphisms such that fT=T´, fB=B´, and ℘T=1,℘B=0. If IL=G,L,ζIL,ξIL,℘ be a LIFSG-3 of G, then JL´=G,L´,foζIL,foξIL,℘´ is also a LIFSG-3 of G. Where(79)foζIL,foξIL:G⟶L⟶L´,
are defined as(80)foζILg=fζILg,foξILg=fξILg,
and(81)℘´l´=supfl=l´℘f−1l´.Proof.
Letg1,g2∈G. Then,(82)foζILg1g2=fζILg1g2≥fζILg1∧ζILg2≥fζILg1∧fζILg2≥foζILg1∧foζILg2.
Similarly,foξILg1g2≤foξILg1∨foξILg2. Let g∈G and(83)℘´foζILg=sup℘f−1foζILg,=M112,℘´foξILg=sup℘f−1foξILg,=M2.
Then,M1 and M2 are of the form(84)M1=℘f−1foζILg=℘ζILg,M2=℘f−1foξILg=℘ξILg.
Thus,(85)℘´foζILg+℘´foξILg=℘ζILg+℘ξILg≤1.
Hence, we get the required result.Theorem 5.
LetG and G´ be any two group and ϑ:G⟶G´ be a group isomorphism. Let L and L´ be any two lattice with top elements T and T´, and bottom elements B and B´. Let f:L⟶L´ and ℘:L⟶0,1 be lattice homomorphisms such that fT=T´, fB=B´, ℘T=1,℘B=0. Let IL=G,L,ζIL,ξIL,℘ be a LIFSG-3 of G. Then,(86)JL´=G´,L´,ζJL´,ξJL´,℘´,
is a LIFSG-3 ofG´, where(87)ζJL´=ϑofoζIL,ξJL´=ϑofoξIL,℘´l´=supfl=l´℘f−1l´.Proof.
IfIL=G,L,ζIL,ξIL,℘ is a LIFSG-3 of G, then from Theorem 4(88)B´L=G,L´,foζIL,foξIL,℘´,
is a LIFSG-3 ofG. From Proposition 9(89)C´L=G´,L´,ϑofoζIL,ϑofoξIL,℘´,
is a LIFSG-3 ofG´.Theorem 6.
For a groupG, lattice L with top and bottom elements T and B, respectively, lattice homomorphism ℘:L⟶0,1. Let IL=G,L,ζIL,ξIL,℘ be a LIFNSG-3 of G. Then, the set(90)GζILξIL=x∈G:ζILx=ζILe,ξILx=ξILe,is normal inG and(91)IL^=GGζILξIL,L,ζIL^,ξIL^,℘.whereζIL^xGζILξIL=ζILx,ξIL^xGζILξIL=ξILxforallx∈G is a LIFNSG-3 of G/GζILξIL.Conversely, ifN⊲G and ILN=G/N,L,ζILN,ξILN,℘ is a LIFNSG-3 of G/N, then INL=G,L,ζINL,ξINL,℘ is a LIFNSG-3 of G, where(92)ζINL:G⟶L,ξINL:G⟶L,are defined as(93)ζINLx=ζINLNxN,ξINLx=ξINLNxN∀x∈G.Proof.
LetIL be a LIFNSG-3 of G. Then, ζILx≤ζILe and ξILx≥ξILe∀x∈G, implies that GζILξIL is a cut subgroup so it is normal in G and allow to construct factor group G/GζILξIL. Now, for the map ζIL^:G/GζILξIL⟶L(94)ζIL^xGζILξIL.yGζILξIL=ζIL^xyGζILξIL,ζIL^xGζILξIL.yGζILξIL=ζILxy,ζIL^xGζILξIL.yGζILξIL≥ζILx∧ζILy,ζIL^xGζILξIL.yGζILξIL≥ζIL^xGζILξIL∧ζIL^yGζILξIL.
Similarly, for the mapξIL^:G/GζILξIL⟶L(95)ξIL^xGζILξIL.yGζILξIL≤ξIL^xGζILξIL∨ξIL^yGζILξIL.
Also(96)℘ζIL^xGζILξIL+℘ξIL^xGζILξIL=℘ζILx+℘ξILx⇒℘ζIL^xGζILξIL+℘ξIL^xGζILξIL≤1.
⇒IL^ is a LIFNSG-3 of G/GζILξIL. Conversely, suppose(97)ILN=G/N,L,ζILN,ξILN,℘,
be a LIFNSG-3 ofG/N. Define ζINL:G⟶L as(98)ζINLx=ζILNxN.
Then,(99)ζINLxy=ζILNxyN,ζINLxy=ζILNxN.yN,ζINLxy≥ζILNxN∧ζILNyN,ζINLxy≥ζINLx∧ζINLy.
DefineξINL:G⟶L as(100)ξINLx=ξILNxN.
Then,(101)ξINLxy≤ξINLx∨ξINLy,℘ζINLx+℘ξINLx=℘ζILNxN+℘ξILxN,⇒℘ζINLx+℘ξINLx≤1,
⇒INL is a LIFNSG-3 of G.


## 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 groupG are exactly the subgroups of G, and conversely, any LIFS-3 of G whose level sets are subgroups of G is LIFSG-3. Lattice-valued intuitionistic fuzzy normal subgroups type-3 and lattice-valued intuitionistic fuzzy factor subgroups type-3 of G are governed by normal subgroups and factor groups of G. 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 p-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 0,1 with a suitable lattice L. The research findings can be utilized for application in algebra [43, 44] and real-life problems [45] to handle uncertainty and ambiguity more accurately.



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