The Fermatean Fuzzy Set (FFS) represents a robust approach for addressing ambiguity, eectively managing issues that remain unresolved by Intuitionistic fuzzy set and Pythagorean fuzzy set concepts. Due to its practical utility and signicant impact on tackling real-world challenges across various domains, FFS has spurred extensive research. This study denes Fermatean fuzzy sublattice and Fermatean fuzzy lattice. Additionally, it introduces Fermatean fuzzy L -ring ideals. The paper explores the concept of homomorphism within Fermatean fuzzy sets. Furthermore, it investigates important ndings concerning the image and pre-image of Fermatean fuzzy L -ring ideals, utilizing properties of inmum and supremum.

The results are illustrated through pertinent numerical examples. Keywords: Intuitionistic fuzzy sets, Pythagorean fuzzy sets, Fermatean fuzzy sets, Fermatean fuzzy lattice, Fermatean fuzzy L -ring ideal.

Fuzzy Sets (FSs), introduced by Zadeh [ 33 ] to model imprecise evaluations, have paved the way for Intuitionistic Fuzzy Sets (IFSs). Atanassov [ 4 ] extended FSs by introducing membership and non-membership degrees, with the constraint that their sum does not exceed 1. The characterization of intuitionistic fuzzy sets as a generalized Corresponding Author: nilkamalbest100 gmail.com Licensee System Analytics. This article is an open-access article distributed under the terms and conditions of the CreativeCommons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0).

, https://doi.org/10.22105/scfa.v1i2.31 form of fuzzy sets is notably intriguing and practical across diverse application domains. Atanassov [5] further advanced this concept by introducing innovative operations dened over intuitionistic fuzzy sets. Some situations where + 1 exists, unlike in IFSs. Because of this necessity in IFSs, Pythagorean fuzzy sets were developed (PFSs). The Pythagorean Fuzzy Set (PFS), design in [ 31 , 30 ] was a unique tool for dealing with ambiguity when evaluating membership grade and nonmembership grade fullling the constraints 0 1 and 0 1 , with the result that 2 + 2 1 . PFSs were better at characterizing unclear data than IFSs.

Fermatean fuzzy sets was the extension Pythagorean fuzzy sets. In Fermatean fuzzy sets the membership grade ( ) and non-membership grade ( ) satisfy the conditions 0 3 + 3 1 , where the values of and lie between 0 and 1. Senapati et al. [ 25 ] denes some new operations over Fermatean fuzzy numbers. Fermatean fuzzy sets were applied in wide range of elds, including articial intelligence, signal processing, management sciences, engineering, mathematics, social sciences, automata theory, medical sciences, biology and statistics etc.

Algebraic patterns nd application in theoretical physics, information science, computer science, control engi - neering, and numerous other domains. This diversity drives scholars to explore various abstract algebra topics and discoveries within the broader framework of fuzzy set theories. Kunchman et al. [ 18 ] introduced the concept of fuzzy prime ideals in gamma near rings. Kim and Jun [ 15 , 16 ] pioneered the intuitionistic fuzzication of multiple semigroup ideals. Kim and Lee [ 17 ] explored the implications of intuitionistic fuzzy bi-ideals in semigroups. Sardar et al. [23] introduced prime ideals, semi-prime ideals, and -semigroups with intuitionistic fuzzy information. Adak et al. [ 1 ] investigated properties of Pythagorean fuzzy ideals in near-rings and presented results on rough Pythagorean fuzzy sets.

The lattice structure holds signicant prominence in fuzzy set theory, extensively discussed and applied across various contexts. R. Natarajan and S. Moganavalli [ 20 ] extended fuzzy sets into lattice theory. Ajmal et al. [ 2 ] explored the lattice of fuzzy rings. Various authors have introduced and investigated fuzzy lattices in distinct manners [ 27 , 28 ]. Similar approaches have been taken in the realm of intuitionistic fuzzy lattices. Thomas et al.

introduced intuitionistic fuzzy lattices by fuzzifying membership elements within lattice structures and examined their properties. Tripathy et al. [ 29 ] proposed intuitionistic fuzzy lattices based on intuitionistic fuzzy order relations introduced by Burillo and Bustine [7, 8]. Furthermore, K. Hur, Y. S. Ahn, and D. S. Kim [ 13 ] expanded the lattice of intuitionistic fuzzy sets in rings. Marashdeh and Salleh [ ? ] introduced and studied intuitionistic fuzzy ideals based on intuitionistic L -fuzzy sets. Sasireka et al. [ 24 ] established properties of intuitionistic fuzzy sub L -rings. G. J. Wang [ 32 ] generalized order-homomorphisms on fuzzy sets. Olson [ 21 ] provided proofs on homomorphisms for hemirings. Palaniappan [22] introduced and studied homomorphisms and anti-homomorphisms of fuzzy and anti-fuzzy ideals.

This paper introduces the concept of Fermatean fuzzy sublattice and lattice, and extends it to dene Fermatean fuzzy L -ideals, exploring their properties. Additionally, it presents insightful characterizations of Fermatean fuzzy L -ring ideals and investigates homomorphisms of these ideals. The study demonstrates that the image and pre-image under homomorphisms of Fermatean fuzzy L -ring ideals remain Fermatean fuzzy L -ring ideals.

The paper is structured as follows: Section 2 provides preliminary denitions and examples. In Section 3, signicant results concerning Fermatean fuzzy L -ring ideals and their homomorphisms are discussed. Section 4 concludes the paper.

We will go over the ideas that are connected to fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets and Fermatean fuzzy sets in this section. Denition 1. A poset ( L; ) is said to form a lattice of for any a; b 2 L , sup f a; b g and inf f a; b g exist in L. In this case we write sup f a; b g = a _ b and inf f a; b g = a ^ b . Throughout this paper we denote a lattice with join ` _ ' and ` ^ ' by simply L.

Denition 2. We dene the fuzzy set M within a universal set S as M = fh ; M ( ) i : 2 S g ; 82 Adak t al. | Soft. Comput. Fusion. Appl. 1(2) (2024) 80-90 where M : S ! [0 ; 1] is a mapping known as the fuzzy membership function. The complement of is dened by ( ) = 1 ( ) for all 2 S and is denoted by . Denition 3. An intuitionistic fuzzy set (IFS) I in S is dened as I = fh ; I ( ) ; I ( ) i : 2 S g ;where the I() is the worth of membership and I() is the worth of non-membership of the element 2 S respectively.

Also I : S ! [0 ; 1] ; I : S ! [0 ; 1] and satisfy the condition 0 I ( ) + I ( ) 1 ; for all 2 S . The degree of indeterminacy h I ( ) = 1 I ( ) I ( ) .In some circumferences 0 I() + I()1 for whatever reason, this may not be the case. We take some situations where 0:9 + 0:4 = 1:3>1 , but 0:92+ 0:42= 0:97<1 . To deal with this problem Yager [31,30]proposed the perspective assumes of Pythagorean fuzzy set in 2013.

Denition 4. A Pythagorean fuzzy set P in universe of discourse S is represented as P = fh ; P ( ) ; P ( ) ij 2 S g ;where P() :S ![0;1] denotes the worth of membership and P() :S ![0;1] represents the worth to whichthe element 2 S is not a member of the set P , with the condition that 0 ( P ( )) 2 + ( P ( )) 2 1 ; for all 2 S . The worth of indeterminacy h P ( ) = p 1 ( P ( ))

2 ( P ( )) 2 .In practice, the condition 0 2 P() + 2 P()1 may not be true for any reason. For example 0:92+ 0:52= 1:06>1 , but 0:93+ 0:53= 0:854<1 , or 0:82+ 0:72= 1:13>1 , but 0:83+ 0:73=:855<1 . To address thisissue, Senapati et. al., [25] proposed the principle of the fermatean fuzzy set in 2021. Denition 5. A fermatean fuzzy set M in a nite universe of discourse S is furnished as M = fh ; M ( ) ; M ( ) ij 2 S g ;where M() :S ![0;1] denotes the worth of membership and M() :S ![0;1] represents the worth to whichthe element 2 S is not a member of the set M , with the predicament that 0 ( M ( ))

3 + ( M ( )) 3 1 ; for all 2 S . The worth of indeterminacy h M ( ) = 3 p 1 ( M ( )) 3 ( M ( )) 3 .Denition 6. A fuzzy subset of a lattice ordered ring (orL-ring in short) R, is called fuzzy subL-ring of R,if the following conditions are satised. (i) ( _ ) min f ( ) ; ( ) g (ii) ( ^ ) min f ( ) ; ( ) g (iii) ( ) min f ( ) ; ( ) g (iv) ( ) min f ( ) ; ( ) g 8 ; 2 L .Denition 7. A fuzzy subset of anL-ring R, is called fuzzyL-ring ideal (or) fuzzyL-ideals of R, if thefollowing conditions are satised.

(i) ( _ ) min f ( ) ; ( ) g (ii) ( ^ ) max f ( ) ; ( ) g (iii) ( ) min f ( ) ; ( ) g (iv) ( ) max f ( ) ; ( ) g 8 ; 2 L . Denition 8. A mapping from ring R to a ring S is called on Homomorphism, if 8 ; 2 R (i) ( + ) = ( ) + ( ) (ii) ( ) = ( ) ( ) .Denition 9. [20] LetRandSbe twoL-rings. A function:R ! Sis called anL-Homomorphism if for all; 2 R (i) ( _ ) = ( ) _ ( )

(ii) ( ^ ) = ( ) ^ ( ) (iii) ( + ) = ( ) + ( ) (iv) ( ) = ( ) ( ) . Denition 10. [ 33 ] Let be a mapping from a set R to a set S and let A be a fuzzy subset in R . Then A is called -invarient if ( ) = ( ) implies A ( ) = A ( ) for all ; 2 R . Clearly, if A is -invarient, then 1 ( ( A )) = A Denition 11. A Fermatean fuzzy set M is said to have sup-property [Inf-property], if for each subset T A there exist t 0 2 T such that, sup t 2 T f ( t ) g = ( t 0 )

inf t 2 T f ( t ) g = ( t 0 )Denition 12. Letbe a mapping from a setXto a setYand leth M ; M ibe Fermatean fuzzy subset inX and Y respectively. (i) ( A ) , the image of A under , is a intuition fuzzy subset in Y for all 2 Y . we dene, ( M )( ) = ( sup 2 1( ) M ( ) ; if 1 ( ) 6 = 0 if 1 ( ) = and ( M )( ) = ( inf 2 1 ( ) M ( ) ; if 1 ( ) 6 = 1 if 1 ( ) = 1 ( B ) is the pre-image of B under , is a Fermatean fuzzy set in X .

i.e, 1 ( M )( ) = M ( ( )) and 1 ( M )( ) = M ( ( )) ; 8 2 R:Denition 13. LetLbe a lattice andM=fh S; M(); M()i: 2 S gbe Fermatean fuzzy set ofL. ThenM is called an Fermatean fuzzy sublattice (Fermatean fuzzy lattice) of L if the following conditions are satised. (i) M ( _ ) min f M ( ) ; M ( ) g (ii) M ( ^ ) min f M ( ) ; M ( ) g (iii) M ( _ ) max f M ( ) ; M ( ) g (iv) M ( ^ ) max f M ( ) ; M ( ) g 8 ; 2 L .Denition 14. A Fermatean fuzzy sublatticeMofLis called an Fermatean fuzzy ideal ofL(Fermatean fuzzyL -ideal) if the following conditions are satised (i) ( _ ) min f ( ) ; ( ) g (ii) ( ^ ) max f ( ) ; ( ) g (iii) ( _ ) max f ( ) ; ( ) g (iv) ( ^ ) min f ( ) ; ( ) g 8 ; 2 L .

83 Denition 15. LetRbe a ring. A Fermatean fuzzy setM=fh ; (); ()i: 2 R gofRis said to beFermatean fuzzy sub L -ring on R if for all ; 2 R , (i) ( ) min f ( ) ; ( ) g (ii) ( ) min f ( ) ; ( ) g (iii) ( _ ) min f ( ) ; ( ) g (iv) ( ^ ) min f ( ) ; ( ) g (v) ( ) max f ( ) ; ( ) g (vi) ( ) max f ( ) ; ( ) g (vii) ( _ ) max f ( ) ; ( ) g (viii) ( ^ ) max f ( ) ; ( ) g .

3|Main ResultsIn this section, we dene Fermatean fuzzyL-ring ideal and dened Fermatean fuzzy homomorphism from aL-ringRintoL-ringS. Also, investigated some related results on image and preimage of homomorphism ofFermatean fuzzy L -ring utilizing supremum and inmum property of Fermatean fuzzy sets.Denition 16. A Fermatean fuzzy subL-ringMonRis said to be a Fermatean fuzzy leftL-ring ideal if forall ; 2 R .

(i) ( ) min f ( ) ; ( ) g (ii) ( ) ( ) (iii) ( _ ) min f ( ) ; ( ) g (iv) ( ^ ) max f ( ) ; ( ) g (v) ( ) max f ( ) ; ( ) g (vi) ( ) ( ) (vii) ( _ ) max f ( ) ; ( ) g (viii) ( ^ ) min f ( ) ; ( ) g .Denition 17. A Fermatean fuzzy subL-ringMonRis said to be a Fermatean fuzzy rightL- ring ideal if forall ; 2 R: (i) ( ) min f ( ) ; ( ) g (ii) ( ) ( )

(iii) ( _ ) min f ( ) ; ( ) g (iv) ( ^ ) max f ( ) ; ( ) g (v) ( ) max f ( ) ; ( ) g (vi) ( ) ( ) (vii) ( _ ) max f ( ) ; ( ) g (viii) ( ^ ) min f ( ) ; ( ) g .Denition 18. A Fermatean fuzzy subL-ringMonRis said to be a Fermatean fuzzyL-ring ideal if it is bothan Fermatean fuzzy left L - ring ideal and an Fermatean fuzzy right L - ring ideal of R .Example 1. Now (R f=f 1 ; 2 ; 3 ; 4 g ;+; ; _ ; ^) is aL-ring under the operations +; ; _ ; and ^dened bythe following tables, + 1 2 3 4 1 1 2 3 4 2 2 1 4 3 3 3 4 1 2 4 4 3 2 1 1 2 3 4 1 1 1 1 1 2 1 2 1 2 3 1 1 3 3 4 1 2 3 4 _ 1 2 3 4 1 1 2 3 4 2 2 2 4 4 3 3 4 3 4 4 4 4 4 4 ^ 1 2 3 4 1 1 1 1 1 2 1 2 1 2 3 1 1 3 3 4 1 2 3 4 84 Adak t al. | Soft. Comput. Fusion. Appl. 1(2) (2024) 80-90 Consider ( ) = 8 < : 0 : 8 if = 1 0 : 6 if = 2 0 : 3 if = 3 ; 4 ; ( ) = 8 < : 0 : 1 if = 1 0 : 2 if = 2 0 : 4 if = 3 ; 4 then ( R f = f 1 ; 2 ; 3 ; 4 g ; + ; ; _ ; ^ ) is a Fermatean fuzzy L -ring ideal.Theorem 1. Letbe a homomorphism from aL-ringRintoL-ringSand leth M ; M ibe a Fermatean fuzzyleft L - ring ideal of S . Then the pre-image h 1 ( M ) ; 1 ( M ) i is a Fermatean fuzzy left L - ring ideal of R .

Proof. Consider a L -ring Homomorphism : R ! S . Let h M ; M i be a Fermatean fuzzy left L - ring ideal of S: 8 ; 2 R . (i) 1 ( M )( ) = M ( ) min f M ( ) ; M ( ) g 1 ( M )( ) min f 1 ( M )( ) ; 1 ( M )( ) g : (ii) 1 ( M )( ) = M ( ) M ( ( ) 1 ( M ( )) 1 ( M )( ) 1 ( M ( ) g : (iii) 1 ( M )( _ ) = M ( _ )

min f M ( ) ; M ( ) g 1 ( M )( _ ) min f 1 ( M )( ) ; 1 ( M )( ) g : (iv) 1 ( M )( ^ ) = M ( ^ ) max f M ( ) ; M ( ) g 1 ( M )( _ ) max f 1 ( M )( ) ; 1 ( M )( ) g : (v) 1 ( M )( ) = M ( ) max f M ( ) ; M ( ) g 1 ( M )( ) max f 1 ( M )( ) ; 1 ( M )( ) g : (vi) 1 ( M )( ) = M ( ) M ( ( )

1 ( M ( )) 1 ( M )( ) 1 ( M ( ) g : (vii) 1 ( M )( _ ) = M ( _ ) max f M ( ) ; M ( ) g 1 ( M )( _ ) max f 1 ( M )( ) ; 1 ( M )( ) g : (viii) 1 ( M )( ^ ) = M ( ^ ) min f M ( ) ; M ( ) g 1 ( M )( _ ) min f 1 ( M )( ) ; 1 ( M )( ) g : ) 1 ( M ) ; 1 ( M ) is a Fermatean fuzzy left L - ring ideal of R .Theorem 2. Let:R ! Sbe a homomorphism from aL-ringRintoS. If M ; Mis a Fermatean fuzzyright L - ring ideal of S , then pre-image 1 ( M ) and 1 ( M ) is a Fermatean fuzzy right L - ring ideal of R .

85 Proof. Consider homomorphism:R ! S, whereRandSbeL-ring. Let M ; Mbe a Fermatean fuzzy rightL -ring ideal of S 8 ; 2 R . (i) 1 ( M )( ) = M ( ) min f M ( ) ; M ( ) g 1 ( M )( ) min f 1 ( M )( ) ; 1 ( M )( ) g : (ii) 1 ( M )( ) = M ( ( )) M ( ( ) = 1 ( M )( )) 1 ( M )( ) 1 (( M )( ) g : (iii) 1 ( M )( _ ) = M ( _ )

min f M ( ) ; M ( ) g 1 ( M )( _ ) min f 1 ( M )( ) ; 1 ( M )( ) g : (iv) 1 ( M )( ^ ) = M ( ( ^ )) max f M ( ) ; M ( ) g 1 ( M )( _ ) max f 1 ( M )( ) ; 1 ( M )( ) g : (v) 1 ( M )( ) = M ( ) max f M ( ) ; M ( ) g 1 ( M )( ) max f 1 ( M )( ) ; 1 ( M )( ) g : (vi) 1 ( M )( ) = M ( ( ))

M ( ( ) 1 ( M ( )) 1 ( M )( ) 1 ( M ( ) g : (vii) 1 ( M )( _ ) = M ( _ ) max f M ( ) ; M ( ) g 1 ( M )( _ ) max f 1 (( M )( )) ; 1 ( M ( )) g : (viii) 1 ( M )( ^ ) = M ( ^ ) min f M ( ) ; M ( ) g 1 ( M )( _ ) min f 1 ( M ( )) ; 1 ( M ( )) g : ) 1 ( M ) ; 1 ( M ) is a Fermatean fuzzy right L - ring ideal of R .Theorem 3. Let:R ! Sbe a homomorphism from aL-ringRintoL-ringS. If M ; Mis a Fermateanfuzzy L - ring ideal of S , then pre-image 1 ( M ) ; 1 ( M ) is a Fermatean fuzzy right L - ring ideal of R .

Proof. It's trivial.Theorem 4. LetRandSbeL-ring and:R ! Sbe a homomorphism. If M ; Mis a Fermatean fuzzy leftL- ring ideal of aL-ringR, with sup property, then the pre-image( M) and( M) is a Fermatean fuzzy leftL -ring ideal of S . 86 Adak t al. | Soft. Comput. Fusion. Appl. 1(2) (2024) 80-90 Proof. Consider homomorphism:R ! S, whereRandSbeL-ring. Let M ; Mbe a Fermatean fuzzy leftL - ring ideal of R 8 ; 2 R: (i) ( M )( ( ) ( )) = ( M ) ( )

min f M ( ) ; M ( ) g ( M )( ( ) ( )) min f ( M )( ) ; ( M )( ) g : (ii) ( M )( ( ) ( )) = ( M )( ( )) M ( ) = ( M ) ( )) ( M )( ( ) ( )) ( M )( ( ) : (iii) ( M )( ( ) _ ( )) = ( M ) ( _ ) min f M ( ) ; M ( ) g ( M )( ( ) _ ( )) min f ( M )( ) ; ( M )( ) g : (iv) ( M )( ( ) ^ ( )) = ( M )( ( ^ ))

max f M ( ) ; M ( ) g ( M )( ( ) ^ ( )) max f ( M )( ) ; ( M )( ) g : (v) ( M )( ( ) ( )) = ( M ) ( ) max f M ( ) ; M ( ) g ( M )( ( ) ( )) max f ( M )( ) ; ( M )( ) g : (vi) ( M )( ( ) ( )) = ( M ) ( ) M ( ( )) ( M ( )) ( M )( ( ) ( )) ( M )( )) : (vii) ( M )( ( ) _ ( )) = ( M ) ( _ )

max f M ( ) ; M ( ) g ( M )( ( ) _ ( )) max f ( M )( )) ; ( M )( ) g : (viii) ( M )( ( ) ^ ( )) = ( M )( ( ^ )) min f M ( ) ; M ( ) g ( M )( ( ) ^ ( )) min f ( M )( ) ; ( M )( ) g : ) ( M ) ; ( M ) is a Fermatean fuzzy right L - ring ideal of S .Theorem 5. Let:R ! Sbe a homomorphism from aL-ringRintoL-ringS. If M ; Mis a Fermateanfuzzy rightL- ring ideal ofR, with sup property, then the pre-image( M) and( M) is a Fermatean fuzzyright L -ring ideal of S .

Proof. Consider a L -ring homomorphism : R ! S . Let M ; M be a Fermatean fuzzy right L - ring ideal of R 8 ; 2 R: (i) ( M )( ( ) ( )) = ( M ) ( ) min f M ( ) ; M ( ) g ( M )( ( ) ( )) min f ( M )( ) ; ( M )( ) g : 87 (ii) ( M )( ( ) ( )) = ( M )( ( )) M ( ) = ( M ) ( )) ( M )( ( ) ( )) ( M )( ( ) : (iii) ( M )( ( ) _ ( )) = ( M ) ( _ )

min f M ( ) ; M ( ) g ( M )( ( ) _ ( )) min f ( M )( ) ; ( M )( ) g : (iv) ( M )( ( ) ^ ( )) = ( M )( ( ^ )) max f M ( ) ; M ( ) g ( M )( ( ) ^ ( )) max f ( M )( ) ; ( M )( ) g : (v) ( M )( ( ) ( )) = ( M ) ( ) max f M ( ) ; M ( ) g ( M )( ( ) ( )) max f ( M )( ) ; ( M )( ) g : (vi) ( M )( ( ) ( )) = ( M ) ( )

M ( ( )) ( M ( )) ( M )( ( ) ( )) ( M )( ( )) : (vii) ( M )( ( ) _ ( )) = ( M ) ( _ ) max f M ( ) ; M ( ) g ( M )( ( ) _ ( )) max f ( M )( ) ; ( M )( ) g : (viii) ( M )( ( ) ^ ( )) = ( M )( ( ^ )) min f M ( ) ; M ( ) g ( M )( ( ) ^ ( )) min f ( M )( ) ; ( M )( ) g : ) ( M ) ; ( M ) is a Fermatean fuzzy right L - ring ideal of S .Theorem 6. Let:R ! Sbe homomorphism fromL-ringRintoL-ringS. If M ; Mis an Fermatean fuzzyL- ring ideal of aL-ringRwith sup property, then pre-image( M); ( M) is a fuzzyL- ring ideal ofL-ringS .

Proof. It's trivial. 4|ConclusionWithin this detailed study, we dene lattice concept of Fermatean fuzzy sets along with their crucial resultsin the Fermatean fuzzy sets context. The left and right ideals of Fermatean fuzzy sets over lattice is initiated.The concept of homomorphism on Fermatean fuzzyL-ring ideal are introduced. With the help of supremumand inmum property of Fermatean fuzzy sets, some important results related to image and pre-image of homomorphism of Fermatean fuzzyL-ring ideal has been investigated. To extend this work, one can discusssome characterization on anti-homomorphism properties of Fermatean fuzzy L -ring ideal.

AcknowledgmentsThe authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which signicantly contributed to the enhancement of thispaper. 88 Adak t al. | Soft. Comput. Fusion. Appl. 1(2) (2024) 80-90

All authors have read and agreed to the published version of the manuscript. FundingThe authors declare that no external funding or support was received for the research presented in this paper,including administrative, technical, or in-kind contributions.

All data supporting the reported ndings in this research paper are provided within the manuscript.

The authors declare that there is no conict of interest concerning the reported research ndings.

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