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 form of fuzzy sets is notably intriguing and practical across diverse application domains. Atanassov [5] further advanced this concept by introducing innovative operations defined over intuitionistic fuzzy sets.

Some situations where µ + ν ≥ 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 fulfilling the constraints 0 ≤ µ ≤ 1 and 0 ≤ ν ≤ 1, with the result that µ + ν ≤ 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 (µ) and non-membership grade (ν) satisfy the conditions 0 ≤ µ + ν ≤ 1, where the values of µ and ν lie between 0 and 1. Senapati et al. [25] defines some new operations over Fermatean fuzzy numbers. Fermatean fuzzy sets were applied in wide range of fields, including artificial intelligence, signal processing, management sciences, engineering, mathematics, social sciences, automata theory, medical sciences, biology and statistics etc.

Algebraic patterns find 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 fuzzification 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 Γ-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 significant 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 define 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 definitions and examples. In Section 3, significant 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.

Definition 1. A poset (L, ≤) is said to form a lattice of for any a, b ∈ L, sup{a, b} and inf{a, b} exist in L.

In this case we write sup{a, b} = a ∨ b and inf{a, b} = a ∧ b.

Throughout this paper we denote a lattice with join ‘∨’ and ‘∧’ by simply L.

Definition 2. We define the fuzzy set M within a universal set S as M = {⟨ξ, αM(ξ)⟩ : ξ ∈ S} , where αM : S → [0, 1] is a mapping known as the fuzzy membership function.

The complement of α is defined by α ¯(ξ) = 1 − α(ξ) for all ξ ∈ S and is denoted by α ¯.

Definition 3. An intuitionistic fuzzy set (IFS) I in S is defined as I = {⟨ξ, αI(ξ), βI(ξ)⟩ : ξ ∈ S}, where the αI(ξ) is the worth of membership and βI(ξ) is the worth of non-membership of the element ξ ∈ S respectively.

Also αI : S → [0, 1], βI : S → [0, 1] and satisfy the condition 0 ≤ αI(ξ) + βI(ξ) ≤ 1, for all ξ ∈ S.

The degree of indeterminacy hI(ξ) = 1 − αI(ξ) − βI(ξ).

In some circumferences 0 ≤ αI(ξ) + βI(ξ) ≤ 1 for whatever reason, this may not be the case. We take some situations where 0.9 + 0.4 = 1.3 > 1, but 0.9 + 0.4 = 0.97 < 1. To deal with this problem„ Yager [31, 30] proposed the perspective assumes of Pythagorean fuzzy set in 2013.

Definition 4. A Pythagorean fuzzy set P in universe of discourse S is represented as P = {⟨ξ, αP (ξ), βP (ξ)⟩|ξ ∈ S}, where αP (ξ) : S → [0, 1] denotes the worth of membership and βP (ξ) : S → [0, 1] represents the worth to which the element ξ ∈ S is not a member of the set P, with the condition that 0 ≤ (αP (ξ)) + (βP (ξ)) ≤ 1, for all ξ ∈ S.

The worth of indeterminacy hP (ξ) = 1 − (αP (ξ)) − (βP (ξ)) .

2 2 2 2

In practice, the condition 0 ≤ αP (ξ) + βP (ξ) ≤ 1 may not be true for any reason. For example 0.9 + 0.5 =

3 3 2 2 3 3

1.06 > 1, but 0.9 + 0.5 = 0.854 < 1, or 0.8 + 0.7 = 1.13 > 1, but 0.8 + 0.7 = .855 < 1. To address this issue, Senapati et. al., [25] proposed the principle of the fermatean fuzzy set in 2021.

Definition 5. A fermatean fuzzy set M in a finite universe of discourse S is furnished as M = {⟨ξ, αM(ξ), βM(ξ)⟩|ξ ∈ S}, where αM(ξ) : S → [0, 1] denotes the worth of membership and βM(ξ) : S → [0, 1] represents the worth to which the element ξ ∈ S is not a member of the set M, with the predicament that 0 ≤ (αM(ξ)) + (βM(ξ)) ≤ 1, for all ξ ∈ S.

3 3 3

The worth of indeterminacy hM(ξ) = 1 − (αM(ξ)) − (βM(ξ)) .

Definition 6. A fuzzy subset α of a lattice ordered ring (or L-ring in short) R, is called fuzzy sub L-ring of R, if the following conditions are satisfied.

(i) α(ξ ∨ η) ≥ min{α(ξ), α(η)} (ii) α(ξ ∧ η) ≥ min{α(ξ), α(η)} (iii) α(ξ − η) ≥ min{α(ξ), α(η)} (iv) α(ξη) ≥ min{α(ξ), α(η)} ∀ ξ, η ∈ L.

Definition 7. A fuzzy subset α of an L-ring R, is called fuzzy L-ring ideal (or) fuzzy L-ideals of R, if the following conditions are satisfied.

(i) α(ξ ∨ η) ≥ min{α(ξ), α(η)} (ii) α(ξ ∧ η) ≥ max{α(ξ), α(η)} (iii) α(ξ − η) ≥ min{α(ξ), α(η)} (iv) α(ξη) ≥ max{α(ξ), α(η)} ∀ ξ, η ∈ L.

Definition 8. A mapping ϕ from ring R to a ring S is called on Homomorphism, if ∀ ξ, η ∈ R (i) ϕ(ξ + η) = ϕ(ξ) + ϕ(η) (ii) ϕ(ξη) = ϕ(ξ) · ϕ(η).

Definition 9. [20] Let R and S be two L-rings. A function ϕ : R → S is called an L-Homomorphism if for all ξ, η ∈ R (i) ϕ(ξ ∨ η) = ϕ(ξ) ∨ ϕ(η) (ii) ϕ(ξ ∧ η) = ϕ(ξ) ∧ ϕ(η) (iii) ϕ(ξ + η) = ϕ(ξ) + ϕ(η) (iv) ϕ(ξη) = ϕ(ξ) · ϕ(η).

Definition 10. [33] Let ϕ be a mapping from a set R to a set S and let A be a fuzzy subset in R. Then A is called ϕ-invarient if ϕ(ξ) = ϕ(η) implies A(ξ) = A(η) for all ξ,η ∈ R. Clearly, if A is ϕ-invarient, then ϕ (ϕ(A)) = A

Definition 11. A Fermatean fuzzy set M is said to have sup-property [Inf-property], if for each subset T ⊂ A there exist t0 ∈ T such that, sup{α(t)} = α(t0)

inf {β(t)} = β(t0)

Definition 12. Let ϕ be a mapping from a set X to a set Y and let ⟨αM, βM⟩ be Fermatean fuzzy subset in X and Y respectively.

(i) ϕ(A), the image of A under ϕ, is a intuition fuzzy subset in Y for all η ∈ Y . we define,

sup αM(ξ), if ϕ (η) ̸= ϕ

ξ∈ϕ−1(η)

ϕ(αM)(η) = 0 if ϕ (η) = ϕ

and

inf βM(ξ), if ϕ (η) ̸= ϕ

ξ∈ϕ−1(η)

ϕ(βM)(η) = 1 if ϕ (η) = ϕ

ϕ (B) is the pre-image of B under ϕ, is a Fermatean fuzzy set in X.

i.e, ϕ (αM)(ξ) = αM(ϕ(ξ)) and ϕ (βM)(ξ) = βM(ϕ(ξ)), ∀ ξ ∈ R.

Definition 13. Let L be a lattice and M = {⟨S, αM(ξ), βM(ξ)⟩ : ξ ∈ S} be Fermatean fuzzy set of L. Then M is called an Fermatean fuzzy sublattice (Fermatean fuzzy lattice) of L if the following conditions are satisfied. (i) αM(ξ ∨ η) ≥ min{αM(ξ), αM(η)} (ii) αM(ξ ∧ η) ≥ min{αM(ξ), αM(η)} (iii) βM(ξ ∨ η) ≤ max{βM(ξ), βM(η)} (iv) βM(ξ ∧ η) ≤ max{βM(ξ), βM(η)} ∀ ξ, η ∈ L.

Definition 14. A Fermatean fuzzy sublattice M of L is called an Fermatean fuzzy ideal of L (Fermatean fuzzy L-ideal) if the following conditions are satisfied (i) α(ξ ∨ η) ≥ min{α(ξ), α(η)} (ii) α(ξ ∧ η) ≥ max{α(ξ), α(η)} (iii) β(ξ ∨ η) ≤ max{β(ξ), β(η)} (iv) β(ξ ∧ η) ≤ min{β(ξ), β(η)} ∀ ξ, η ∈ L.

Definition 15. Let R be a ring. A Fermatean fuzzy set M = {⟨ξ, α(ξ), β(ξ)⟩ : ξ ∈ R} of R is said to be Fermatean fuzzy sub L-ring on R if for all ξ, η ∈ R, (i) α(ξ − η) ≥ min{α(ξ), α(η)} (ii) α(ξη) ≥ min{α(ξ), α(η)} (iii) α(ξ ∨ η) ≥ min{α(ξ), α(η)} (iv) α(ξ ∧ η) ≥ min{α(ξ), α(η)} (v) β(ξ − η) ≤ max{β(ξ), β(η)} (vi) β(ξη) ≤ max{β(ξ), β(η)} (vii) β(ξ ∨ η) ≤ max{β(ξ), β(η)} (viii) β(ξ ∧ η) ≤ max{β(ξ), β(η)}.

In this section, we define Fermatean fuzzy L-ring ideal and defined Fermatean fuzzy homomorphism from a L-ring R into L-ring S. Also, investigated some related results on image and preimage of homomorphism of Fermatean fuzzy L-ring utilizing supremum and infimum property of Fermatean fuzzy sets.

Definition 16. A Fermatean fuzzy sub L-ring M on R is said to be a Fermatean fuzzy left L-ring ideal if for all ξ, η ∈ R.

(i) α(ξ − η) ≥ min{α(ξ), α(η)} (ii) α(ξη) ≥ α(η) (iii) α(ξ ∨ η) ≥ min{α(ξ), α(η)} (iv) α(ξ ∧ η) ≥ max{α(ξ), α(η)} (v) β(ξ − η) ≤ max{β(ξ), β(η)} (vi) β(ξη) ≤ β(η) (vii) β(ξ ∨ η) ≤ max{β(ξ), β(η)} (viii) β(ξ ∧ η) ≤ min{β(ξ), β(η)}.

Definition 17. A Fermatean fuzzy sub L-ring M on R is said to be a Fermatean fuzzy right L- ring ideal if for all ξ, η ∈ R.

(i) α(ξ − η) ≥ min{α(ξ), α(η)} (ii) α(ξη) ≥ α(ξ) (iii) α(ξ ∨ η) ≥ min{α(ξ), α(η)} (iv) α(ξ ∧ η) ≥ max{α(ξ), α(η)} (v) β(ξ − η) ≤ max{β(ξ), β(η)} (vi) β(ξη) ≤ β(ξ) (vii) β(ξ ∨ η) ≤ max{β(ξ), β(η)} (viii) β(ξ ∧ η) ≤ min{β(ξ), β(η)}.

Definition 18. A Fermatean fuzzy sub L-ring M on R is said to be a Fermatean fuzzy L-ring ideal if it is both an Fermatean fuzzy left L- ring ideal and an Fermatean fuzzy right L- ring ideal of R.

Example 1. Now (R{= {ξ1, ξ2, ξ3, ξ4}, +, ·, ∨, ∧) is a L-ring under the operations +, ·, ∨, and ∧ defined by the following tables,

+ ξ1 ξ2 ξ3 ξ4 · ξ1 ξ2 ξ3 ξ4 ∨ ξ1 ξ2 ξ3 ξ4 ∧ ξ1 ξ2 ξ3 ξ4 ξ1 ξ1 ξ2 ξ3 ξ4 ξ1 ξ1 ξ1 ξ1 ξ1 ξ1 ξ1 ξ2 ξ3 ξ4 ξ1 ξ1 ξ1 ξ1 ξ1 ξ2 ξ2 ξ1 ξ4 ξ3 ξ2 ξ1 ξ2 ξ1 ξ2 ξ2 ξ2 ξ2 ξ4 ξ4 ξ2 ξ1 ξ2 ξ1 ξ2 ξ3 ξ3 ξ4 ξ1 ξ2 ξ3 ξ1 ξ1 ξ3 ξ3 ξ3 ξ3 ξ4 ξ3 ξ4 ξ3 ξ1 ξ1 ξ3 ξ3 ξ4 ξ4 ξ3 ξ2 ξ1 ξ4 ξ1 ξ2 ξ3 ξ4 ξ4 ξ4 ξ4 ξ4 ξ4 ξ4 ξ1 ξ2 ξ3 ξ4 Consider

 0.8 if ξ = ξ1  0.1 if ξ = ξ1 α(ξ) = 0.6 if ξ = ξ2 β(ξ) = 0.2 if ξ = ξ2 0.3 if ξ = ξ3, ξ4, 0.4 if ξ = ξ3, ξ4 then (R{= {ξ1, ξ2, ξ3, ξ4}, +, ·, ∨, ∧) is a Fermatean fuzzy L-ring ideal.

Theorem 1. Let ϕ be a homomorphism from a L-ring R into L-ring S and let ⟨αM, βM⟩ be a Fermatean fuzzy

−1 −1

left L- ring ideal of S. Then the pre-image ⟨ϕ (αM), ϕ (ηM)⟩ is a Fermatean fuzzy left L- ring ideal of R.

Proof. Consider a L-ring Homomorphism ϕ : R → S.

Let ⟨αM, βM⟩ be a Fermatean fuzzy left L- ring ideal of S. ∀ ξ, η ∈ R.

(i) ϕ (αM)(ξ − η) = αMϕ(ξ − η) ≥ min{αMϕ(ξ), αMϕ(η)}

−1 −1 −1

ϕ (αM)(ξ − η) ≥ min{ϕ (αM)(ξ), ϕ (αM)(η)}.

(ii) ϕ (αM)(ξη) = αMϕ(ξη) ≥ αM(ϕ(η) ≥ ϕ (αMϕ(η))

−1 −1

ϕ (αM)(ξη) ≥ ϕ (αM(η)}.

(iii) ϕ (αM)(ξ ∨ η) = αMϕ(ξ ∨ η) ≥ min{αMϕ(ξ), αMϕ(η)}

−1 −1 −1

ϕ (αM)(ξ ∨ η) ≥ min{ϕ (αM)(ξ), ϕ (αM)(η)}.

(iv) ϕ (αM)(ξ ∧ η) = αMϕ(ξ ∧ η) ≥ max{αMϕ(ξ), αMϕ(η)}

−1 −1 −1

ϕ (αM)(ξ ∨ η) ≥ max{ϕ (αM)(ξ), ϕ (αM)(η)}.

(v) ϕ (βM)(ξ − η) = βMϕ(ξ − η) ≤ max{βMϕ(ξ), βMϕ(η)}

−1 −1 −1

ϕ (βM)(ξ − η) ≤ max{ϕ (βM)(ξ), ϕ (βM)(η)}.

(vi) ϕ (βM)(ξη) = βMϕ(ξη) ≤ βM(ϕ(η) ≤ ϕ (βMϕ(η))

−1 −1

ϕ (αM)(ξη) ≥ ϕ (βM(η)}.

(vii) ϕ (βM)(ξ ∨ η) = βMϕ(ξ ∨ η) ≤ max{βMϕ(ξ), βMϕ(η)}

−1 −1 −1

ϕ (βM)(ξ ∨ η) ≤ max{ϕ (βM)(ξ), ϕ (βM)(η)}.

(viii) ϕ (βM)(ξ ∧ η) = βMϕ(ξ ∧ η) ≤ min{βMϕ(ξ), βMϕ(η)}

−1 −1 −1

ϕ (βM)(ξ ∨ η) ≤ min{ϕ (βM)(ξ), ϕ (βM)(η)}.

−1 −1

∴ ϕ (αM), ϕ (βM) is a Fermatean fuzzy left L- ring ideal of R.

Theorem 2. Let ϕ : R → S be a homomorphism from a L-ring R into S. If αM, βM is a Fermatean fuzzy

−1 −1

right L- ring ideal of S, then pre-image ϕ (αM) and ϕ (βM) is a Fermatean fuzzy right L- ring ideal of R. Proof. Consider homomorphism ϕ : R → S, where R and S be L-ring. Let αM, βM be a Fermatean fuzzy right L-ring ideal of S ∀ ξ, η ∈ R.

(i) ϕ (αM)(ξ − η) = αMϕ(ξ − η) ≥ min{αMϕ(ξ), αMϕ(η)}

−1 −1 −1

ϕ (αM)(ξ − η) ≥ min{ϕ (αM)(ξ), ϕ (αM)(η)}.

(ii) ϕ (αM)(ξη) = αM(ϕ(ξη)) ≥ αM(ϕ(ξ)

−1 −1

= ϕ (αM)(ξ))ϕ (αM)(ξη) ≥ ϕ ((αM)(ξ)}.

(iii) ϕ (αM)(ξ ∨ η) = αMϕ(ξ ∨ η) ≥ min{αMϕ(ξ), αMϕ(η)}

−1 −1 −1

ϕ (αM)(ξ ∨ η) ≥ min{ϕ (αM)(ξ), ϕ (αM)(η)}.

(iv) ϕ (αM)(ξ ∧ η) = αM(ϕ(ξ ∧ η)) ≥ max{αMϕ(ξ), αMϕ(η)}

−1 −1 −1

ϕ (αM)(ξ ∨ η) ≥ max{ϕ (αM)(ξ), ϕ (αM)(η)}.

(v) ϕ (βM)(ξ − η) = βMϕ(ξ − η) ≤ max{βMϕ(ξ), βMϕ(η)}

−1 −1 −1

ϕ (βM)(ξ − η) ≤ max{ϕ (βM)(ξ), ϕ (βM)(η)}.

(vi) ϕ (βM)(ξη) = βM(ϕ(ξη)) ≤ βM(ϕ(ξ) ≤ ϕ (βMϕ(ξ))

−1 −1

ϕ (αM)(ξη) ≥ ϕ (βM(η)}.

(vii) ϕ (βM)(ξ ∨ η) = βMϕ(ξ ∨ η) ≤ max{βMϕ(ξ), βMϕ(η)}

−1 −1 −1

ϕ (βM)(ξ ∨ η) ≤ max{ϕ ((βM)(ξ)), ϕ (βM(η))}.

(viii) ϕ (βM)(ξ ∧ η) = βMϕ(ξ ∧ η) ≤ min{βMϕ(ξ), βMϕ(η)}

−1 −1 −1

ϕ (βM)(ξ ∨ η) ≤ min{ϕ (βM(ξ)), ϕ (βM(η))}.

−1 −1

∴ ϕ (αM), ϕ (βM) is a Fermatean fuzzy right L- ring ideal of R.

Theorem 3. Let ϕ : R → S be a homomorphism from a L-ring R into L-ring S. If αM, βM is a Fermatean

−1 −1

fuzzy L- ring ideal of S, then pre-image ϕ (αM), ϕ (βM) is a Fermatean fuzzy right L- ring ideal of R.

Proof. It’s trivial.

Theorem 4. Let R and S be L-ring and ϕ : R → S be a homomorphism. If αM, βM is a Fermatean fuzzy left L- ring ideal of a L-ring R, with sup property, then the pre-image ϕ(αM) and ϕ(βM) is a Fermatean fuzzy left L-ring ideal of S.

Proof. Consider homomorphism ϕ : R → S, where R and S be L-ring. Let αM, βM be a Fermatean fuzzy left L- ring ideal of R ∀ ξ, η ∈ R.

(i) ϕ(αM)(ϕ(ξ) − ϕ(η)) = ϕ(αM)ϕ(ξ − η) ≥ min{αM(ξ), αM(η)} ϕ(αM)(ϕ(ξ) − ϕ(η)) ≥ min{ϕ(αM)(ξ), ϕ(αM)(η)}.

(ii) ϕ(αM)(ϕ(ξ)ϕ(η)) = ϕ(αM)(ϕ(ξη)) ≥ αM(η) = ϕ(αM)ϕ(η)) ϕ(αM)(ϕ(ξ)ϕ(η)) ≥ ϕ(αM)(ϕ(η).

(iii) ϕ(αM)(ϕ(ξ) ∨ ϕ(η)) = ϕ(αM)ϕ(ξ ∨ η) ≥ min{αM(ξ), αM(η)} ϕ(αM)(ϕ(ξ) ∨ ϕ(η)) ≥ min{ϕ(αM)(ξ), ϕ(αM)(η)}.

(iv) ϕ(αM)(ϕ(ξ) ∧ ϕ(η)) = ϕ(αM)(ϕ(ξ ∧ η)) ≥ max{αM(ξ), αM(η)} ϕ(αM)(ϕ(ξ) ∧ ϕ(η)) ≥ max{ϕ(αM)(ξ), ϕ(αM)(η)}.

(v) ϕ(βM)(ϕ(ξ) − ϕ(η)) = ϕ(βM)ϕ(ξ − η) ≤ max{βM(ξ), βM(η)} ϕ(βM)(ϕ(ξ) − ϕ(η)) ≤ max{ϕ(βM)(ξ), ϕ(βM)(η)}.

(vi) ϕ(βM)(ϕ(ξ)ϕ(η)) = ϕ(βM)ϕ(ξη) ≤ βM(ϕ(η)) ≤ ϕ(βMϕ(η)) ϕ(βM)(ϕ(ξ)ϕ(η)) ≥ ϕ(βM)(η)).

(vii) ϕ(βM)(ϕ(ξ) ∨ ϕ(η)) = ϕ(βM)ϕ(ξ ∨ η) ≤ max{βM(ξ), βM(η)} ϕ(βM)(ϕ(ξ) ∨ ϕ(η)) ≤ max{ϕ(βM)(ξ)), ϕ(βM)(η)}.

(viii) ϕ(βM)(ϕ(ξ) ∧ ϕ(η)) = ϕ(βM)(ϕ(ξ ∧ η)) ≤ min{βM(ξ), βM(η)} ϕ(βM)(ϕ(ξ) ∧ ϕ(η)) ≤ min{ϕ(βM)(ξ), ϕ(βM)(η)}.

∴ ϕ(αM), ϕ(βM) is a Fermatean fuzzy right L- ring ideal of S.

Theorem 5. Let ϕ : R → S be a homomorphism from a L-ring R into L-ring S. If αM, βM is a Fermatean fuzzy right L- ring ideal of R, with sup property, then the pre-image ϕ(αM) and ϕ(βM) is a Fermatean fuzzy right L-ring ideal of S.

Proof. Consider a L-ring homomorphism ϕ : R → S.

Let αM, βM be a Fermatean fuzzy right L- ring ideal of R ∀ ξ, η ∈ R.

(i) ϕ(αM)(ϕ(ξ) − ϕ(η)) = ϕ(αM)ϕ(ξ − η) ≥ min{αM(ξ), αM(η)} ϕ(αM)(ϕ(ξ) − ϕ(η)) ≥ min{ϕ(αM)(ξ), ϕ(αM)(η)}.

(ii) ϕ(αM)(ϕ(ξ)ϕ(η)) = ϕ(αM)(ϕ(ξη)) ≥ αM(ξ) = ϕ(αM)ϕ(ξ)) ϕ(αM)(ϕ(ξ)ϕ(η)) ≥ ϕ(αM)(ϕ(ξ).

(iii) ϕ(αM)(ϕ(ξ) ∨ ϕ(η)) = ϕ(αM)ϕ(ξ ∨ η) ≥ min{αM(ξ), αM(η)} ϕ(αM)(ϕ(ξ) ∨ ϕ(η)) ≥ min{ϕ(αM)(ξ), ϕ(αM)(η)}.

(iv) ϕ(αM)(ϕ(ξ) ∧ ϕ(η)) = ϕ(αM)(ϕ(ξ ∧ η)) ≥ max{αM(ξ), αM(η)} ϕ(αM)(ϕ(ξ) ∧ ϕ(η)) ≥ max{ϕ(αM)(ξ), ϕ(αM)(η)}.

(v) ϕ(βM)(ϕ(ξ) − ϕ(η)) = ϕ(βM)ϕ(ξ − η) ≤ max{βM(ξ), βM(η)} ϕ(βM)(ϕ(ξ) − ϕ(η)) ≤ max{ϕ(βM)(ξ), ϕ(βM)(η)}.

(vi) ϕ(βM)(ϕ(ξ)ϕ(η)) = ϕ(βM)ϕ(ξη) ≤ βM(ϕ(ξ)) ≤ ϕ(βM(ξ)) ϕ(βM)(ϕ(ξ)ϕ(η)) ≥ ϕ(βM)(ϕ(ξ)).

(vii) ϕ(βM)(ϕ(ξ) ∨ ϕ(η)) = ϕ(βM)ϕ(ξ ∨ η) ≤ max{βM(ξ), βM(η)} ϕ(βM)(ϕ(ξ) ∨ ϕ(η)) ≤ max{ϕ(βM)(ξ), ϕ(βM)(η)}.

(viii) ϕ(βM)(ϕ(ξ) ∧ ϕ(η)) = ϕ(βM)(ϕ(ξ ∧ η)) ≤ min{βM(ξ), βM(η)} ϕ(βM)(ϕ(ξ) ∧ ϕ(η)) ≤ min{ϕ(βM)(ξ), ϕ(βM)(η)}.

∴ ϕ(αM), ϕ(βM) is a Fermatean fuzzy right L- ring ideal of S.

Theorem 6. Let ϕ : R → S be homomorphism from L-ring R into L-ring S. If αM, βM is an Fermatean fuzzy L- ring ideal of a L-ring R with sup property, then pre-image ϕ(αM), ϕ(βM) is a fuzzy L- ring ideal of L-ring

Proof. It’s trivial.

Within this detailed study, we define lattice concept of Fermatean fuzzy sets along with their crucial results in the Fermatean fuzzy sets context. The left and right ideals of Fermatean fuzzy sets over lattice is initiated. The concept of homomorphism on Fermatean fuzzy L-ring ideal are introduced. With the help of supremum and infimum property of Fermatean fuzzy sets, some important results related to image and pre-image of homomorphism of Fermatean fuzzy L-ring ideal has been investigated. To extend this work, one can discuss some characterization on anti-homomorphism properties of Fermatean fuzzy L-ring ideal.

The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.

All authors have read and agreed to the published version of the manuscript.

The 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 findings in this research paper are provided within the manuscript.

The authors declare that there is no conflict of interest concerning the reported research findings.