Bipolar Fuzzy Sublattices and Ideals

. In this article, we introduce and study the theory of bipolar fuzzy sublattices (BFLs) and bipolar fuzzy ideals (BFIs) of a lattice, and some interesting properties of these BFLs and BFIs are established. Moreover, we study the properties of BFIs under lattice homomorphisms and also an application of BFLs.


Introduction
Zadeh [4] introduced the concept of fuzzy sets (FSs) in 1965, and it has become a thriving area of research in a variety of fields. Following that, several researchers applied this concept to various algebraic structures. The fuzzy set theory has various expansions, such as vague sets (VSs), intervalvalued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFS) and so on. The IFS was introduced by Atanassov [2] in 1986 as a generalization of the FS. In both the FS and IFS, the membership value range is in [0,1]. Later, Ajmal and Thomas [5] specifically applied the concept of FSs in lattice theory and developed the theory of fuzzy sublattices (FSLs). Thereafter, Thomas and Nair [3] introduced the concept of intuitionistic fuzzy sublattices (IFSLs) in 2011. Characterization of intuitionistic fuzzy ideals and filters based on lattice operations were studied by Milles [11] in 2017. Later, rough vague lattices were studied by Rao [13] in 2019. Vague lattices were introduced by Rao [12] in 2020. In 2020, Milles [8,9] researched on the principal intuitionistic fuzzy ideals and filters on a lattice and the lattice of intuitionistic fuzzy topologies generated by intuitionistic fuzzy relations. Zhang [10] studied intuitionistic fuzzy filters on residuated lattices. Nowadays, bipolarity is playing a vital role in many areas. This has become a thriving area of research in many fields like artificial intelligence (AI), machine learning (ML) etc. Lee [1] introduced the concept of bipolar fuzzy sets (BFSs) in 2000, with membership values ranging from [-1,1]. Eswarlal and Kalyani [6,7] investigated bipolar vague cosets, homomorphism, and anti homomorphism in bipolar vague normal groups (BVNGs), and used bipolar vague sets (BVSs) to solve MCDM problems.
In this paper, we introduce the concepts of BFLs and BFIs of a lattice. Some interesting characterizations and properties of these BFLs and BFIs are established. In addition, we study the properties of BFIs under lattice homomorphisms and also an application of BFLs.
Here, we will review a few standard definitions that are relevant to this work.
The intersection of B δ and B ω , denoted by B δ ∩ B ω , is a BFS in X defined as: for each T ∈ ). (iv) The union of B δ and B ω , denoted by B δ ∪ B ω , is a BFS in X defined as: for each T ∈ X, (B δ ∪

Bipolar fuzzy sublattices and ideals
In this section, we introduce and study BFLs and BFIs and their characterizations.
Then for all T , k ∈ L, the following conditions are equivalent: Proof. For any T , k ∈ L, we have T ∧ k ≤ T and T ∧ k ≤ k.
Suppose that (ii) is true. Let T , k ∈ L be such that T ≤ k.
. Hence, the proof is completed. Similar to Theorem 3.1, we get the following theorem.
Then for all T , k ∈ L, the following conditions are equivalent: Then B δ is called a bipolar fuzzy sublattice (BFL) of L if the following conditions are satisfied for all T , k ∈ L, We can routinely prove that B δ is a BFL of L.
Then B δ is called a bipolar fuzzy ideal (BFI) of L if the following conditions are satisfied for all T , k ∈ L, Similarly, we get Similarly, we get Every BFI of L is a BFL, but the converse need not be true. Consider the lattice L given in Example 3.1. Then the BFI given by is a BFL of L. But the BFL given by  Then for all T , k ∈ L, the following four statements hold: Proof. Let T , k ∈ L.

Bipolar fuzzy ideals under lattice homomorphisms
Definition 4.1. Let θ : L → L 1 be a mapping and B δ = (B P δ , B N δ ) be a BFS in L. Then the image Hence, θ(B δ ) is a BFI of L 1 . .
Theorem 4.3. Let θ : L → L 1 be a homomorphism and let µ and η be BFLs of L and L 1 , respectively.
Proof. The proof is omitted since it follows the same proof of Theorems 4.1 and 4.2.
Theorem 4.4. If θ : L → L 1 is an surjection and B η , B δ are BFSs of L and L 1 , respectively, then    and Ψ(B 1 δ ) = f −1 (B 1 δ ). By Theorems 4.1 and 4.2, we have ς and Ψ are well-defined. Also by Theorems 4.4 and 4.5, we have ς and Ψ are the inverse to each other which gives that the oneto-one correspondence. Also by Theorem 4.6, we get B δ ⊆ B η ⇒ f (B δ ) ⊆ f (B η ). Hence, the correspondence is order preserving.

An application of bipolar fuzzy sublattices
The single pattern: the one-minute microwave [15].
The one-minute microwave is a simple system with the following requirements: between B and D then it is considered the button is unpressed. In this case, there is no question about whether the timer is initiated or not. So, B ∧ D = D.
Similarly, if we take the operator 'ON' between T and D then it is considered the timer is initiated and oven-door is closed. Then cooking will be done. So, T ∨ D = T .
If we take the operator 'OFF' between T and D then it is considered the timer is uninitiated and door is open. So, T ∧ D = B (here the minimum considered to be B, because the possibility that the timer is uninitiated is that the button is unpressed). We can routinely prove that B δ is a BFL of L.

Conclusion and future work
In this article, we have introduced the concepts of BFLs and BFIs of a lattice. Interesting properties of these BFLs and BFIs are developed. Moreover, we investigated the properties of BFIs under lattice homomorphism and an application of BFLs is given.
Our future work is to develop the bipolar fuzzy prime ideals, bipolar fuzzy principal ideals, quotient ideals, bipolar fuzzy filters, and bipolar fuzzy prime filters of a lattice.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.