Hyperstructures are crucial in mathematics for both theoretical and practical applications. The two-element composition forms a set, which is a specialty in algebraic hyperstructures because, in classical algebraic structures, two-element compositions form other elements again. To improve hyperstructure theory, Marty [1] introduced the notion of hypergroup, which was the first step in hyperstructure theory. Plenty of concepts developed under algebraic hyperstructures, such as weak hyperstructures, hypergroups, hyperrings, and hypermodules [2–4]. Konstantinidou and Mittas [5] and Mittas introduced the concept of hyperlattice and established the theory of modular, distributive, and complementary hyperlattices [6,7]. Rasouli and Davvaz [8] defined the fundamental relations in hyperlattice and obtained a lattice from the hyperlattice.

In 1965, Zadeh [9] introduced fuzzy set theory as a versatile solution for addressing variability in problems. They presented fuzzy algebraic operations. Fuzzy set theory is crucial for establishing relations in 0, 1, and multiple domains within hyperstructure theory. This has led to broader research for algebraic hyperstructures and fuzzy-set combinations [10,11]. Many scientists investigated the perception of associating a fuzzy set with a hyperstructure [12–14]. He and Xin [15] introduced the concept of a fuzzy hyperlattice and derived the connections between hyperlattices and fuzzy hyperlattices. In 1998, Rosenberg [16] considered connections between hyperstructures and binary relationships. To connect fuzzy hyperstructures and binary relations, Preethi et al. [17] introduced a concept called the fuzzy hyperlattice-ordered group (FHLOG) and studied the properties and applications of FHLOGs [17–20].

Objectives of the study are as follows:

• • Fuzzy theory is immensely useful to people involved in research and development in fields such as mathematicians, researchers, computer software developers and engineers. Comparison of many domains is achievable through hyperstructure theory. Therefore, we can promote a broad range of practical applications under the topic hyperlattice-ordered group (HLOG) and the FHLOG.

• • By analysing the relation between FHLOG and HLOG, we can develop both concepts in detail. This provides a vast idea for extending applications to these topics.

The remainder of this paper is organized as follows: Section 2 covers some basic concepts, Section 3 introduces the new concept of HLOG with a practical example, Section 4 explores the relation between the FHLOG and HLOG, and Section 5 concludes the paper.

Algebraic hyperstructures are generalizations of classical algebraic structures [2, 15, 21]. Let ℋ be a non-empty set, and ℘^*(ℋ) be the set of all non-empty subsets of ℋ. Now, consider the maps f_k : ℋ×ℋ → ℘^*(ℋ), where k is a positive integer ranging from 1 to n. Maps f_k are called (binary) hyperoperations. An algebraic system, (ℋ, f₁, .., f_n) is called a (binary) hyperstructure.

Let ℒ be a non-empty set, and ℘^*(ℒ) be the set of all non-empty subsets of ℒ. A hyperoperation on ℒ is a map ⊕ : ℒ×ℒ →℘^*(ℒ), which associates a non-empty subset i ⊕ j with any pair (i, j) of elements of ℒ×ℒ.

If and ℬ are non-empty subsets of ℒ for i, j, u ∈ ℒ, then

(1) ;

(2) .

Definition 2.1 [15, 22]

Let ℒ be a non-empty set with two hyperoperations “⊕” and “⊗”. A triplet (ℒ, ⊕, ⊗) is called a hyperlattice if ∀ i, j, k ∈ ℒ

• (1) (Idempotent laws) i ∈ i ⊗ i, i ∈ i ⊕ i;

• (2) (Commutative laws) i ⊗ j = j ⊗ i and i ⊕ j = j ⊕ i;

• (3) (Associative laws) (i ⊗ j) ⊗ k = i ⊗ (j ⊗ k), (i ⊕ j) ⊕ k = i ⊕ (j ⊕ k);

• (4) (Absorption laws) i ∈ i ⊗ (i ⊕ j), i ∈ i ⊕ (i ⊗ j).

Remark 1 [22]

Each lattice is a hyperlattice.

Definition 2.2 [23]

Let (ℒ,∨,∧) be a hyperlattice. We call (ℒ,≤) an ordered hyperlattice if: ≤ is a partial-order relationship and u ≤ v implies that u∨w ≤ v∨w and u∧w ≤ v∧w.

Definition 2.3 [15]

Let ℒ be a non-empty set endowed with two fuzzy hyperoperations ⊗ and ⊕ (where ℱ^*(ℒ) is the set of all nonzero fuzzy subsets of ℒ. The mapping of fuzzy hyperoperation is of the form ◦ : ℒ × ℒ → ℱ^*(ℒ) ). The triplet (ℒ, ⊗, ⊕) is called a fuzzy hyperlattice if ∀ i, j, k ∈ ℒ

• (i) (i ⊗ i)(i) > 0, (i ⊕ i)(i) > 0.

• (ii) i ⊗ j = j ⊗ i, i ⊕ j = j ⊕ i.

• (iii) (i ⊗ j) ⊗ k = i ⊗ (j ⊗ k) and (i ⊕ j) ⊕ k = i ⊕ (j ⊕ k).

• (iv) (i ⊗ (i ⊕ j))(i) > 0, (i ⊕ (i ⊗ j))(i) > 0.

Note [15]

If and are two nonzero fuzzy subsets ∀i, u ∈ ℒ then we define

• (i) ,

• (ii) .

Theorem 2.4 [15]

Consider a fuzzy hyperlattice (ℒ, ×, +) and define the hyperoperations on ℒ as ∀ i, j ∈ ℒ, i ⊗ j = {u ∈ ℒ|(i × j)(u) > 0} and i ⊕ j = {u ∈ ℒ|(i + j)(u) > 0}. If (ℒ, ×, +) is a fuzzy hyperlattice, (ℒ, ⊗, ⊕) is a hyperlattice.

Theorem 2.5 [15]

Consider a hyperlattice (ℒ, ⊗, ⊕) and define the fuzzy hyperoperations on ℒ as ∀ i, j ∈ ℒ, i × j = χ_i⊗j and i + j = χ_i⊕j. If (ℒ, ⊗, ⊕) is a hyperlattice and (ℒ, ×, +) is a fuzzy hyperlattice.

Definition 2.6 [17]

A non-empty set is called a FHLOG iff ∀i, u,

• (i) is the group.

• (ii) denotes a fuzzy hyperlattice.

• (iii) R[{i} + (u ⊕ v), {i + u} ⊕ {i + v}] = 1 and R[{i} + (u ⊗ v), {i + u} ⊗ {i + v}] = 1.

Here, denotes the fuzzy partial-order relationship.

Example 2.7 [17]

(Q⁺ \ {0}, *, ⊗, ⊕) with the fuzzy hyperoperations defined by i ⊕ j = χ_{i,j} and i ⊗ j = χ_(i∧j), ∀i, j ∈ Q⁺ \ {0} and R : Q⁺ \ {0} × Q⁺ \ {0} → [0, 1] is a fuzzy partial-order relation. (Q⁺ \ {0}, *, ⊗, ⊕) is an example of FHLOG.

In this section, we define a HLOG and discuss its fundamental properties. In this section, we use symbols ⊙, ⊖, ∨, and ∧ to represent hyperoperations. We define the hyperoperations ∨, ∧ on ℋ as follows: For all i, j ∈ ℋ, i∨j = {i ∨ j}, i∧j = {i ∧ j}.

Definition 3.1

A non-empty set ℋ is called a HLOG iff ∀i, u, v ∈ ℋ

• (i) (ℋ, +) denotes the group.

• (ii) (ℋ, ⊙, ⊖) is a hyperlattice.

• (iii) {i} + (u ⊙ v) = {i + u} ⊙ {i + v} and {i} + (u ⊖ v) = {i + u} ⊖ {i + v}.

Example 3.2

(ℝ, *, ⊖, ⊙) with the hyperoperations defined by i ⊙ j = {u ∈ ℝ|u ≤ i, u ≤ j} and i ⊖ j = {u ∈ ℝ|i ≤ u, j ≤ u}, ∀i, j ∈ ℝ. (ℝ, *, ⊖, ⊙) is an example of HLOG.

Example 3.3

Consider a flat square card in real 3D space (ℝ³-space) to construct a rotation group. Rotate the card by π radians, i.e., 180^°, around the , and axes. Denote these rotations as (r₁, r₂, r₃), with the do-nothing operation denoted by e. If we rotate the card by r₁, followed by r₂ rotation, we obtain r₃ rotation. Table 1 shows the Cayley table for card rotations in ℝ³-space. Consider (r₁, r₂, r₃, e) to be a set ℋ. Figure 1 shows the rotation in ℝ³-space. Here, the rotation group is an example of an HLOG with hyperoperations defined by ∀a, b ∈ ℋ, a ⊖ b = {a, b} and r₁ ⊙ r₃ = {ℋ}. (i.e.). (ℋ, *, ⊖, ⊙) is the HLOG.

Remark 2

• (i) HLOG is distributive.

• (ii) Let ℋ be any HLOG. Then, ∀ i, j, k ∈ ℋ, {i∧j} – {i∧k} ≤ {j –k}.

• (iii) Let ℋ be any HLOG. Then, ∀ i, j ∈ ℋ {i ⊙ j} = ({i –j} ⊙ {0}) + {j} and {i ⊖ j} = ({i –j} ⊖ {0}) + {j}.

• (iv) Let ℋ be any HLOG. Then, ∀ i, j, k ∈ ℋ {i ⊙ j} – {k} = {i –k} ⊙ {j –k} and {i ⊖ j} – {k} = {i –k} ⊖ {j –k}.

Definition 3.4

A non-empty set ℋ is called an HLOG iff ∀i, u, v, w ∈ ℋ

• (i) (ℋ, .) is a group.

• (ii) (ℋ, ⊙, ⊖) is a hyperlattice.

• (iii) u ≤ v ⇒ (i.u) ⊙ w ≤ (i.v) ⊙ w

  
  

  u ≤ v ⇒ (i.u) ⊖ w ≤ (i.v) ⊖ w.

Theorem 3.5

The two definitions of HLOG are equivalent.

Theorem 3.6

Let ℋ be any HLOG, then ∀ i, j, k ∈ ℋ, {i} – (j∨k) = {i –j}∧{i –k}.

Theorem 3.7

Let ℋ be any HLOG, then ∀ i, j ∈ ℋ, {i} – {i∧j} + {j} = {j}∨{i}.

Corollary 3.8

Letℋ be a HLOG, ∀ i, j ∈ ℋ. The following are equivalent.

• (i) {i∧j} = {0}.

• (ii) {i + j} = {i∨j}.

Corollary 3.9

Let ℋ be any HLOG, then ∀ i, j ∈ ℋ, {i} + {j} = {i∨j} + {j∧i}.

In this section, the concepts of FHLOGs and HLOG are related by using suitable theorems.

Theorem 4.1

Let ∀ i, j ∈ ℋ, the fuzzy hyperoperations are defined by i ⊗ j = χ_{i,j} and i ⊕ j = χ_{i∧j} imply that (ℋ, ⊗, ⊕) is a fuzzy hyperlattice. In ℋ the hyperoperations are defined by ∀i, j ∈ ℋ,

and

If (ℋ, *, ⊗, ⊕) is a FHLOG, then (ℋ, *, ⊖, ⊙) is a HLOG.

Theorem 4.2

Let (ℋ, *, ⊖, ⊙) be a HLOG and the fuzzy hyperoperations on ℋ defined by i ⊗ j = χ_{i⊖j} and i ⊕ j = χ_{i⊙j}, ∀ i, j ∈ ℋ, If (ℋ, *, ⊖, ⊙) is an HLOG then (ℋ, *, ⊗, ⊕) is an FHLOG.

Theorem 4.3

Let (ℋ, *, ⊗, ⊕) be FHLOG and let the hyperoperations on ℋ be defined by ∀i, j ∈ ℋ,

and

If (ℋ, *, ⊗, ⊕) is a FHLOG then (ℋ, *, ⊖, ⊙) is a HLOG.

In this study, we proposed a concept called HLOG with a practical example and conjointly studied the relationship between FHLOG and HLOG. We conclude that under certain hyperoperations and fuzzy hyperoperations, FHLOG acts as HLOG and vice versa. However, a limitation of this study is that the general structures of hyperoperations and fuzzy hyperoperations are yet to be defined. In future work, we will extend this theory by introducing a general case of hyperstructures. Moreover, the theory can be applied to solve decision-making problems.

Table 1. Cayley table.