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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 418-424

Published online December 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.4.418

© The Korean Institute of Intelligent Systems

Relation Amidst Fuzzy Hyperlattice-Ordered Group and Hyperlattice Ordered Group

Vimala Jayakumar1, D. Preethi2, S. Rajareega3, and Seyyed Ahmad Edalatpanah4

1Department of Mathematics, Alagappa University, Karaikudi, India
2Department of Mathematics, Mohamed Sathak Engineering College, Kilakarai, India
3Department of Basic Sciences and Humanities, GMR Institute of Technology, Rajam, India
4Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran

Correspondence to :
Seyyed Ahmad Edalatpanah (saedalatpanah@gmail.com)

Received: October 30, 2022; Revised: February 6, 2023; Accepted: October 16, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper introduces a new concept called hyperlattice-ordered group with a practical example and some properties. Additionally, we explore the features of algebraic hyperstructures and fuzzy algebraic hyperstructures. Furthermore, the relationship between the fuzzy hyperlattice-ordered group and the hyperlattice-ordered group is investigated.

Keywords: Hyperlattice, Fuzzy hyperlattice, Hyperlattice-ordered group, Fuzzy hyperlattice-ordered group

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 [24]. 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 [1214]. 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 [1720].

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 fk : ℋ×ℋ → ℘*(ℋ), where k is a positive integer ranging from 1 to n. Maps fk are called (binary) hyperoperations. An algebraic system, (ℋ, f1, .., fn) 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 ij 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) iii, iii;

• (2) (Commutative laws) ij = ji and ij = ji;

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

• (4) (Absorption laws) ii ⊗ (ij), ii ⊕ (ij).

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 uv implies that uwvw and uwvw.

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) (ii)(i) > 0, (ii)(i) > 0.

• (ii) ij = ji, ij = ji.

• (iii) (ij) ⊗ k = i ⊗ (jk) and (ij) ⊕ k = i ⊕ (jk).

• (iv) (i ⊗ (ij))(i) > 0, (i ⊕ (ij))(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, ij = {u ∈ ℒ|(i × j)(u) > 0} and ij = {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 = χij and i + j = χij. 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} + (uv), {i + u} ⊕ {i + v}] = 1 and R[{i} + (uv), {i + u} ⊗ {i + v}] = 1.

Here, denotes the fuzzy partial-order relationship.

Example 2.7 [17]

(Q+ \ {0}, *, ⊗, ⊕) with the fuzzy hyperoperations defined by ij = χ{i,j} and ij = χ(ij), ∀i, jQ+ \ {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, ij = {ij}, ij = {ij}.

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

Example 3.2

(ℝ, *, ⊖, ⊙) with the hyperoperations defined by ij = {u ∈ ℝ|ui, uj} and ij = {u ∈ ℝ|iu, ju}, ∀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, ab = {a, b} and r1r3 = {ℋ}. (i.e.). (ℋ, *, ⊖, ⊙) is the HLOG.

Remark 2

• (i) HLOG is distributive.

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

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

• (iv) Let ℋ be any HLOG. Then, ∀ i, j, k ∈ ℋ {ij} – {k} = {i –k} ⊙ {j –k} and {ij} – {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) uv ⇒ (i.u) ⊙ w ≤ (i.v) ⊙ w

uv ⇒ (i.u) ⊖ w ≤ (i.v) ⊖ w.

Theorem 3.5

The two definitions of HLOG are equivalent.

Proof

Here, we define the hyperoperations as uv = {uv} and uv = {uv}. For all u, v ∈ ℋ.

To prove, the following two conditions are equivalent:

• (1) uv ⇒ (i.u) ⊙ w ≤ (i.v) ⊙ w.

• (2) {i}.(uv) = {i.u} ⊙ {i.v} for all i, u, v, w ∈ ℋ.

(1) ⇒ (2)

Assume that uv ⇒(i.u) ⊙ w ≤ (i.v) ⊙ w ⇒(i.u) ≤ (i.v).

Now, {i}.(uv) = {i.v}; {i.u} ⊙ {i.v} = {i.v}
∴ {i}.(uv) = {i.u} ⊙ {i.v}
(2) ⇒ (1)

We assume that {i}.(uv) = {i.u} ⊙ {i.v} and uv

{i}.(uv)={i.u}{i.v}{i}.{v}={i.u}{i.v}{i.v}={i.u}{i.v}{i.u}{i.v}{i.u}w{i.v}w.

Theorem 3.6

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

Theorem 3.7

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

Corollary 3.8

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

• (i) {ij} = {0}.

• (ii) {i + j} = {ij}.

Corollary 3.9

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

4. Relation Between Fuzzy Hyperlattice-Ordered Group and Hyperlattice-Ordered Group

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

ij={uij(u)>0},

and

ij={uij(u)>0}.

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

Proof

(i) Condition one of Definition 3.1 holds.

(ii) Proving that (ℋ, ⊖, ⊙) forms a hyperlattice.

• (a) Idempotent law ∀i ∈ ℋ

ii={uii(u)>0}={uχ{i,i}(u)>0}={i},iii.

Similarly, the idempotent law holds for ⊙.

• (b) Commutative law: ∀i, j ∈ ℋ

ij={uij(u)>0}={uχ{i,j}(u)>0}={uχ{j,i}(u)>0}={uji(u)>0}=ji.

Similarly, commutative law holds for ⊙.

• (c) Associative law: ∀i, j, k, u ∈ ℋ

If ui ⊖ (jk), then ∃ vjk such that uiv. This implies that χ{i,v}(u) > 0 and χ{j,k}(v) > 0⇒(iv)(u) > 0 and (jk)(v) > 0. Hence,

(i(jk))(u)=Supv{(iv)(u)(jk)(v)}(iv)(u)(jk)(v)>0.

Since i ⊗ (jk) = (ij) ⊗ k ⇒ ((ij) ⊗ k)(u) > 0

Supv∈ℋ{(ij)(v) ∧ (vk)(u)} > 0

Supv∈ℋ{χ{i,j}(v) ∧ χ{v,k}(u)} > 0.

Hence, v′ ∈ ℋ exists such that χ{i,j}(v′) > 0 and χ{v′,k}(u) >

0. ⇒ (ij)(v′) > 0 and (v′k)(u) > 0

v′ij, uv′k

u ∈ (ij) ⊖ k

i ⊖ (jk) ⊆ (ij) ⊖ k

Similarly, we have that (ij) ⊖ ki ⊖ (jk)

⇒ (ij) ⊖ k = i ⊖ (jk)

Similarly, the associative law holds true for ⊙.

• (d) Absorption law: ∀i, j ∈ ℋ

Since, (i ⊗ (ij))(i)= Supv∈ℋ{(iv)(i)∧(ij)(v)} > 0

Supv∈ℋ{χ{i,v}(i)∧χ{ij}(v)} > 0

v′ ∈ ℋ such that χ{i,v′}(i) > 0 and χ{ij}(v′) > 0

⇒ (iv′)(i) > 0 and (ij)(v′) > 0.

This shows that ∃ iiv′ and v′ij

ii ⊖ (ij).

Similarly, ii ⊙ (ij),

Therefore, (ℋ,⊖, ⊙) denotes a hyperlattice.

(iii) Assume that R[i * (uv), (i * u) ⊕ (i * v)] = 1 ⇒ i * (uv) = (i * u) ⊕ (i * v) and

R[i*(uv),(i*u)(i*v)]=1i*(uv)=(i*u)(i*v).

Then, (i * u) ⊕ (i * v) = χ{(i*u)(i*v)}.

v′ ∈ ℋ such that χ{(i*u)(i*v)}(v′) > 0 = (i * u) ⊙ (i * v)(I)

i * (uv) = (i * u) ⊕ (i * v) > 0 for any v′ ∈ ℋ.

u′ ∈ ℋ such that (i * (uv))(u′) > 0 = i * (uv) (II)

From (I) and (II), i * (uv) = (i * u) ⊙ (i * v). Similarly,

i * (uv) = (i * u) ⊖ (i * v).

Hence (ℋ, *, ⊖, ⊙) is HLOG.

Theorem 4.2

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

Proof

(i) Condition one of Definition 2.6 holds.

(ii) Proving that (ℋ, ⊗, ⊕) is a fuzzy hyperlattice.

The proof is obvious by Theorem 2.5.

(iii) Assume that i * (jk) = (i * j) ⊖ (i * k) and i * (jk) = (i * j) ⊙ (i * k), ∀ i, j, k ∈ ℋ.

To prove that R[i * (jk), (i * j) ⊗ (i * k)] = 1 and R[i * (jk), (i * j) ⊕ (i * k)] = 1

i*(jk)=i*χ{jk}=χ{(i*j)(i*k)}=(i*j)(i*k).

Therefore, R[i * (jk), (i * j) ⊗ (i * k)] = 1.

Similarly,

i*(jk)=i*χ{jk}=χ{(i*j)(i*k)}=(i*j)(i*k).

Therefore, R[i * (jk), (i * j) ⊕ (i * k)] = 1

Hence, (ℋ, *, ⊗, ⊕) is FHLOG.

Theorem 4.3

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

ij={uij(u)>0},

and

ij={uij(u)>0}.

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

Proof

(i) Condition one of Definition 3.1 holds.

(ii) Proving that (ℋ, ⊖, ⊙) is a hyperlattice.

The proof is obvious by Theorem 2.4.

(iii) Proving that i * (jk) = (i * j) ⊙ (i * k) and i * (jk) = (i * j) ⊙ (i * k).

Assume that ∀ i, j, k ∈ ℋ,

R[i*(jk),(i*j)(i*k)]=1i*(jk)=(i*j)(i*k),

and

R[i*(jk),(i*j)(i*k)]=1i*(jk)=(i*j)(i*k).i*(jk)={u(i*(j*k))(u)>0}={u((i*j)(i*k))(u)>0}by (III)=(i*j)(i*k)

Next,

i*(jk)={u(i*(jk))(u)>0}={u((i*j)(i*k))(u)>0}by (IV)=(i*j)(i*k)

Hence, (ℋ, *, ⊖, ⊙) is 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.

The article has been written with the joint financial support for RUSA-Phase 2.0 grant sanctioned vide letter No. F 24-51/2014-U, Policy (TN Multi-Gen), Department of Edn. Govt. of India, Dt. 09.10.2018, UGC-SAP (DRS-I) vide letters. No. F.510/8/DRS-I/2016 (SAP-I) Dt. 23.08.2016, DST-PURSE 2nd Phase programme (vide letters No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FST level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17 Dt. 20.12.2018.

No potential conflict of interest relevant to this article was reported.

Table. 1.

Table 1. Cayley table.

*r1r2r3e
r1er3r2r1
r2r3er1r2
r3r2r1er3
er1r2r3e

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Vimala Jayakumar received the Ph.D. degree in Mathematics from Alagappa University, Karaikudi, Tamil Nadu, India, in 2007. She has 19 years of teaching experience. She has presented many papers in various international conferences. She published three books. She has completed eight Ph.D.’s and currently five ongoing under her guidance. She has published more than 50 research articles in various reputed journals. Her research interests include algebra, lattice theory, fuzzy mathematics, algebraic hyperstructures, fuzzy algebraic hyperstructures, group theory, soft computing, neutrosophic sets, and multicriteria decision-making. She was awarded the “Best Researcher Award” in mathematics 2015–2016 by International Multidisciplinary Research Foundation (IMRF) and “Distinguished Women in Science 2017” by VIWA. She won the “Women Researcher Award” which was organized by International Organization of Scientific Research and Development. She has professionally visited Malaysia, Dubai, Singapore.

D. Preethi was born in Tamilnadu, India in 1996. She received the M.Sc. degree in Mathematics from the Alagappa University, Tamilnadu, India, in 2018 and the Ph.D. in Mathematics from Alagappa University, Taminadu, India, in 2022. She is currently an assistant professor in Department of Science and Humanities, Mohamed Sathak Engineering College, Kilakarai, Ramanathapuram District, Taminadu, India. She published articles in various international journals, many of which are indexed by Scopus and Web of Science, such as Journal of Intelligent & Fuzzy Systems, AIMS-Mathematics, MDPIMathematics, Journal of Discrete Mathematical Sciences and Cryptography. Her current research interests include fuzzy algebraic hyperstructures, lattice theory, complex fuzzy sets, decision support systems, fuzzy sets and its application, soft computing, multiple-attribute decision making, fuzzy algebra, and multiple-criteria group decision making.

S. Rajareega was born in Tamilnadu, India in 1996. She received the M.Sc. degree in Mathematics from the Alagappa University, Tamilnadu, India, in 2018 and the Ph.D. in Mathematics from Alagappa University, Taminadu, India, in 2022. She is currently an assistant professor in Department of Basic Science and Humanities, GMR Institute of Technology, Rajam, Andra Pradesh, India. She published articles in various international journals, many of which are indexed by Scopus and Web of Science, such as Journal of Intelligent & Fuzzy Systems, AIMS-Mathematics, MDPI-Mathematics, Journal of Discrete Mathematical Sciences and Cryptography. Her current research interests include complex fuzzy sets, decision support systems, fuzzy sets and its application, soft computing, multiple-attribute decision making, fuzzy algebra, and multiple-criteria group decision making.

Seyyed Ahmad Edalatpanah is an associate professor at the Ayandegan Institute of Higher Education, Tonekabon, Iran. He received his Ph.D. in Applied Mathematics from the University of Guilan, Rasht, Iran. He is currently working as the Chief of R&D at the Ayandegan Institute of Higher Education, Iran. He is also an academic member of Guilan University and the Islamic Azad University of Iran. Dr. Edalatpanah’s fields of interest include numerical computations, operational research, uncertainty, fuzzy set and its extensions, numerical linear algebra, soft computing, and optimization. He has published over 150 journal and conference proceedings papers in the above research areas. He serves on the editorial boards of several international journals. He is also the Directorin-Charge of the Journal of Fuzzy Extension & Applications (http://www.journal-fea.com/). Currently, he is president of International Society of Fuzzy Set Extensions and Applications (https://isfsea.org). Edalatpanah’s research is widely recognized internationally, he has been featured in the list of the Top 2% scientists in the world published by Stanford University from 2021 to present.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 418-424

Published online December 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.4.418

Copyright © The Korean Institute of Intelligent Systems.

Relation Amidst Fuzzy Hyperlattice-Ordered Group and Hyperlattice Ordered Group

Vimala Jayakumar1, D. Preethi2, S. Rajareega3, and Seyyed Ahmad Edalatpanah4

1Department of Mathematics, Alagappa University, Karaikudi, India
2Department of Mathematics, Mohamed Sathak Engineering College, Kilakarai, India
3Department of Basic Sciences and Humanities, GMR Institute of Technology, Rajam, India
4Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran

Correspondence to:Seyyed Ahmad Edalatpanah (saedalatpanah@gmail.com)

Received: October 30, 2022; Revised: February 6, 2023; Accepted: October 16, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper introduces a new concept called hyperlattice-ordered group with a practical example and some properties. Additionally, we explore the features of algebraic hyperstructures and fuzzy algebraic hyperstructures. Furthermore, the relationship between the fuzzy hyperlattice-ordered group and the hyperlattice-ordered group is investigated.

Keywords: Hyperlattice, Fuzzy hyperlattice, Hyperlattice-ordered group, Fuzzy hyperlattice-ordered group

1. Introduction

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 [24]. 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 [1214]. 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 [1720].

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.

2. Preliminaries

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 fk : ℋ×ℋ → ℘*(ℋ), where k is a positive integer ranging from 1 to n. Maps fk are called (binary) hyperoperations. An algebraic system, (ℋ, f1, .., fn) 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 ij 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) iii, iii;

• (2) (Commutative laws) ij = ji and ij = ji;

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

• (4) (Absorption laws) ii ⊗ (ij), ii ⊕ (ij).

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 uv implies that uwvw and uwvw.

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) (ii)(i) > 0, (ii)(i) > 0.

• (ii) ij = ji, ij = ji.

• (iii) (ij) ⊗ k = i ⊗ (jk) and (ij) ⊕ k = i ⊕ (jk).

• (iv) (i ⊗ (ij))(i) > 0, (i ⊕ (ij))(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, ij = {u ∈ ℒ|(i × j)(u) > 0} and ij = {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 = χij and i + j = χij. 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} + (uv), {i + u} ⊕ {i + v}] = 1 and R[{i} + (uv), {i + u} ⊗ {i + v}] = 1.

Here, denotes the fuzzy partial-order relationship.

Example 2.7 [17]

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

3. Hyperlattice-Ordered Group

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, ij = {ij}, ij = {ij}.

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

Example 3.2

(ℝ, *, ⊖, ⊙) with the hyperoperations defined by ij = {u ∈ ℝ|ui, uj} and ij = {u ∈ ℝ|iu, ju}, ∀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, ab = {a, b} and r1r3 = {ℋ}. (i.e.). (ℋ, *, ⊖, ⊙) is the HLOG.

Remark 2

• (i) HLOG is distributive.

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

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

• (iv) Let ℋ be any HLOG. Then, ∀ i, j, k ∈ ℋ {ij} – {k} = {i –k} ⊙ {j –k} and {ij} – {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) uv ⇒ (i.u) ⊙ w ≤ (i.v) ⊙ w

uv ⇒ (i.u) ⊖ w ≤ (i.v) ⊖ w.

Theorem 3.5

The two definitions of HLOG are equivalent.

Proof

Here, we define the hyperoperations as uv = {uv} and uv = {uv}. For all u, v ∈ ℋ.

To prove, the following two conditions are equivalent:

• (1) uv ⇒ (i.u) ⊙ w ≤ (i.v) ⊙ w.

• (2) {i}.(uv) = {i.u} ⊙ {i.v} for all i, u, v, w ∈ ℋ.

(1) ⇒ (2)

Assume that uv ⇒(i.u) ⊙ w ≤ (i.v) ⊙ w ⇒(i.u) ≤ (i.v).

Now, {i}.(uv) = {i.v}; {i.u} ⊙ {i.v} = {i.v}
∴ {i}.(uv) = {i.u} ⊙ {i.v}
(2) ⇒ (1)

We assume that {i}.(uv) = {i.u} ⊙ {i.v} and uv

${i}.(u⊙v)={i.u}⊙{i.v}⇒{i}.{v}={i.u}⊙{i.v}⇒{i.v}={i.u}⊙{i.v}⇒{i.u}≤{i.v}⇒{i.u}⊙w≤{i.v}⊙w.$

Theorem 3.6

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

Theorem 3.7

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

Corollary 3.8

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

• (i) {ij} = {0}.

• (ii) {i + j} = {ij}.

Corollary 3.9

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

4. Relation Between Fuzzy Hyperlattice-Ordered Group and Hyperlattice-Ordered Group

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

$i⊖j={u∈ℋ∣i⊗j(u)>0},$

and

$i⊙j={u∈ℋ∣i⊗j(u)>0}.$

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

Proof

(i) Condition one of Definition 3.1 holds.

(ii) Proving that (ℋ, ⊖, ⊙) forms a hyperlattice.

• (a) Idempotent law ∀i ∈ ℋ

$i⊖i={u∈ℋ∣i⊗i(u)>0}={u∈ℋ∣χ{i,i}(u)>0}={i},∴i∈i⊖i.$

Similarly, the idempotent law holds for ⊙.

• (b) Commutative law: ∀i, j ∈ ℋ

$i⊖j={u∈ℋ∣i⊗j(u)>0}={u∈ℋ∣χ{i,j}(u)>0}={u∈ℋ∣χ{j,i}(u)>0}={u∈ℋ∣j⊗i(u)>0}=j⊖i.$

Similarly, commutative law holds for ⊙.

• (c) Associative law: ∀i, j, k, u ∈ ℋ

If ui ⊖ (jk), then ∃ vjk such that uiv. This implies that χ{i,v}(u) > 0 and χ{j,k}(v) > 0⇒(iv)(u) > 0 and (jk)(v) > 0. Hence,

$(i⊗(j⊗k))(u)=Supv∈ℋ{(i⊗v)(u)∧(j⊗k)(v)}≥(i⊗v)(u)∧(j⊗k)(v)>0.$

Since i ⊗ (jk) = (ij) ⊗ k ⇒ ((ij) ⊗ k)(u) > 0

Supv∈ℋ{(ij)(v) ∧ (vk)(u)} > 0

Supv∈ℋ{χ{i,j}(v) ∧ χ{v,k}(u)} > 0.

Hence, v′ ∈ ℋ exists such that χ{i,j}(v′) > 0 and χ{v′,k}(u) >

0. ⇒ (ij)(v′) > 0 and (v′k)(u) > 0

v′ij, uv′k

u ∈ (ij) ⊖ k

i ⊖ (jk) ⊆ (ij) ⊖ k

Similarly, we have that (ij) ⊖ ki ⊖ (jk)

⇒ (ij) ⊖ k = i ⊖ (jk)

Similarly, the associative law holds true for ⊙.

• (d) Absorption law: ∀i, j ∈ ℋ

Since, (i ⊗ (ij))(i)= Supv∈ℋ{(iv)(i)∧(ij)(v)} > 0

Supv∈ℋ{χ{i,v}(i)∧χ{ij}(v)} > 0

v′ ∈ ℋ such that χ{i,v′}(i) > 0 and χ{ij}(v′) > 0

⇒ (iv′)(i) > 0 and (ij)(v′) > 0.

This shows that ∃ iiv′ and v′ij

ii ⊖ (ij).

Similarly, ii ⊙ (ij),

Therefore, (ℋ,⊖, ⊙) denotes a hyperlattice.

(iii) Assume that R[i * (uv), (i * u) ⊕ (i * v)] = 1 ⇒ i * (uv) = (i * u) ⊕ (i * v) and

$R[i*(u⊗v),(i*u)⊗(i*v)]=1⇒i*(u⊗v)=(i*u)⊗(i*v).$

Then, (i * u) ⊕ (i * v) = χ{(i*u)(i*v)}.

v′ ∈ ℋ such that χ{(i*u)(i*v)}(v′) > 0 = (i * u) ⊙ (i * v)(I)

i * (uv) = (i * u) ⊕ (i * v) > 0 for any v′ ∈ ℋ.

u′ ∈ ℋ such that (i * (uv))(u′) > 0 = i * (uv) (II)

From (I) and (II), i * (uv) = (i * u) ⊙ (i * v). Similarly,

i * (uv) = (i * u) ⊖ (i * v).

Hence (ℋ, *, ⊖, ⊙) is HLOG.

Theorem 4.2

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

Proof

(i) Condition one of Definition 2.6 holds.

(ii) Proving that (ℋ, ⊗, ⊕) is a fuzzy hyperlattice.

The proof is obvious by Theorem 2.5.

(iii) Assume that i * (jk) = (i * j) ⊖ (i * k) and i * (jk) = (i * j) ⊙ (i * k), ∀ i, j, k ∈ ℋ.

To prove that R[i * (jk), (i * j) ⊗ (i * k)] = 1 and R[i * (jk), (i * j) ⊕ (i * k)] = 1

$i*(j⊗k)=i*χ{j⊖k}=χ{(i*j)⊖(i*k)}=(i*j)⊗(i*k).$

Therefore, R[i * (jk), (i * j) ⊗ (i * k)] = 1.

Similarly,

$i*(j⊕k)=i*χ{j⊙k}=χ{(i*j)⊙(i*k)}=(i*j)⊕(i*k).$

Therefore, R[i * (jk), (i * j) ⊕ (i * k)] = 1

Hence, (ℋ, *, ⊗, ⊕) is FHLOG.

Theorem 4.3

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

$i⊖j={u∈ℋ∣i⊗j(u)>0},$

and

$i⊙j={u∈ℋ∣i⊕j(u)>0}.$

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

Proof

(i) Condition one of Definition 3.1 holds.

(ii) Proving that (ℋ, ⊖, ⊙) is a hyperlattice.

The proof is obvious by Theorem 2.4.

(iii) Proving that i * (jk) = (i * j) ⊙ (i * k) and i * (jk) = (i * j) ⊙ (i * k).

Assume that ∀ i, j, k ∈ ℋ,

$R[i*(j⊗k),(i*j)⊗(i*k)]=1⇒i*(j⊗k)=(i*j)⊗(i*k),$

and

$R[i*(j⊕k),(i*j)⊕(i*k)]=1⇒i*(j⊕k)=(i*j)⊕(i*k).$$i*(j⊖k)={u∈ℋ∣(i*(j*k))(u)>0}={u∈ℋ∣((i*j)⊗(i*k))(u)>0} by (III)=(i*j)⊖(i*k)$

Next,

$i*(j⊙k)={u∈ℋ∣(i*(j⊕k))(u)>0}={u∈ℋ∣((i*j)⊕(i*k))(u)>0} by (IV)=(i*j)⊙(i*k)$

Hence, (ℋ, *, ⊖, ⊙) is HLOG.

5. Conclusion

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.

Fig 1.

Figure 1.

Rotation in ℝ3-space.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 418-424https://doi.org/10.5391/IJFIS.2023.23.4.418

Cayley table.

*r1r2r3e
r1er3r2r1
r2r3er1r2
r3r2r1er3
er1r2r3e

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