Distinct features and validation of δ⋇-algebras: ananalytical exploration Текст научной статьи по специальности «Математика»

VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3

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Distinct features and validation of £*-algebras: an analytical exploration

Prakasam Muralikrishnaa, Perumal Hemavathib,

Raja Vinodkumar0, Perumal Chanthinid,

Kaliyaperumal Palanivele, Seyyed Ahmad Edalatpanahf

aMuthurangam Government Arts College (Autonomus), PG and Research Department of Mathematics, Vellore, Republic of India, e-mail: [email protected],

ORCID iD: ©https://orcid.org/0000-0003-0652-2224 bSaveetha Institute of Medical and Technical Sciences (SIMATS), Saveetha School of Engineering, Department of Mathematics, Thandalam, Republic of India,

e-mail: [email protected], corresponding author, ORCID iD: ©https://orcid.org/0000-0003-0607-2817 cPrathyusha Engineering College (Autonomous),

Department of Mathematics, Thiruvallur, Republic of India, e-mail: [email protected],

ORCID iD: ©https://orcid.org/0000-0001-7847-4933 dCollege of Science & Humanities - SRMIST, Department of Computer Applications, Potheri Campus, Republic of India, e-mail: [email protected],

ORCID iD: ©https://orcid.org/0000-0003-0245-3065 eVellore Institute of Technology (VIT), School of Advanced Sciences, Department of Mathematics, Vellore, Republic of India, e-mail: [email protected],

ORCID iD: ©https://orcid.org/0000-0001-6389-7992 fAyandegan Institute of Higher Education, Department of Applied Mathematics, Tonekabon, Islamic Republic of Iran, e-mail: [email protected],

ORCID iD: ©https://orcid.org/0000-0001-9349-5695

doi https://doi.org/10.5937/vojtehg72-50294

FIELD: mathematics

ARTICLE TYPE: original scientific paper

Abstract:

Introduction/purpose: This research introduces the concept of a 6*-algebra, a unique structure in the field of abstract algebra. The study aims to explore the defining features and distinct properties of 6*-algebras, distinguishing them from other algebraic systems and examining their interrelations with other types of algebras.

Methods: The methodology includes the formal definition and characterization of 6*-algebras, a comparative analysis with the existing algebraic

structures, and an exploration of their interconnections. An algorithm is developed to verify whether a given structure meets the conditions of a S*-algebra.

Results: The results reveal that S* -algebras possess unique properties not found in other algebraic systems. The comparative study clarifies their distinctive place within the algebraic landscape and highlights significant interrelations with other structures. The verification algorithm proves effective in identifying S*-algebras, providing a systematic approach for further study.

Conclusions: In conclusion, S* -algebras represent a significant addition to abstract algebra, offering new theoretical insights and potential for future research. The study’s findings enhance the understanding of algebraic systems and their interconnections, opening new avenues for exploration in the field.

Key words: S*-algebra, Fuzzy algebra, Fuzzy logic, Fuzzy sets.

Introduction

BCI and BCK algebras are foundational algebraic structures in universal algebra, first introduced by Iseki and Tanaka (Iseki & Tanaka, 1978). In 1999, the author pioneered the concept of QS-algebra, which is closely linked to BCI/BCK-algebras, and further explored the G-part of QS-algebra in the same context (Ahn & Kim, 1999). The author also delved into the concept of BP-algebra, examining its relationship with other associated algebras (Ahn & Han, 2013). Within the same study, the exploration of quadratic BP-algebra and its corresponding algebras was undertaken.

Akram and Kim (Akram & Kim, 2007) conducted research on BCI-algebra and K-algebra, presenting various studies and insights. A novel algebraic concept named Z-algebra was introduced in 2017 (Chan-dramouleeswaran et al., 2017), where the properties of this new notion were thoroughly reviewed and discussed. Kaviyarasu et al. (Kaviyarasu et al., 2017) introduced INK-algebras, which represents a significant development in algebraic theory. The notion of a J-algebra was initially introduced by Iseki et al. (Iseki et al., 2006). It was subsequently shown that a variety of d-algebras can be constructed from minimal sharp J-algebras. The study also delved into the disjointness digraph within J-algebras and explored Smarandache disjointness. Kim and Kim (Kim & Kim, 2008) extended the concept of B-algebras to BG-algebras by utilizing a non-group-derived, non-empty set as a foundation for constructing a BG-algebra. Furthermore,

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several BG-algebra isomorphism theorems and associated properties were unveiled through the application of the concept of normal subalgebras.

As an extension of the BCK-algebra concept, a BE-algebra was introduced by Kim and Kim (Kim & Kim, 2006). Within BE-algebras, the concept of upper sets was leveraged to establish an equivalent condition for filters.

Kim and So (Kim & So, 2012) delved into the properties of в-algebras and their interconnections with -algebras. They notably illustrated that (в-, +) forms a semigroup with identity 0 when (в-, -, +, 0) is a -algebra. Specific constructions related to linear algebra within the field were also discussed.

Kim and Kim (Kim & Kim, 2006) introduced the notion of limited BM-algebras and delved into their properties. The concept of BO-algebra was initially introduced by Kim and Kim (Kim & Kim, 2012), highlighting that every BO-algebra is 0-commutative.

Expanding on dual BCK/BCI/BCH algebras and BE-algebras, Meng (Meng, 2010) proposed CI-algebras. This work explored the connections between BE-algebras and the core properties of CI-algebras, establishing that in transitive BE-algebras, the concept of ideals aligns with that of filters. Megalai and Tamilarasi (Megalai & Tamilarasi, 2010) introduced TM-algebra, offering comprehensive insights into its relationship with various algebraic structures. A group of algebras related to BCH, BCI, and BCK algebras, along with other notable groups, were introduced by Neggers and Kim (Neggers & Kim, 2002a). This class showcased an intriguing link between groups and B-algebras (Neggers & Kim, 2002b).

Neggers and Kim (Neggers & Kim, 1999) discussed a series of algebras that naturally bridge groups and sets. While this class encompasses various objects, it remains amenable to analysis using traditional methods.

Furthermore, after exploring the relationships between d-algebras and BCK-algebras, the concept of d-algebras emerged as another generalization of BCK-algebras. Jun et al. (Jun et al., 1998) introduced a BH-algebra which signifies a broader concept that encompasses BCH, BCI, and BCK-algebras. This generalization likely extends the understanding and application of these algebraic structures, offering a unified framework to study their properties and relationships.In 2007, BF-algebras were introduced as an extension of B-algebras, incorporating the concepts of an ideal and a normal ideal (Walendziak, 2007). This analytical exploration

aims to delve deeper into the unique attributes and defining characteristics of £*-algebras.

Preliminaries

This section presents some essential definitions with relevant examples that are needed to this article. Hereafter E is the Universal Set, * is the binary operation on E, and 0 is the constant element in E unless otherwise specified.

Definition 1. The structure (E, *, 0) is called a B-algebra, if

• Ф * Ф = 0

• Ф * 0 = Ф

• (Ф * А) * ф = Ф * (Ф * (o * A)), уФ, Л,ф £ e.

Example 1. From the following table let E = {0, аК1, aa„2} be a Б-algebra.

Table 1-B-algebra

* 0 aKi аК2

0 0 аК2 ак1

axi aKi 0 аК2

aK2 аК2 aKi 0

Definition 2. The structure (E, *, 0) is referred to as a BF-algebra, if

• Ф * Ф = 0

• Ф * 0 = Ф

• 0 * (Ф * Л) = Л * Ф У Ф,А £ E.

Definition 3. The structure (E, *, 0) is called an AMR-algebra, if

• Ф * 0 = Ф

• (Ф * A) * ф = А * (Ф * Ф) у Ф,А,ф £ e.

Let us define a binary relation Ф < A iff Ф * Л = 0

Example 2. Let E = {0, аК1 ,аК2 ,аКъ} be a set with a binary operation * defined by:

Table 2-AMR-algebra

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* 0 ак1 аК2 акз

0 0 aKi аК2 акз

axi аК2 акз 0

аК2 акз 0 аК1

акз 0 ак1 аК2

Then (E, *, 0) is an AMR-algebra.

Definition 4. The structure (E, *, 0) is called a Z-algebra, if

• & * & = 0 • 0 * & = &

• & * & = &

• Hr * Л = Л * &, when& = 0andA = 0, V&, Л e E.

Definition 5. The structure (E, *, 0) is called a BCK-algebra, if

• ((& * A) * (& * ф)) * (Ф * A) = 0

• 0 * & = 0 • & * & = 0

• (& * (& * A)) * Л = 0

• & * A = 0 & A * & = 0, imply & = A, V&, A e e.

Definition 6. The structure (E, *, 0) is called a Q-algebra, if

• & * 0 = &

• & * & = 0

• (& * A) * ф = (& * Ф) * A, V&, A e e.

Definition 7. The structure (E, *, 0) is called a TM-algebra, if

•Ф*0 = Ф

• (Ф * A) * (Ф * ф) = (ф * A), VФ, A e E.

Definition 8. The structure (E, *, 0) is called a BH-algebra, if

• Ф* 0 = Ф

• Ф*Ф = 0

• Ф * A = 0& A * Ф = 0, implies Ф = A, V Ф, A e E.

The structure of a £*-algebra

This section examines the features of a 8*-algebra, a novel algebraic structure.

Definition 9. The structure (E, *, 0) is called a 8*-algebra, if

(I) 0 * Ф = Ф

(II) Ф * Ф = 0

(III) (Ф * (Л * Ф)) * Ф = (Ф * Л) * (Ф * Ф), V Ф,Л,Ф e E.

Example 3. It is clear that E = ({0, аК1 ,aK2,аКз}, *, 0) is a 8*-algebra from the following table.

Table 3-8*-algebra

* 0 ак1 аК2 акз

0 0 ак1 аК2 акз

ак1 ак1 0 ак1 аК2

аК2 аК2 ак1 0 акз

акз акз аК2 акз 0

Example 4. Consider the set E = {0, аК1, aK2, aK3} with a binary operation * defined by

Table 4-8*-algebra

* 0 ак1 аК2 акз

0 0 ак1 аК2 акз

ак1 ак1 0 акз аК1

аК2 аК2 акз 0 аК2

акз акз аК1 аК2 0

Then (E, *, 0) is a 8*-algebra.

Definition 10. Let E be a non empty subset of a 8*-algebra of E. Then E is referred to as a 8*-subalgebra of E, if e\ * e2 e E, Ve\,e2 e E.

Example 5. For the 8*-algebra in Example 3.1, the subsets A =

{аК1 ,аК2} c E & B = {аК2 ,аКз} c E are the 8*-subalgebras of E, but the subset C = {аК1, аК2, аКз} c E is not a 8*-subalgebra of E.

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Proposition 1. Let (E, *, 0) be a 5*-algebra. Then it is not a BCK-algebra (Iseki & Tanaka, 1978), INK-algebra (Kaviyarasu et al., 2017), BE-algebra (Kim & Kim, 2006), BF-algebra (Walendziak, 2007), QS-algebra (Ahn & Kim, 1999), BP-algebra (Ahn & Han, 2013), Z-algebra (Chan-dramouleeswaran etal., 2017), BM-algebra (Kim & Kim, 2006), BG-algebra (Kim & Kim, 2008), B-algebra (Neggers & Kim, 2002a) or BH-algebra (Jun etal., 1998).

Proof. In every aforementioned algebra mentioned here with the exception of a 5*-algebra, observe that (Ф * (А * Ф)) * Ф = (Ф * A) *

(Ф * Ф), V Ф,Л,Ф e E. This Condition has been successfully introduced and implemented in example 3 and this type of condition was not used in any of the above cited algebras. □

Remark 1.The 5*-algebra (E, *, 0) provided in example 4 is not

a B-algebra, since (аК2 * аК1) * aK3 = aK3 * aK3 = 0 and

aK2 * (aK3 * (0 * aKl)) = aK2 * (aK3 * aKl) = aK2 * aKl = aK3

imply (ак2 * ®xi ) * ак3 = ак2 * (ак3 * (0 * ®xi )).

Lemma 1. If (E, *, 0) is a 5*-algebra, then (A * Ф) * Ф = A * (Ф * Ф) for

any Ф,А,Ф e E.

Proof. This follows from the axioms (I) and (II)

Lemma 2. If (E, *, 0) is a 5*-algebra, then (Ф * (A * (0 * Л))) * Ф = Ф for

any Ф,А e E.

Proof. From axioms (III) Ф = 0 * A, it is found that

(Ф * (A * (0 * A))) * Ф = ((Ф * Ф) * (A * Ф))

ie)(A * Ф) * Ф = 0 * ((А * Ф) * Ф)

by (I) by(II) by(i)

= (0 * A) * (ф * Ф) = Al * (Ф * Ф)

□

(0 * (A * Ф)) * A

(Л * Ф) * Л Ф * (Л * Л)

Ф * 0

Ф asclaimed.

by(II) by(i)

□

Lemma 3. If (E, *, 0) is a 6*-algebra, then Ф * Ф = Ф * Л implies Ф = Л for

any Ф,ф,Ф e E.

Proof. If Ф * Ф = Ф * A, then (Ф * (Л* (0 * Л))) * Ф = (Ф * (A* (0 * Л)) * A and thus by lemma 1 it follows that Ф = A. □

Lemma 4. If (E, *, 0) is a 6*-algebra, then for any Ф, A e E it follows that

(i) Ф * Ф = 0 imples Ф = Ф

(ii) 0 * Ф = 0 * Ф imples Ф = Ф (Ш)Ф * (0 * Ф) = i1.

Proof.(i) Since Ф * Ф = 0 implies Ф * Ф = Ф * Ф, it follows that Ф = Ф.

(ii) If 0 * Ф = 0 *Ф, then 0 = Ф * Ф = (0 * (0 *Ф)) * Ф = (0 * 0) * (Ф * Ф) = 0 * (Ф * Ф) = (Ф * Ф) and thus by (i), Ф = Ф.

(iii) For any Ф e E, it is obtained that

Ф * (0 * Ф) = (Ф * (0 * Ф)) * Ф = (Ф * 0) * (Ф * Ф) by axioms (I) and (II) it follows that i1 = i1 * (0 * ф) as claimed.

Note that: Let (E, *, 0) be a 6*-algebra and let e E. Define in =

ьф1-1 *(0*i*)(n > 1) and 14 = i*. Note that *(0*s) = s*(0*i+) =

l* by lemma 2.

Lemma 5. If (E, *, 0) is a 6*-algebra and let s e E. Then sn * sm = sn-m, where n > m.

Proof. Let E is a 6*-algebra. It is noted that by lemma 3 it follows that

s2 * s = s1 * (0 * s) * s = (s * 0) * (s * s) = s * 0 = s.

Assume that s(n+1) * s = sn(n > 1). Then

s

n+2

* s = (sn+1 * (0 * s)) * g = (sn+1 * 0) * (s * s)

= sn+l * 0

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= sn+l.

Assume sn * sm = sn-m, where n - m > 1. Then

sn * sm+1 = sn * ((0 * sm) * s)

= ((0 * sm) * s) * sn = (0 * sm) * (s * sn)

= sm * sn-1 = sn-1 * sm

= sn-m+1 (n - m - 1 > 0).

□

Lemma 6. If (£, *, 0) is a 6*-algebra and let s e £. Then sn * sm = sn-m, where n > m.

Proof. Let £ is a 6*-algebra then note that by lemma 3 it follows that s2 *

s = s1 * (0 * s) * s = (s * 0) * (s * s) = s * 0 = s.

Assume that s(n + 1) * s = sn(n > 1). Then

sn+2 * s = (sn+1 * (0 * s)) * g = (sn+1 * 0) * (s * s)

= sn+1 * 0 = sn+1.

Assume sn * sm = sn-m, where n - m > 1. Then

sn * sm+1 = gn * ((0 * sm) * s)

= ((0 * sm) * s) * sn = (0 * sm) * (s * sn)

= sm * sn-1 = sn-1 *sm

= sn-m+1) (n - m - 1 > 0).

□

Lemma7. If (E, *, 0) isa 6* -algebra and let s e E. Then sm *sn = sn-1 *0, where n > m.

Proof. Let E is a 6*-algebra, By applying (I), (III) and lemma 3, it is follows that s * s2 = s * (s1 * (0 * s)) = (s * 0) * (s1 * s) = s * 0 = 0 * s. Assume that g * sn = s(n — 1) where (n > 1). Then

s * sn-1 = g * (gn * (0 * s))

= (sn * (0 * s)) * s = (sn * 0) * (s * s)

= sn * 0.

Assume that sm * sn = sn - m where n-m> 1. Then

sn * sm+1 = sn * ((sm * (0 * s))

= (sm * (0 * s)) * sn = (sm * 0) * (s * sn)

= sm * sn-1 = sn-1 *sm

= 0 * s(n-m-1) (n — m — 1 > 0).

□

It is summarized that the above lemmas:

Theorem 1. Let (E, *, 0) is a 6*-algebra and let s e E. Then

s

n

* sm

{s(n-m) :

0 * s(n-m)

ifn > m otherwise.

Proposition 2. Let (E, *, 0) be a 6*-algebra. Consequently, the subsequent outcomes are valid, for all ex,ey e E.

(i) (ex * (ex * (£y * ex))) = £x, if £y = 0.

(ii) (£y * ex) = (ey * 0) * (x * 0).

(iii) (ex * ey) * [(ey * ex) * (ex * ey)] = ex * ey.

(iv) 0 * (ex * ey) = (0 * ex) * (0 * ey).

(Y)(E! *! 0) У * (0 * ex) = ey * ex, ex = ey.

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Proof. Let (E, *, 0) be a 8*-algebra, ex,ey e E.

Suppose y= 0. Then,

(i) (£x * (£x * (t£y * £x)) = (£x * (£x * (0 * £x)))

= (£x * (£x * £x)) by axiom(I)

= (£x * 0) by axiom(II)

= £x by axiom(I).

(ii) (£y * 0) * (£x * 0)) = (£y * (0 * £x)) * 0 by axiom(III)

= (£y * £x) * 0 by axiom(I)

= (£y * £x) by axiom(I).

(iii)(£x * £y) * [(£y * £x) * (£x * y)]

= (£x * £y) * [(£y * (£x * £x)) * £y] by axiom(III). = (£x * £y) * ((£y * 0) * £y) by axiom(II)

= (£x * £y) * (£y * £y) by axiom(I)

= (£x * £y) * 0 by axiom(I)

= £x * £y by axiom(I).

(iv) (0 * (£x * £y) = £x * £y by axiom(I)

= (0 * £x) * (0 * £y) by axiom(II).

(v) (£y * (0 * £x) = (0 * £y) * (0 * £x) by axiom(I)

= (£y * £x) by axiom(I).

□

Proposition 3. Let (E, *, 0) and (E', *', 0') be two 8*-algebras. A mapping z : E ^ E' of a 8*-algebras is referred to as a homomorphism, if

z(£x * £y) = z(£x) *' z(£y), ^£x, £y € E.

Definition 11. Let (Z, *, 0) and (Z', *', 0') be two 5*-algebras. A mapping z : Z ^ Z' of 5*-algebras is called a homomorphism. Then the kernel of z is the subset of Z, defined by ker(z) = {ex £ Z : z(ex) = 0'}

Lemma 8. Let z : Z ^ Z' be a homormorphism of a 5*-algebra. Then

z(0) = 0', 0 £ Z.

Proof.Let z : Z ^ Z' be an homomorphism of 5*-algebras. Then z(0) = z(0 * 0) = z(0) *' z(0) = 0'.

Theorem 2. Let (Z, *, 0) and (Z', *', 0') be two 5*-algebras. let z : Z ^ Z' be a surjective 5* - homomorphism. If A is a 5*-subalgebra of Z, then z(A) is a 5*-subalgebra of Z'.

Proof. Let (Z, *, 0) and (Z', *', 0') be two 5*-algebras. Let z : Z ^ Z' be a homomorphism and A be a 5*-subalgebra of Z.

Now, ea,eb £ A ^ ea * eb £ A л z(a),z(eb) £ z(A)

^ z(ea) * z(eb) = z(ea * eb) £ z(A)

Hence z(A) is a 5*-algebra of Z'.

Theorem 3. Let (Z, *, 0) and (Z', *', 0') be two 5*-algebras. Let z : Z ^ Z' be a surjective 5*- homomorphism. If B is a 5*-subalgebra of Z', then z-l(B) is a 5*-subalgebra of Z.

Proof. It is known that z-l(B) = {x £ Z : z(x) = y for some y £ B} Assume that x and y £ z-1(B). Then z(ea),z(eb) £ (B).

Since B is a 5*-subalgebra of Y,

^ z(ex) *' z(ey) £ B. Also, since z is a homomorphism, z(ex * ey) =

z(ex) * z(ey) £ B, л ex * ey £ z 1 (B) .

Hence z-l(B) is a 5*-algebra of Z.

□

Algorithm for a 5*-algebra

In this section, it is demonstrated that an algorithm to check the conditions of 5*-algebras uses the values in between 0 and 1.

def generate_table(rows, cols):

# Create a list of labels for the rows and columns

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labels = [’0’] + [chr(ord(’a’) + i) for i in range(cols -1)]

# Print the header row header = ” ”.join(labels[:cols]) print(header)

# Loop through the rows for i in range(rows):

# Initialize the row with the label row = [labels[i]]

# Loop through the columns for j in range(1, cols):

# Fill in the cells based on the table pattern if i == 0:

row.append(labels[j])

else:

row.append(labels[j] if j == i else labels[i])

# Print the row print(” ”.join(row))

# Call the function with 4 rows and 4 columns generate_table(4, 4)

Where a = aXl ,b = aX2,c = aX3 from the above pattern, the pattern according to the equation is the original table.

To prove the algebraic expression, it is necessary to show that the left-hand side (LHS) is equal to the right-hand side (RHS) for all possible combinations of the values 'a',' b', and 'c'.

Let us break down the LHS and RHS step by step:

* 0 a b c 00abc aa0ab bba0c ccbc0

LHS: (a * (b * c)) * a

Start with 'a'.

Find the value at the intersection of the row 'a' and the column corresponding to the value of (b * c). In this case, (b * c) can be found in the cell at the intersection of the row b and the column c .

Finally, find the value at the intersection of the row corresponding to the result of (a * (b * c)) and the column 'a'. In this case, the result of (a * (b * c)) can be found in the cell at the intersection of the row 'a' and the column corresponding to the value of (b * c).

RHS: (a * b) * (c * a)

Start with 'a'. Find the value at the intersection of the row 'a' and the column 'b'. Find the value at the intersection of the row 'c' and the column 'a'. Finally, find the value at the intersection of the row corresponding to the result of (a*b) and the column corresponding to the result of (c*a).

Now, let us go through the computations for each case:

LHS:

(a * (b * c)) * a = (a * (b * c)) * a = (a * (b * c))

RHS:

(a * b) * (c * a) = (a * b) * (c * a) = (a * (b * c))

It is evident that the LHS and RHS are both equal to (a * (b * c)), which means that the algebraic expression (a * (b * c)) * a = (a * b) * (c * a) is true for all possible combinations of 'a','b', and 'c'.

Therefore, the algebraic expression is proven to be true using the given table.

Conclusion

This study introduced a novel algebraic class, a £*-algebra which is deeply rooted in the foundational set theory principles. Through careful

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analysis, it became evident that a £*-algebra stands apart from the existing algebraic structures, showcasing distinct characteristics and properties. Employing a unique methodology, the study meticulously formalized the concept of a £*-algebra, providing clarity and insight into its inner workings. As a result, a host of new results emerged from this exploration, each bolstered by relevant examples to illustrate its significance. This conceptual framework not only enriches our understanding of algebraic structures but also opens doors for further exploration and expansion into other algebraic substructures in future research endeavors.

References

Ahn, S.S. & Kim, H.S. 1999. On QS-algebras. Journal of the Chungcheong Mathematical Society, 12(1), pp.33-41 [online]. Available at: https://koreascience.kr/article/JAKO199917069750921.page [Accessed: 15 May 2024].

Ahn, S.S. &Han, J.S. 2013 On BP-Algebras. Hacettepe Journal of Mathematics and Statistics, 42(5), pp.551-557 [online]. Available at:

https://dergipark.org.tr/en/pub/hujms/issue/7746/101253 [Accessed: 15 May 2024].

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Caracteristicas distintivas y validacion de £*-algebras: una ex-ploracion analrtica

Prakasam Muralikrishnaa, Perumal Hemavathib,

Raja Vinodkumarc, Perumal Chanthinid,

Kaliyaperumal Palanivel®, Seyyed Ahmad Edalatpanahf

a Muthurangam Escuela de Artes del Gobierno (Autonoma),

PG y Departamento de Investigation de Matematicas,

Vellore, RepOblica de la India

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b Instituto Saveetha de Ciencias Medicas y Tecnicas (SIMATS), Escuela de Ingenieria Saveetha, Departamento de Matematicas, Thandalam, RepOblica de la India, autor de correspondencia

c Prathyusha Facultad de Ingenieria (Autonoma), Departamento de Matematicas, Thiruvallur, RepOblica de la India d Facultad de Ciencias y Humanidades - SRMIST, Departamento de Aplicaciones Informaticas, Campus Potheri, RepOblica de la India e Instituto de Tecnologia Vellore (VIT), Facultad de Ciencias Avanzadas, Departamento de Matematicas,

Vellore, RepOblica de India

f Instituto Ayandegan de Educacion Superior, Departamento de Matematicas Aplicadas, Tonekabon, RepOblica Islamica de Iran

CAMPO: matematicas

TIPO DE ARTiCULO: articulo cientifico original Resumen:

Introduccion/objetivo: Esta investigacion introduce el concepto de S*-algebra, una estructura unica en el campo del algebra abs-tracta. El estudio tiene como objetivo explorarlas caracterfsticas determinates y las propiedades distintivas de S*- algebras, dis-tinguiendolos de otros sistemas algebraicos y examinando sus interrelaciones con otros tipos de algebras.

Metodos: La metodologfa incluye la definicion formal y caracte-rizacion de S* -algebras, un analisis comparativo con las estruc-turas algebraicas existentes y una exploracion de sus intercone-xiones. Se desarrolla un algoritmo para verificar si una estructura determinada cumple las condiciones de un S*-algebra.

Resultados: Los resultados revelan que las S*-algebras poseen propiedades unicas que no se encuentran en otros sistemas algebraicos. El estudio comparativo aclara su lugar distintivo den-tro del panorama algebraico y destaca interrelaciones significa-tivas con otras estructuras. El algoritmo de verificacion resulta eficaz para identificar S*-algebras, proporcionando un enfoque sistematico para estudios posteriores.

Conclusion: En conclusion, las S*-algebras representan una adicion significativa al algebra abstracta, ofreciendo nuevos co-nocimientos teoricos y potencial para investigaciones futuras. Los hallazgos del estudio mejoran la comprension de los sistemas algebraicos y sus interconexiones, abriendo nuevas vfas para la exploracion en este campo.

Palabras claves: S* -algebra, Algebra difusa, Logica difusa, Con-juntos difusos.

Отличительные особенности и валидация З*-алгебры: аналитическое исследование

Пракасам Мураликришнаа, Перумал Хемаватиб,

Раджа Винодкумарв, Перумалр Чантиниг,

Калияперумал Паланивелд, Сейед Ахмад Эдалатпанах®

a Государственный колледж искусств Мутурангам (автономный), Научно-исследовательский институт математического факультета, Веллор, Республика Индия б Институт медицинских и технических наук Савиты (SIMATS), инженерная школа Савиты, математический факультет,

Тандалам, Республика Индия, корреспондент

в Инженерный колледж Пратьюша (автономный), математический факультет, Тируваллур, Республика Индия

г Колледж естественных и гуманитарных наук - SRMIST, кафедра компьютерных приложений, кампус Потери, Республика Индия д Технологический институт Веллора (VIT), Школа передовых наук, математический факультет, Веллор, Республика Индия е Аяндеганский институт высшего образования, факультет прикладной математики, Тонекабон, Исламская Республика Иран

РУБРИКА ГРНТИ: 27.17.00 Алгебра ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: В данном исследовании вводится понятие З*-алгебры с уникальной структурой в области абстрактной алгебры. Целью исследования является изучение определяющих особенностей и отличительных свойств З*-алгебр, отличающих ее от других алгебраических систем и изучение их взаимосвязи с другими типами алгебр.

Методы: Методология включает формальное определение и характеристику З*-алгебры, сравнительный анализ с существующими алгебраическими структурами и изучение их взаимосвязей. Разработан алгоритм для проверки, насколько данная структура удовлетворяет условия З*-алгебры.

Результаты: Результаты показали, что З* -алгебра обладает уникальными свойствами, которых нет в других алгебраических системах. Сравнительное исследование проясняет ее особое место в алгебраическом царстве и подчеркивает важные взаимосвязи с другими структура-

Prakasam, M. et al., Distinct features and validation of J*-algebras: an analytical exploration, pp.1046-1065

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ми. В статье также подтверждено, что алгоритм верификации эффективен в идентификации З*-алгебр и, таким образом, предоставляет систематический подход в дальнейших исследованиях.

Выводы: В заключении подчеркивается важная роль З*-алгебры в качестве значительного дополнения к абстрактной алгебре, представляя новые теоретические идеи и потенциал для будущих исследований. Результаты исследования расширяют понимание алгебраических систем и их взаимосвязей, открывая новые возможности для исследований в этой области.

Ключевые слова: З* -алгебра, нечеткая алгебра, нечеткая логика, нечеткие множества.

Изразите одлике и валидаци]а З*-алгебри: аналитичко истраживане

Пракасам Мураликришнаа, Перумал Хемавати6,

Раца Винодкумарв, Перумал Чантиниг,

Кал^аперумал Паланивелд, Се]'ед Ахмад Едалатпанах* 4

a Државни уметнички колец Мутурангам (аутономни), Оде^ене математике за последипломске и истраживачке студне,

Велор, Република Инди]а

б Институт медицинских и техничких наука Савита (SIMATS), Електротехнички факултет Савита, Департман за математику, Тхандалам, Република Инди]а, ауторза преписку в Инженерски колец Притуш]а (аутономни), Одсекза математику, Тирувалур, Република Инди]а

г Факултет за науку и хуманистичке науке - SRMIST, Оде^ене за рачунарске апликаци]е, Кампус Потхери, Република Инди]а

д Технолошки институт Велор (VIT), Факултет напредних наука, Одсек за математику, Велор, Република Инди]а

4 Институт за високо образоване А]андеган, Одсек за применену математику, Тонекабон, Исламска Република Иран

ОБЛАСТ: математика

КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:

Увод/ция>: Ово истраживане уводи концепт З* -алгебре, }единствене структуре у области апстрактне алгебре. Цин студи}е ]е да истражи карактеристичне црте и из-

разите одлике 5*-алгебри да би се показало по чему се ра-злику}у од осталих алгебри и да би се испитали ме^усобни односи са другим врстама алгебри.

Методе: Методологи]а обухвата формалну дефиници]у и карактеризаци}у 5* -алгебри, компаративну анализу са по-сто]еЯим алгебарским структурама као и истраживале лихових ме^усобних веза. Разви]ен}е алгоритам да потвр-ди да дата структура испулава услове 5*-алгебре. Резултати: Резултати показу]у да 5* -алгебре карактери-шу}единствене одлике щих нема у другим алгебарским си-стемима. Упоредна студи}а по}ашлава лихово посебно место у алгебарском царству и истиче важне ме^усобне везе са другим структурама. Показано ]е да ]е верификациони алгоритам ефикасан у идентификации 5*-алгебри, чиме се обезбе^у/е систематски приступ далем истраживалу. Заклучак: Може се заклучити да су 5*-алгебре знача]ан додатак апстрактно] алгебри и да нуде нове теориске увиде и потенциале за дала истраживала. Налази ове студи/е проширу]у наше разумевале алгебарских система и лихових ме^усобних веза, отвара}уЬи истовремено нове путеве истраживала у ово] области.

Клучне речи: 5* -алгебра, фази алгебра, фази логика, фази скупови.

Paper received on: 07.04.2024.

Manuscript corrections submitted on: 24.09.2024.

Paper accepted for publishing on: 25.09.2024.

© 2024 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier (http://vtg.mod.gov.rs, http://BTr.M0.ynp.cp6}. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).

Prakasam, M. etal., Distinct features and validation of J*-algebras: an analytical exploration, pp.1046-1065