Int. J. Anal. Appl. (2023), 21:60

Fuzzy Soft Boolean Rings

Gadde Sambasiva Rao1, D. Ramesh1, Aiyared Iampan2,∗, B. Satyanarayana3

1Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education

Foundation, Vaddeswaram, Andhra Pradesh-522302, India
2Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School

of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand
3Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Andhra

Pradesh-522510, India

∗Corresponding author: aiyared.ia@up.ac.th

Abstract. This article introduces the idea of (fuzzy soft Boolean rings) FSBRs and investigates their

algebraic properties. The concepts of fuzzy soft ideals (FSIs) of FSBRs, and idealistic fuzzy soft

Boolean rings (IFSBRs) are then defined and discussed.

1. Introduction

In classical mathematics, exact solutions to mathematical models are required. If the model is so

complex that an exact solution cannot be determined, we can get a rough estimate. In 1999, Russian

researcher Molodtsov [8] pioneered the idea behind soft set (SS) theory and began developing the

foundations of the corresponding theory as a novel approach to modelling uncertainty. The SS is an

approximate representation of an object. There are numerous potential applications for SS theory. SS

theory and its applications are currently advancing at a rapid pace. Maji et al. [7] proposed new SS

definitions. Pei and Miao [9] looked into how SSs and information systems interact. By fusing SS and

fuzzy set (FS) designs, Maji et al. [6] developed the concept of fuzzy soft sets (FSSs) in 2001. To

continue the investigation, Ahmad and Kharal [2] obtainable some more properties of FSSs. There

has been a surge of interest in the algebraic structure of SSs in recent years. Soft groups were defined

Received: Apr. 10, 2023.

2020 Mathematics Subject Classification. 03E72, 03G05, 28A60, 06D72.
Key words and phrases. boolean ring; fuzzy soft set; fuzzy soft boolean ring; fuzzy soft sub boolean ring; fuzzy ideal;

fuzzy soft ideal; idealistic fuzzy soft boolean ring.

https://doi.org/10.28924/2291-8639-21-2023-60
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-60


2 Int. J. Anal. Appl. (2023), 21:60

by Aktaş and Çag̃man [3], and some properties were derived from them. Additionally, they contrasted

SSs with rough and FSs, two concepts that are related. The fundamental ideas of soft rings were first

introduced by Acar et al. [1] in 2010, which are a generalized family of subrings. Liu [5] proposed the

fuzzy ring concept in 1982. Following that, Dixit et al. [4] investigated the fuzzy ring and discovered

some theoretical analogues. The algebraic characteristics of FSSs in Boolean ring (BR) theory are

examined in this paper.

2. Preliminaries

To begin, we will present Maji et al. [6] and Ahmad and Kharal [2]’s fundamental definitions and

notations.

Definition 2.1. Let Ê denote a set of parameters, Z denote an initial universe, and Î represents the

closed unit interval. Q(Z) represents Z’s power set. Then the pair (M,Ê) over Z is a soft set, where

M : Ê → Q(Z) is a set valued function.

Definition 2.2. Let Ê denote a set of parameters, Z denote an initial universe, and Î represents the

closed unit interval, i.e., Î = [0,1]. Q(Z) represents Z’s power set, where M : C → IZ is set-valued
function and IZ represent the collection of all fuzzy sets on Z.

Definition 2.3. Consider (M,C) and (N,D) to be FSSs. Then (M,C) is a FSS of (N,D) and we can

write (M,C) ⊆ (N,D) if (i) C ⊆ D (ii) for each α ∈ C,Mα ≤ Nα implying that, Mα is fuzzy subset
of Nα.

Definition 2.4. Let us assume (M,C) and (N,D) be two FSSs, with C∩D 6= ∅. Then the FSS (O,E)
is formed by the intersection of (M,C) and (N,D), where E = C ∩D and Oα = Mα ∧Nα,∀α ∈ E.
We can write (M,C)e (N,D)= (O,E).

Definition 2.5. Let us assume (M,C) and (N,D) are two FSSs. The FSS (O,E) is formed by the

union of (M,C) and (N,D), where E = C ∪D and

(∀α ∈ E)


Oα =



Mα if α ∈ C −D

Nα if α ∈ D−C

Mα,∨Nα if α ∈ C ∩D


 . (2.1)

Then we write (M,C)d (N,D)= (O,E).

Definition 2.6. Let (Mj,Cj)j∈J be a family of FSSs. The union of these FSSs is a FSS (O,E), where

E =∪j∈JCj and O(α)=∨j∈JMj(α),∀α ∈ E. Then we can write dj∈J(Mj,Cj)= (O,E).

Definition 2.7. Let (Mj,Cj)j∈J be a family of FSSs, with ∩j∈JCj 6= ∅. A FSS is formed by the
intersection of these FSSs (O,E), where E =∩j∈JCj and O(α)=∧j∈JMj(α),∀α ∈ E. Then we can
write ej∈J(Mj,Cj)= (O,E).



Int. J. Anal. Appl. (2023), 21:60 3

Definition 2.8. Let (M,C) and (N,D) be two FSSs, subsequently, (M,C) AND (N,D) are represented

by (M,C)∧̂(N,D) and it’s indicated by (O,C × D), where O(α,β) = Oα,β = Mα ∧ Nβ for every
(α,β)∈ C ×D.

Definition 2.9. Let (M,C) and (N,D) be two FSSs. Then (M,C) OR (N,D) are represented by

(M,C)∨̂(N,D) and it’s indicated by (O,C × D), where O(α,β) = Oα,β = Mα ∨ Nβ for every
(α,β)∈ C ×D.

Definition 2.10. Consider (M,C) to be a FSS. The set Supp(M,C)= {α ∈ C : M(α)= Mα 6= ∅} is
known support of the FSS (M,C). If the support of a FSS is greater than the empty set, it is said to

be non-null.

3. Fuzzy Soft Boolean Rings

The concept of soft rings was proposed by Acar et al. [1]. In this section contains, we define FSBRs

and discuss some of their fundamental properties. R denotes a BR from now on, and all FSSs are

preferred over R.

Definition 3.1. Let us assume (M,C) is a non-null SS. Then (M,C) is referred to as a soft Boolean

ring (SBR) over R if for each α ∈ C,M(α) is a sub-BR of R.

Definition 3.2. Let us assume (M,C) is a non-null FSS. Then (M,C) is referred to as a FSBR over

R if for each γ ∈ C, M(γ) = Mγ is a F-sub-BR of R, i.e., Mγ(α − β) ≥ min(Mγ(α),Mγ(β)) and
Mγ(α ·β)≥min(Mγ(α),Mγ(β)),∀α,β ∈ R.

Example 3.1. Let R = {0, j, t, r} be a nonempty set with two binary operations + and · defined as
follows:

+ 0 j t r

0 0 j t r

j j 0 r t

t t r 0 j

r r t j 0

· 0 j t r
0 0 0 0 0

j 0 j r t

t 0 r t j

r 0 t j r



4 Int. J. Anal. Appl. (2023), 21:60

Let A = {e11,e
1
2,e

1
3} be the set of parameters and now define a FSS (M,C) on a BR R by

M(e11)= {(0,0.9),(j,0.8),(t,0.6),(r,0.4)},

M(e12)= {(0,0.8),(j,0.5),(t,0.3),(r,0.1)},

M(e13)= {(0,0.9),(j,0.6),(t,0.5),(r,0.4)}.

Here (M,C) is a FSS over R, which is also a F-sub-BR of R, for all α ∈ C. Hence (M,C) is a FSBR
over R.

Example 3.2. Because each SS can be thought of as a FSS, and each characteristic function of a

BR is a F-sub-BR of R, we can think of an SBR as a FSBR.

Theorem 3.1. Let (M,C) and (N,D) are two FSBRs over R. If (M,C)∧̂(N,D) is non-null, then it’s
a FSBR over R.

Proof. Let (M,C)∧̂(N,D) = (O,C × D), where O(α,β) = Mα ∧ Nβ,∀(α,β) ∈ C × D. Since
(O,C ×D), is non-null, then there exists the pair (α,β)∈ C ×D such that Oα,β = Mα ∧Nβ 6= OR.
We already know that Mα,∀α ∈ C and Nβ,∀β ∈ D are F-sub-BR of R. Since then the intersection
of two F-sub-BRs of R is also a F-sub-BR of R, then O(α,β) = Oα,β is a F-sub-BR of R. Hence

(O,C ×D)= (M,C)∧̂(N,D) is a FSBR over R. �

Theorem 3.2. Let (M,C) and (N,D) are two FSBRs over R. If (M,C)e(N,D) is non-null, then it’s

a FSBR over R.

Proof. Let (M,C)e (N,D) = (O,E), where E = C ∩D and Oα = Mα ∧Nα,∀α ∈ E. Since (O,E)
is non-null, then there exists α ∈ E such that Oα(β) 6=0 for some β ∈ R. We know Mα ∧Nα is a F-
sub-BR of R, because Oα 6=0R and Mα,Nα are F-sub-BR of R. Therefore, (O,E)= (M,C)e(N,D)
is a FSBR over R. �

Theorem 3.3. Let (Mj,Cj)j∈J be a family of FSBRs over R. There are also the following:

(i) If ∧̂j∈J(Mj,Cj) is non-null, then it’s a FSBR over R.
(ii) If ej∈J(Mj,Cj) is non-null, it’s a FSBR over R.

Proof. (i) Let ∧̂j∈J(Mj,Cj) = (O,E), where E = ej∈JCj and Oα = ∧j∈JMj(ej),∀α = (αj)j∈J ∈ E.
Suppose that the FSS (O,E) is non-null. If α =(αj)j∈J ∈Supp(O,E), then Oα =∧j∈JMj(αj) 6=0R.
Since (Mj,Cj) is a FSBR over R, ∀j ∈ J, Mj(αj) is a F-sub-BR of R. As a result Oα is a F-sub-BR
of R for all α ∈Supp(O,E). Consequently, ∧̂j∈J(Mj,Cj)= (O,E) is a FSBR over R.

(ii) Let ej∈J(Mj,Cj) = (O,E), where E = ∩j∈JCj and Oα = ∧j∈JMj(αj),∀α ∈ E. Suppose that
the FSS (O,E) is non-null. If α ∈ Supp(O,E), then Oα = ∧j∈JMj(αj) 6= 0R. Since (Mj,Cj) is a
FSBR over R, then Mj(αj) is a F-sub-BR of R for all j ∈ J. As a result Oα is a F-sub-BR of R for
all α ∈Supp(O,E). Therefore ej∈J(Mj,Cj)= (O,E) is a FSBR over R. �



Int. J. Anal. Appl. (2023), 21:60 5

Definition 3.3. Let (M,C) and (N,D) be two FSBRs over R. Then (N,D) is referred to as a FSSBR

of (M,C) if the circumstances listed below are true:

(i) D ⊆ C,
(ii) Nα is a F-sub-BR of Mα,∀α ∈Supp(N,E).

Theorem 3.4. Let (M,C) and (N,D) be two FSBRs over R. If (M,C)e(N,D) is non-null, then it’s

a FSSBR of (M,C) and (N,D).

Proof. (M,C)e(N,D)= (O,E), where E = C∩D and Oα = Mα∧Nα,∀α ∈ E. Since E = C∩D ⊆ C
and Oα = Mα ∧Nα, is a F-sub-BR of Mα, then (O,E) is a FSSBR of (M,C). Similarly, we obtain
that (O,E) is a FSSBR of (N,D). �

4. Fuzzy Soft Ideals of Fuzzy Soft Boolean Rings

Definition 4.1. Assume (M,C) is a FSBR over R. A FSS (N,D) is a FSI of (M,C), as indicated by

(N,D)/̂(M,C). If it meets the following criteria:

(i) D ⊆ C,
(ii) Nγ is a fuzzy ideal (FI) of a fuzzy BR Mγ for all γ ∈ Supp(N,D), i.e., Nγ is a FI, for each
γ ∈Supp(N,D),

(a) Nγ(α−β)≥ Nγ(α)∧Nγ(β),
(b) Nγ(αβ)≥ Nγ(α)∧Nγ(β),
(c) Nγ(α)≤ Mγ(α),∀α,β ∈ R.

Example 4.1. Take a look at the BR (R,+, ·) established in Example 3.1.

M(e11)= {(0,0.9),(j,0.8),(t,0.6),(r,0.4)},

M(e12)= {(0,0.8),(j,0.5),(t,0.3),(r,0.1)},

M(e13)= {(0,0.9),(j,0.6),(t,0.5),(r,0.4)}.

Here (M,C) is a FSS over R, which is also a F-sub-BR of R, ∀α ∈ C. Hence (M,C) is a FSBR over
R. Let D = {e12} and N : D → Q(R) be a function with a set of values defined by

N(e12)= {(0,0.4),(1,0.3),(2,0.2),(3,0.2)}.

Obviously (N,D) is a FSS of R. We also see that D ⊆ C and N(γ) is a FI of M(γ),∀γ ∈ I. As a
result, (N,D) is a FSI of (M,C).

Theorem 4.1. Let (N1,D1) and (N2,D2) be FSIs of a FSBR (M,C). Then (N1,D1)e (N2,D2) is a

FSI of (M,C) if it is non-null.

Proof. Let (N1,D1)/̂(M,C),(N2,D2)/̂(M,C). By the Definition 2.4, we write (N1,D1)e((N2,D2)=

(N,D), where D = D1∩D2 and N(γ)= N1(γ)∧N2(γ),∀γ ∈ D. Since D1 ⊆ C and D2 ⊆ C, we have



6 Int. J. Anal. Appl. (2023), 21:60

D1∩D2 = D ⊆ C. Suppose that (N,D) is non-null. If γ ∈Supp(N,D), then N(γ)= N1(γ)∧N2(γ) 6=
0R. Since (N1,D1)/̂(M,C) and (N2,D2)/̂(M,C), N1(γ) and N2(γ) are both FIs of M(γ), we conclude

M(γ). As a result, N(γ) is a FI of M(γ),∀γ ∈Supp(N,D). Therefore, (N1,D1)e(N2,D2)= (N,D)
is a FSI of (M,C). �

Theorem 4.2. Let (N1,D1) and (N2,D2) be FSIs of a FSBR (M,C). If D1 and D2 are disjoint, then

(N1,D1)d (N2,D2) is a FSI of (M,C).

Proof. Let (N1,D1)/̂(M,C),(N2,D2)/̂(M,C). By the Definition 2.5, we write (N1,D1)d(N2,D2)=

(N,D), where D = D1 ∪D2 and ∀γ ∈ D,

(∀α ∈ E)


Nγ =



N1(γ) if α ∈ D1 −D2

N2(γ) if α ∈ D2 −D1

N1(γ)∨N2(γ) if α ∈ D1 ∩D2


 . (4.1)

Obviously, we have D ⊆ C. Since D1 and D2 are disjoint, γ ∈ D1 − D2 or γ ∈ D2 − D1,∀γ ∈
Supp(N,D). Let γ ∈ D1−D2. Since (N1,D1)/̂ is a FI of M(γ). Thus, ∀γ ∈Supp(N,D), (N1,D1)⊆
(M,C). Consequently, (N,D) is a FSI of (M,C). �

5. Idealistic Fuzzy Soft Boolean Rings

Definition 5.1. Let (M,C) be a non-null FSS. Then (M,C) is referred to as an IFSBR over R, if Mγ
is a FI of R, ∀γ ∈ Supp(M,C). In other words, for each γ ∈ Supp(M,C),Mγ is a FI of R defined
in [4], i.e., Mγ(α−β)≥ Mγ(α)∧Mγ(β) and Mγ(α ·β)≥ Mγ(α)∨Mγ(β),∀α,β ∈ R.

Example 5.1. Let R = {0, j, t, r} be a set with two binary operations + and · as shown:

+ 0 j t r

0 0 j t r

j j 0 r t

t t r 0 j

r r t j 0

· 0 j t r
0 0 0 0 0

j 0 j r t

t 0 r t j

r 0 t j r

Then (R,+, ·) is a BR. Let A = {e11,e
1
2} represent the set of parameters.

M(e11)= {(0,0.9),(j,0.7),(t,0.6),(r,0.4)},

M(e12)= {(0,0.8),(j,0.5),(t,0.3),(r,0.1)}.



Int. J. Anal. Appl. (2023), 21:60 7

Here (M,C) is a FSS over R. Also, we can also see that M(γ) is a FI of R, ∀γ ∈ C. As a result,
(M,C) is an IFSBR over R.

Theorem 5.1. Assume (M,C) and (N,D) are two IFSBRs over R. Then (M,C)e(N,D) is an IFSBR

over R if it is non-null.

Proof. Let (M,C)e (N,D) = (O,E), where O = C ∩D and Oγ = Mγ ∧Nγ,∀γ ∈ E. Suppose that
(O,E) is non-null. If γ ∈ Supp(O,E), then Oγ = Mγ ∧Nγ 6= 0R. As a result, Mγ and Nγ are both
FIs of R. As a result, Oγ is a FI of R, ∀γ ∈ Supp(O,E). Hence, (O,E) = (M,C)e (N,D) is an
IFSBR over R. �

Theorem 5.2. Assume (M,C) and (N,D) are two IFSBRs over R. If C and D are disjoint, then

(M,C)d (N,D) is an IFSBR over R.

Proof. Let (M,C)d (N,D)= (O,E), where E = C ∪D and

(∀γ ∈ E)


Oγ =



Mγ if γ ∈ C −D

Nγ if ∈ D−C

Mγ ∨Nγ if γ ∈ C ∩D


 . (5.1)

Let us suppose that C ∩D = ∅. Then either γ ∈ C −D or Nγ ∈ D−C,∀γ ∈Supp(O,E).
If γ ∈ C −D,Oγ = Mγ is a FI of R. Because (M,C) is an IFSBR over R.
If γ ∈ D−C,Oγ = Nγ is a FI of R. Because (N,D) is an IFSBR over R.

Thus, for all γ ∈ Supp(O,D),Oγ is a FI of R. Consequently, (O,E)= (M,C)d (N,D) is an IFSBR
over R. �

Theorem 5.2 is false generally if and only if C and D are not disjoint. Consequently, the theorem

is not generally true. Because a ring’s FI may not be the union of two different FIs of a ring R.

Theorem 5.3. Assume (M,C) and (N,D) are two IFSBRs over R. Then (M,C)∧̂(N,D) is an IFSBR
over R if it is non-null.

Proof. Let (M,C)∧̂(N,D) = (O,C × D), where O(α,β) = Oα,β = Mα ∧ Nβ,∀(α,β) ∈ (C × D).
Assume (O,C×D) is non-null. If (α,β)∈Supp(O,C×D), then Oα,β = Mα∧Nβ 6= ∅. Since (M,C)
and (N,D) are IFSBRs over R, we can conclude that Mα and Nβ are both FIs of R. As a result, Oα,β
is a FI of R, ∀(α,β)∈Supp(O,C×D). Thus, (O,C×D)= (M,C)∧̂(N,D) is an IFSBR over R. �

6. Conclusion

The concept of FSBRs is introduced and its individual properties are studied in this paper. The

concepts of FSIs of a FSBR and an IFSBR are also introduced. This research could be expanded to

investigate the properties of FSSs in other algebraic structures.



8 Int. J. Anal. Appl. (2023), 21:60

Acknowledgment: This research project was supported by the Thailand Science Research and Inno-

vation Fund and the University of Phayao (Grant No. FF66-UoE017).

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

References

[1] U. Acar, F. Koyuncu, B. Tanay, Soft Sets and Soft Rings, Computers Math. Appl. 59 (2010), 3458-3463. https:

//doi.org/10.1016/j.camwa.2010.03.034.

[2] B. Ahmad, A. Kharal, On Fuzzy Soft Sets, Adv. Fuzzy Syst. 2009 (2009), 586507. https://doi.org/10.1155/

2009/586507.

[3] H. Aktaş, N. Çag̃man, Soft Sets and Soft Groups, Inform. Sci. 177 (2007), 2726-2735. https://doi.org/10.

1016/j.ins.2006.12.008.

[4] V.N. Dixit, R. Kumar, N. Ajmal, On Fuzzy Rings, Fuzzy Sets Syst. 49 (1992), 205-213. https://doi.org/10.

1016/0165-0114(92)90325-x.

[5] W. Liu, Fuzzy Invariant Subgroups and Fuzzy Ideals, Fuzzy Sets Syst. 8 (1982), 133-139. https://doi.org/10.

1016/0165-0114(82)90003-3.

[6] P.K. Maji, R. Biswas, A.R. Roy, Fuzzy Soft Sets, J. Fuzzy Math. 9 (2001), 589-602.

[7] P.K. Maji, R. Biswas, A.R. Roy, Soft Set Theory, Computers Math. Appl. 45 (2003), 555-562. https://doi.org/

10.1016/s0898-1221(03)00016-6.

[8] D. Molodtsov, Soft Set Theory–First Results, Computers Math. Appl. 37 (1999), 19-31. https://doi.org/10.

1016/s0898-1221(99)00056-5.

[9] D. Pei, D. Miao, From Soft Sets to Information Systems, in: 2005 IEEE International Conference on Granular

Computing, IEEE, Beijing, China, 2005: pp. 617-621. https://doi.org/10.1109/GRC.2005.1547365.

https://doi.org/10.1016/j.camwa.2010.03.034
https://doi.org/10.1016/j.camwa.2010.03.034
https://doi.org/10.1155/2009/586507
https://doi.org/10.1155/2009/586507
https://doi.org/10.1016/j.ins.2006.12.008
https://doi.org/10.1016/j.ins.2006.12.008
https://doi.org/10.1016/0165-0114(92)90325-x
https://doi.org/10.1016/0165-0114(92)90325-x
https://doi.org/10.1016/0165-0114(82)90003-3
https://doi.org/10.1016/0165-0114(82)90003-3
https://doi.org/10.1016/s0898-1221(03)00016-6
https://doi.org/10.1016/s0898-1221(03)00016-6
https://doi.org/10.1016/s0898-1221(99)00056-5
https://doi.org/10.1016/s0898-1221(99)00056-5
https://doi.org/10.1109/GRC.2005.1547365

	1. Introduction
	2. Preliminaries
	3. Fuzzy Soft Boolean Rings
	4. Fuzzy Soft Ideals of Fuzzy Soft Boolean Rings
	5. Idealistic Fuzzy Soft Boolean Rings
	6. Conclusion
	References