Rough soft hyperrings and corresponding decision making

Abstract

If I is a normal hyperideal of a hyperring R, then an equivalence relation on R can be defined as follows: ≡I by x ≡ Iy if and only if x-y∩ I ≠ ∅. Hence a normal hyperideal of a hyperring is acted as an equivalence relation. In this paper we introduce a kind of novel rough soft hyperrings with respect to a normal hyperideal of a hyperring, in Pawlak approximation spaces. Some new operations of lower and upper rough soft hyperrings are studied. In particular, lower and upper rough soft hyperideals with respect to a normal hyperideal are investigated, respectively. Some good examples are also given. Finally, we propose a novel kind of decision making method to rough soft hyperrings.

Full text

Journal of Intelligent & Fuzzy Systems 33 (2017) 1479–1489
DOI:10.3233/JIFS-162041
IOS Press
1479
Rough soft hyperrings and corresponding
decision making
Xueling Maa,∗, Jianming Zhanaand Violeta Leoreanu-Foteab
aDepartment of Mathematics, Hubei University for Nationalities, Enshi, Hubei Province, China
bFaculty of Mathematics, University “Alexandru Ioan Cuza”, Iasi, Romania
Abstract.IfIis a normal hyperideal of a hyperring R, then an equivalence relation on Rcan be defined as follows: ≡I
by x≡Iyif and only if x−y∩I/=∅. Hence a normal hyperideal of a hyperring is acted as an equivalence relation. In
this paper we introduce a kind of novel rough soft hyperrings with respect to a normal hyperideal of a hyperring, in Pawlak
approximation spaces. Some new operations of lower and upper rough soft hyperrings are studied. In particular, lower and
upper rough soft hyperideals with respect to a normal hyperideal are investigated, respectively. Some good examples are also
given. Finally, we propose a novel kind of decision making method to rough soft hyperrings.
Keywords: Hyperring, (normal) hyperideal, rough soft hyperring (hyperideal), decision making
1. Introduction
Rough set theory was first proposed in 1982, by
Pawlak [37], as an important mathematical approach
to deal with imprecision, vagueness and uncertainty
in data analysis. It is well known that an equivalence
relation on a set partitions it into some disjoint classes
and vice versa. However, equivalence relations in
Pawlak rough sets are too restrictive for theoretical
and practical aspects. Based on this reason, many
researchers generalized them to some other mod-
els, for examples, [41, 49]. At the same time, some
researchers applied rough sets to algebras, topologies,
logic and others. In particular, Davvaz [11] investi-
gated rough ideals (rings) by means of an ideal of
a ring. She [38] studied rough consistency measures
of logic theories and approximate reasoning in rough
logic. Zhang [48] studied union and intersection oper-
ations of rough sets. It is pointed out that rough sets
have been applied to many research fields, such as
∗Corresponding author. Xueling Ma, Department of Math-
ematics, Hubei University for Nationalities, Enshi, Hubei
Province 445000, China. Tel.: +86 0718 8437732; E-mail:
zjmhbmy@126.com.
machine learning, expert systems, intelligent deci-
sion, pattern recognition, image processing, knowl-
edge discovering, three-way decision and others.
Soft set theory was first put forth by Molodtsov
[36] as a novel mathematical tool for dealing with
uncertainties from the viewpoint of parametrization.
In 2003, Maji et al. [29] proposed some operations
on soft sets. Further, Ali et al. [3] gave some new
operations and reduction of parameters on soft sets.
A new definition of soft set parametrization reduction
was proposed by Chen [7]. Ali [2] studied reduction
of parameters on soft sets. Feng [17] investigated
attribute analysis of information systems based on
elementary some implications. In particular, Li [26]
studied soft covering and its parameter reduction.
Recently, the algebraic structures of soft sets have
been studied increasingly. Aktas [1] introduced soft
groups and derived their basic operations. For exam-
ples, Feng [18] characterized soft semirings. Zhan
[45] investigated soft BL-algebras based on fuzzy
sets. At the same time, soft sets have potential appli-
cations in many fields, such as decision making,
information systems, description logic, forecasting,
data analysis, and so on (see [5, 6, 31, 50].
1064-1246/17/$35.00 © 2017 – IOS Press and the authors. All rights reserved

1480 X. Ma et al. / Rough soft hyperrings
It is well known that uncertainties which arise from
various domains have many different natures so that
we can not capture them by a single mathematical
tool. Based on this reason, some researchers put forth
some hybrids, such as fuzzy soft sets, fuzzy rough
sets, rough soft sets and soft rough fuzzy sets, and so
on. In 1990, Dubois and Prade introduced the con-
cepts of rough fuzzy sets and fuzzy rough sets. In
2001, Maji [30] proposed the concept of fuzzy soft
sets. In particular, Feng [19, 20] put forth rough soft
sets and soft rough sets, respectively. In 2011, Meng
[34] modified soft rough fuzzy sets and soft fuzzy
rough sets. In 2014, Li [27] investigated the relation-
ships among soft sets, soft rough sets and topologies.
Recently, Sun [39] applied soft fuzzy rough sets to
decision making.
On the other hand, algebraic hyperstructures
introduced by Marty [32] in 1934, is a natural
generalization of algebraic structures. Nowadays, it
has been applied to many areas, such as geometry,
lattices, automata, cryptography, combinatorics, arti-
ficial intelligence, probabilities, and so on. For more
details, see the books, [8, 9, 14, 40]. Hypergroups,
as basic hyperalgebras, have been investigated by
some researchers, see [10, 23, 24, 47]. A well-known
type of hyperrings, called the Krasner hyperring [22],
has been investigated by many researchers. In 2004,
Davvaz [12] established three isomorphism theorems
of hyperrings and derived the Jordan-holder theorems
for hyperrings. Further, Ma and Zhan [28, 43] investi-
gated (fuzzy) isomorphism theorems of -hyperrings
and hypermodules, respectively. We also notice that
the relationships among fuzzy sets, rough sets and
hyperrings have been investigated by Corsini, Cristea,
Davvaz, Leoreanu-Fotea, Zhan, for examples, see
[13, 25, 44]. Some classical hyperings were inves-
tigated by Davvaz, Ameri, Jafarpour, and so on. For
more details, see [4, 15, 21, 33, 35].
Recently, by means of Feng’s idea in [20], Zhan
[46] firstly applied rough soft sets to algebraic
structures–hemirings, and described some charac-
terizations of rough soft hemirings. In the present
paper, we put forth a novel rough soft algebraic
structure–hyperrings by another way. Let Ibe a nor-
mal hyperideal of a hyperring R. Then we define the
relation ≡Iby x≡Iyif and only if x−y∩I/=∅.It
is clear that the relation ≡Iis an equivalence relation
on R. By this idea, we propose the concept of rough
soft hyperrings based on a normal hyperideal of a
hyperring, which is different from Zhan’s idea in [46].
This paper is organized as follows. We recall some
concepts and results on hyperrings, rough sets and
soft sets in Section 2. In Section 3, we put forth
the concept of rough soft hyperrings w.r.t. a normal
hyperideal of a hyperring and investigate some rough
strong approximation operations. In Section 4, we
propose rough soft hyperideal w.r.t a normal hyper-
ideal. In particular, some good examples are explored
to support our arguments. In Section 5, we give a
novel kind of decision making method to rough soft
hyperrings.
2. Preliminaries
By a hyperoperation on Rwe mean a mapping
“+”: R×R→P∗(R), where P∗(R) is the set of all
non-empty subsets in R, written as (x, y)→ x+y.
Rtogether with a hyperoperation “ +" is called a
hypergroupoid.
If x∈Rand A, B ∈P∗(R), then we denote A+
B=a∈A,b∈Ba+b, x +A={x}+Aand A+
x=A+{x}.
We say that a hypergroupoid (R, +) is a canonical
hypergroup if it satisfies the following conditions:
(i) (x+y)+z=x+(y+z) for all x, y, z ∈R;
(ii) x+y=y+xfor all x, y ∈R;
(iii) for each x∈R, there exists an element 0 ∈R
such that 0 +x=x+0={x}(0 is called the
zero element of (R, +));
(iv) for each x∈R, there exists a unique element
x∈Rsuch that 0 ∈x+x=x+x,(xis
called the inverse of xand is denoted by −x);
(v) z∈x+y⇒x∈z−yand y∈z−xfor all
x, y, z ∈R.
Throughout this paper, by a hyperring we mean a
Krasner hyperring.
Definition 2.1. [22] An algebraic structure (R, +,·)is
called a Krasner hyperring if it satisfies the following
conditions:
(1) (R, +) is a canonical hypergroup;
(2) Relating the multiplication, (R, ·) is a semi-
group having zero as a bilaterally absorbing
element, that is, 0 ·x=x·0=0 for all x∈R;
(3) The multiplication is distributive with respect
to the hyperoperation “+”, that is, z·(x+y)=
z·x+z·yand (x+y)·z=x·z+y·zfor
all x, y, z ∈R.
From the definition, the following elementary
facts are immediate: −(−x)=x,−0=0, where 0
is unique and −(x+y)=−x−yfor all x, y ∈R.

X. Ma et al. / Rough soft hyperrings 1481
A non-empty subset Ain Ris said to be a subhyper-
ring of Rif (A, +,.) is itself a Krasner hyperring. A
subhyperring Ais called a hyperideal of Rif x·y∈A
and y·x∈Afor all x∈Rand y∈A. Moreover,
A hyperideal Ais called normal if x+A−x⊆A
for all x∈R.
Definition 2.2. [36] A pair S=(F, A) is called a
soft set over U, where A⊆Eand f:A→P(U)is
a set-valued mapping.
Definition 2.3. Let (F, A) be a non-null soft set over
R. Then
(1) (F, A) is called a soft hyperring over Rif
F(x) is a subhyperring of Rfor all x∈
Supp(F, A)={x∈A|F(x)/=∅};
(2) (F, A) is called a soft (resp., left, right) hyper-
ideal over Rif F(x) is a (resp., left, right)
hyperideal of Sfor all x∈Supp(F, A).
Definition 2.4. [3] Let (F, A) and (G, B) be two soft
sets over U.
(1) The extended intersection of (F, A) and (G, B),
denoted by (F, A)(G, B), is defined as the
soft set (H, C), where C=A∪B, and ∀e∈C,
H(e)=⎧
⎪
⎨
⎪
⎩
F(e)if e ∈A\B,
G(e)if e ∈B\A,
F(e)∩G(e)if e ∈A∩B.
(2) The restricted intersection of (F, A) and
(G, B), denoted by (F, A)(G, B), is defined
as the soft set (H, C), where C=A∩Band
H(c)=F(c)∩G(c) for all c∈C.
(3) The extended union of (F, A) and (G, B),
denoted by (F, A)
∪(G, B), is defined as the
soft set (H, C), where C=A∪B, and ∀e∈C,
H(e)=⎧
⎪
⎨
⎪
⎩
F(e)if e ∈A\B,
G(e)if e ∈B\A,
F(e)∪G(e)if e ∈A∩B.
(4) The restricted union of (F, A) and (G, B),
denoted by (F, A)∪R(G, B), is defined as the
soft set (H, C), where C=A∩Band H(c)=
F(c)∪G(c) for all c∈C.
Note that restricted intersection was also known
as bi-intersection in Feng et al. [18], and extended
union was at first introduced and called union by Maji
et al. [29].
Definition 2.5. Let S=(F, A) and T=(G, B)be
two soft sets over a common hemiring S, then:
(1) The “addition” of Sand T, denoted by S+
Tis defined by S+T=(F, A)+(G, B)=
(H, A ∗B), where H(x, y)=F(x)+G(y) for
all (x, y)∈A∗B;
(2) The “multiplication” of Sand T, denoted by
S·Tis defined by S·T=(F, A)·(G, B)=
(H, A ∗B), where H(x, y)=F(x)·G(y) for
all (x, y)∈A∗B.
Definition 2.6. Let Mbe a subset of Sand S=
(F, A) a soft set over S. Then the “addition” of M
and S, denoted by M+Sis defined by M+S=
M+F(x), for all x∈A.
Definition 2.7. [37] For an approximation space
(U, ρ), by a rough approximation in (U, ρ ) we mean
a mapping Apr :P(U)→P(U)×P(U) defined for
every X∈P(U)byApr(X)=(Apr (X),Apr(X)),
where
Apr (X)={x∈U:[x]ρ⊆X},
and
Apr (X)={x∈U:[x]ρ∩X/=∅}.
Apr (X) is called a lower rough approximation of X
and Apr (X) is called an upper rough approximation
of Xin (U, ρ). Moreover, Xis called definable if
Apr (X)=Apr(X).
By combining rough sets and soft sets, Feng [20]
put forth rough soft sets as follows:
Definition 2.8. [20] Let (U, ρ) be a Pawlak approx-
imation space and S=(F, A) a soft set over U.
The lower and upper rough approximations of S=
(F, A) w.r.t. (U, ρ) are denoted by ρ(S)=(F,A)
and ρ(S)=(F,A), which are soft sets over Uwith
F(x)=ρ(F(x)) ={y∈U|[y]ρ⊆F(x)}and F(x)=
ρ(F(x)) ={y∈U|[y]ρ∩F(x)/=∅}, for all x∈A.
If ρ(S)=ρ(S), Sis called definable; otherwise
Sis called a rough soft set.
3. Rough soft hyperrings w.r.t. a normal
hyperideal
Let Ibe a normal hyperideal of a hyperring R.
Then we define a relation ≡Iby
x≡Iyif and only if x−y∩I/=∅.
It is clear that the relation ≡Iis an equivalence
relation on R(see [12]).

1482 X. Ma et al. / Rough soft hyperrings
Let [x]Idenote the equivalence class of xw.r.t. I.
Then [x]I=x+Ifor all x∈R(see [12]).
From now on, we use the pair (R, I) instead of the
approximation space (U, ρ).
Definition 3.1. Let Ibe a normal hyperideal of R,
(R, I) a Pawlak approximation space and S=(F, A)
a soft set over R. The lower and upper rough approx-
imations of S=(F, A) w.r.t. (R, I) are denoted
by:
Apr I(S)=(FI,A) and AprI(S)=(FI,A),
which are soft sets over Rwith FI(x)=
Apr I(F(x)) ={y∈R|y+I⊆F(x)}and FI(x)=
Apr I(F(x)) ={y∈R|(y+I)∩F(x)/=∅}, for all
x∈A.
(i) Apr I(S)=AprI(S), the soft set (S) is said to
be definable;
(ii) Apr I(S)/=AprI(S), Apr I(S)(AprI(S)) is
called a lower (upper) rough soft hyperring
w.r.t. Iover R,ifFI(x)(FI(x)) is a subhyper-
ring of R, for all x∈Supp(F, A). Moreover, S
is called a rough soft hyperring w.r.t. Iover R,
if FI(x) and FI(x) are subhyperrings of R, for
all x∈Supp(F, A).
Example 3.2. Let R={0,a,b,c}be a set with the
hyperoperation (+) and multiplication (·) as follows:
+0abc
00abc
a a {0,b,c}aa
bba b{0,b,c}
cca{0,b,c}b
·0abc
00000
a0abb
b0b00
c0b00
Then Ris a hyperring. Let I={0,b,c}, then it is
a normal hyperideal of Rand [0]I=[b]I=[c]I=
{0,b,c}and [a]I={a}.
Define a soft set S=(F, A) over R, where A=
{0,a},byF(0) ={0,b}and F(a)={0,a}.
By calculations, FI(0) =∅, F I(0) =
{0,b,c},FI(a)=∅and FI(0) ={0,b,c}. Thus, S
is a rough soft hyperring w.r.t. Iover R.
Example 3.3. Let R={0,a,b,c}be a set
with the operation (+) and multiplication (·)as
follows:
+0abc
0 0 abc
aa0cb
bbca0
ccb0a
·0abc
00000
a0000
b0000
c000 a
Then Ris a hyperring. Let I={0,a}, then it is a
normal hyperideal of Rand [0]I=[a]I={0,a}and
[b]I=[c]I={b, c}.
Define a soft set S=(F, A) over R, where A=
{0,a},byF(0) ={0,a,b}and F(a)={0,b}.
By calculations, FI(0) ={0,a}, FI(0) =
R, F I(a)=∅ and FI(0) =R. Thus, Sis a
rough soft hyperring w.r.t. Iover R.
Theorem 3.4. Suppose that (R, I)is a Pawlak
approximation space, S=(F, A)and T=(G, B)
are soft sets over R. Then we have:
(1) Apr I(S)⊆S⊆AprI(S);
(2) Apr I(ST)=AprI(S)Apr I(T);
(3) Apr I(SET)=AprI(S)EApr I(T);
(4) Apr I(ST)⊆AprI(S)Apr I(T);
(5) Apr I(SET)⊆AprI(S)EApr I(T);
(6) Apr I(S∪RT)⊇AprI(S)∪RApr I(T);
(7) Apr I(S
∪T)⊇Apr I(S)
∪Apr I(T);
(8) Apr I(S∪RT)=AprI(S)∪RApr I(T);
(9) Apr I(S
∪T)=Apr I(S)
∪Apr I(T);
(10) S⊆T⇒Apr I(S)⊆AprI(T),Apr I(S)⊆
Apr I(T).
Proof. These proofs are similar to those of Theorems
3 and 4 in [20]. 
Theorem 3.5. Suppose that (R, I)is a Pawlak
approximation space, AprI(S)=(FI,A)and
Apr I(T)=(GI,B)are lower rough soft hyper-
rings over R.IfSTis a non-null soft set, then
Apr I(ST)and AprI(SET)are lower rough
soft heyperrings over R.
Proof. By the hypothesis, ∀x∈Supp(F, A),y ∈
Supp(G, B), FI(x) and GI(y) are subhyperrings of
R. Since STis a non-null soft set, so ∀x∈A∩B,
FI(x)∩GI(x) is a subhyperring of S. That is to say,
Apr I(S)AprI(T) is a lower rough soft hyperring
over S. By Theorem 3.4, we have AprI(ST)is
a lower rough soft hyperring over S. The case for
Apr I(SET) can be similarly proved. 
Remark 3.6. In general, Apr I(ST) and
Apr I(SET) are not upper rough soft hyper-
rings over Reven if AprI(S)=(FI,A) and

X. Ma et al. / Rough soft hyperrings 1483
Apr I(T)=(GI,A) are upper rough soft hyperrings
over R.
Example 3.7. Let R={0,a,b,c}be a set with
the hyperoperation (+) and multiplication (·)as
follows:
+0abc
0 0 abc
aa{0,a}{b, c}{b, c}
bb{b, c}a{0,a}
cc{b, c}{0,a}a
·0abc
00000
a0a0a
b0000
c000 c
Then Ris a hyperring. Let I={0,a}, then it is a
normal hyperideal of Rand [0]I=[a]I={0,a}and
[b]I=[c]I={b, c}.
Define two soft sets S=(F, A) and T=
(G, B) over R, where A={e1,e
2},B={e2,e
3}
and F(e1)={0,a},F(e2)={a, b},G(e2)={0,b},
G(e3)=R.
By calculations, FI(e1)={0,a},FI(e2)=
GI(e2)=GI(e3)=R.
Since A∩B={e2}and F(e2)∩G(e2)={b},we
have AprI(ST)=Apr I({b})={b, c}, this is not
an upper rough soft hyperring over R. At the same
time, Apr I(SET) is also not an upper rough soft
hyperring over R.
Remark 3.8. In general, S∪RTand S
∪Tare not
lower (upper) rough soft hyperrings over Reven if
S=(F, A) and S=(G, B) are lower (upper) rough
soft hyperrings over R.
4. Rough soft hyperideals with a normal
hyperideal
Definition 4.1. Let Ibe a normal hyperideal of R,
(R, I) a Pawlak approximation space and S=(F, A)
a soft set over R, then Sis called a lower (upper)
rough soft hyperideal w.r.t. Iover Rif FI(x)(FI(x))
is a hyperideal of R, for all x∈Supp(F, A). In par-
ticular, Sis called a rough soft hyperideal w.r.t. I
over Rif FI(x) and FI(x) are hyperideals of R, for
all x∈Supp(F, A).
Example 4.2. Let R={0,a,b,c}be a set with
the hyperoperation (+) and multiplication (·)as
follows:
+0abc
0 0 abc
aa{0,c}c{a, b}
bbc{0,c}{a, b}
c c {a, b}{a, b}{0,c}
·0abc
00000
a0aac
b0abc
c0cc0
Then Ris a hyperring. Let I={0,c}, then it is a
normal hyperideal of Rand [0]I=[c]I={0,c}and
[a]I=[b]I={a, b}.
Define a soft set S=(F, A) over R, where A=
{0,a},byF(0) ={0,a,c}and F(a)={0,b,c}.
By calculations, FI(0) ={0,c}, FI(0) =
R, F I(a)={0,c}and FI(0) =R. Thus, Sis a
rough soft hyperideal w.r.t. Iover R.
Lemma 4.3. Let Ibe a normal hyperideal of R
and S=(F, A)a soft set over R. Then AprI(S)=
I+S.
Proof. Let ybe any element of FI(x). Then (y+I)∩
F(x)/=∅,∀x∈A, and so there exists a∈Rsuch
that a∈(y+I)∩F(x), that is, y∈a+Iand a∈
F(x), hence y∈I+F(x). This means that Apr I(S)
⊆I+S.
Conversely, if y∈I+F(x), then there exist a∈
Iand b∈F(x) such that y∈a+b, that is, b∈
−a+y⊆y+I, and so, b∈(y+I)∩F(x), that is,
y∈FI(x). This means that I+S⊆Apr I(S). Thus,
Apr I(S)=I+S.
Now, we discuss some properties of lower and
upper approximations in hyperrings.
Theorem 4.4. Let Ibe a normal hyperideal of R,
S=(F, A)and T=(G, B)any two non-null soft
sets over R. Then
Apr I(S)·AprI(T)⊆Apr I(S·T).
Proof. By Lemma 4.3, ∀x∈Supp(F, A),y ∈
Supp(F, B), we have FI(x)GI(y)=(I+F(x))
(I+G(y)) =I2+IG(y)+F(x)I+F(x)G(y)⊆I+
F(x)G(y)=Apr I(F(x)G(y)). This completes the
proof. 
The following example shows that inclusion sym-
bol “⊆” in above theorem may not be replaced an
equal sign.
Example 4.5. Let R={0,a,b,c}be a set with
the hyperoperation (+) and multiplication (·)as
follows:

1484 X. Ma et al. / Rough soft hyperrings
+0abc
0 0 abc
a a {0,b}{a, c}b
bb{a, c}{0,b}a
ccb a0
·0abc
00000
a0cbc
b0b0b
c0cb c
Then Ris a hyperring. Let I={0,b}, then it is a
normal hyperideal of Rand [0]I=[b]I={0,b}and
[a]I=[c]I={a, c}.
Define two soft sets S=(F, A) and T=(G, B)
over R, where A={0},B={a},byF(0) ={0,a,b},
G(a)={0,b,c}.
By calculations, FI(0) =Rand GI(a)=R.
Hence, FI(0) ·GI(a)={0,b,c}. This means that
Apr I(S)·AprI(T)={0,b,c}.
On the other hand, Apr I(F(0) ·G(a)) =
Apr I({0,b,c})=R, that is, Apr I(S·T)=R.
Thus, Apr I(S)·AprI(T)(Apr I(S·T).
If we strength the condition, we can obtain the
following result:
Theorem 4.6. Let Ibe a normal hyperideal of R,
S=(F, A)and T=(G, B)any two non-null soft
sets over R.IfI2=I, then
(1) Apr I(S)·AprI(T)=Apr I(S·T);
(2) Apr I(S)·AprI(T)⊆Apr I(S·T).
Proof. (1) By Lemma 4.3, ∀x∈Supp(F, A),y ∈
Supp(F, B), we have FI(x)GI(y)=(I+F(x))
(I+G(y)) =I2+IG(y)+F(x)I+F(x)G(y)=
I+F(x)G(y)=Apr I(F(x)G(y)). This means that
Apr I(S)·AprI(T)=Apr I(S·T).
(2) ∀x∈Supp(F, A),y ∈Supp(F, B ), suppose c
be any element of FI(x)GI(y). Then c=n
i=1aibi
for some ai∈FI(x) and bi∈GI(y). Therefore
ai+I⊆F(x) and bi+I⊆G(y) for 1 ≤i≤n.
Since n
i=1(ai+I)(bi+I)⊆F(x)G(y), we have
n
i=1(aibi+aiI+Ibi+I2)=n
i=1(aibi+I)=
c+I⊆F(x)G(y). Hence c∈Apr I(F(x)G(y)).
This completes the proof. 
Remark 4.7. Note that if I2/=I, then Theorem
4.6(2) may not be true as shown in the following
example.
Example 4.8. Let R={0,a,b,c}be a set with
the hyperoperation (+) and multiplication (·)as
follows:
+0abc
0 0 abc
aa{0,c}c{a, b}
bbc{0,c}{a, b}
ccc{a, b}{0,c}
·0abc
00000
a0aa0
b0aa0
c000 0
Then Ris a hyperring. Let I={0,c}, then it is a
normal hyperideal of R,butI2/=I. Also, we have
[0]I=[c]I={0,c}and [a]I=[b]I={a, b}.
Define two soft sets S=(F, A) and T=(G, B)
over R, where A={0},B={a},byF(0) ={0,a,c},
G(a)={0,b,c}.
By calculations, FI(0) ={0,c}and GI(a)=
{0,c}. Hence, FI(0) ·GI(a)={0}. This means that
Apr I(S)·AprI(T)={0}.
On the other hand, Apr I(F(0) ·G(a)) =
Apr I({0,a})=∅, that is, AprI(S·T)=∅.
Thus, Apr I(S)·AprI(T)*Apr I(S·T).
The next example shows that the inclusion symbol
“⊆” in Theorem 4.6 (2) may not be replaced by an
equal sign.
Example 4.9. Let R={0,a,b,c}be a set with the
hyperoperation (+) and multiplication (·) as follows:
+0abc
00abc
a a {0,c}c{a, b}
bbc0a
cc{a, b}a{0,c}
·0abc
00000
a0ccc
b0cac
c0cc c
Then Ris a hyperring. Let I={0,c}, then it is a
normal hyperideal of Rand I2=I. Also, we have
[0]I=[c]I={0,c}and [a]I=[b]I={a, b}.
Define two soft sets S=(F, A) and T=(G, B)
over R, where A={0},B={a},byF(0) =
{0,b},G(a)={0,b,c}.
By calculations, FI(0) =∅,G
I(a)={0,c}.
Hence, FI(0) ·GI(a)=∅. This means that
Apr I(S)·AprI(T)=∅.
On the other hand, Apr I(F(0) ·G(a)) =
Apr I({0,a,c})={0,c}, that is, AprI(S·T)=
{0,c}.
Thus, Apr I(S)·AprI(T)(Apr I(S·T).

X. Ma et al. / Rough soft hyperrings 1485
Theorem 4.10. Let Ibe a normal hyperideal of R,
S=(F, A)and T=(G, B)any two non-null soft
sets over R. Then
(1) Apr I(S)+AprI(T)=Apr I(S+T);
(2) Apr I(S)+AprI(T)⊆Apr I(S+T).
Proof. (1) By Lemma 4.3, we have
Apr I(S)+AprI(T)=I+S+I+T=I+S
+T=Apr I(S+T).
(2) ∀x∈A, y ∈B, suppose cbe any element of
FI(x)+GI(y). Then c∈a+bfor some a∈FI(x)
and b∈GI(y). Hence a+I⊆F(x) and b+I⊆
G(y), we have c+I⊆a+b+I=(a+I)+(b+
I)⊆F(x)+G(y) and so c∈Apr I(F(x)G(y)). This
completes the proof. 
The next example shows that the inclusion symbol
“⊆” in Theorem 4.10(2) may not be replaced by an
equal sign.
Example 4.11. Consider Example 4.8. We know that
FI(0) +GI(a)={0,c},butAprI(F(0) +G(a)) =
{0,a,b,c}. This means that Ap rI(S)+Apr I(T)(
Apr I(S+T).
Definition 4.12. A hyperring Ris called idempotent
if xx =xfor all x∈R.
Remark 4.13. Clearly, if Ris an idempotent hyper-
ring, then A2=Afor any hyperideal Aof R.
Example 4.14. Let R={0,a,b,c}be a set with the
hyperoperation (+) and multiplication (·) as follows:
+0abc
0 0 abc
a a {0,c}ba
bbb{0,a,c}b
cca b 0
·0abc
00000
a0a00
b00bc
c00cc
Then Ris an idempotent hyperring.
Theorem 4.15. Let Rbe an idempotent hyperring, Ia
normal hyperideal of R,S=(F, A)and T=(G, B)
any two non-null soft sets over R. Then
(1) Apr I(S)AprI(T)⊆Apr I(S·T);
(2) Apr I(S)AprI(T)⊆Apr I(S·T).
Proof. (1) ∀x∈A, y ∈B, suppose cbe any ele-
ment of FI(x)∩GI(y). Then c+I∩F(x)/=∅
and c+I∩G(y)/=∅, and so there exist
a∈c+I∩F(x) and b∈c+I∩G(y). That is
a∈c+d,a∈F(x), b∈c+eand b∈G(y) for
some d, e ∈I. Thus ab ∈(c+d)(c+e)=cc +
ce +dc +de ⊆c2+cI +Ic +I2=c+I. Hence
c∈ab +I⊆F(x)G(y)+I=Apr I(F(x)G(y)).
Therefore Apr I(S)AprI(T)⊆Apr I(S·T).
(2) ∀x∈A, y ∈B, let cbe any element of FI(x)∩
GI(y). Then c+I⊆F(x) and c+I⊆G(y). There-
fore, (c+I)(c+I)=cc +cI +Ic +I2=c+I⊆
F(x)G(y), that is, c∈Apr I(F(x)G(y)). Thus,
Apr I(S)AprI(T)⊆Apr I(S·T).
The next example shows that the inclusion symbol
“⊆” in above theorem may not be replaced by an
equal sign.
Example 4.16. Let R={0,a,b,c}be a set with
the hyperoperation (+) and multiplication (·)as
follows:
+0abc
0 0 abc
a a {0,c}c{a, b}
bbc{0,c}{a, b}
c c {a, b}{a, b}{0,c}
·0abc
00000
a0abc
b0bbc
c0cc c
Then Ris an idempotent hyperring. Let I={0,c},
then it is a normal hyperideal of Rand I2=I.
Also, we have [0]I=[c]I={0,c}and [a]I=[b]I=
{a, b}.
(1) Define two soft sets S=(F, A) and T=
(G, B) over R, where A={0},B={0},byF(0) =
{b},G(a)={0,a}.
By calculations, FI(0) ={a, b}, GI(0) =R.
Hence, FI(0) ∩GI(0) ={a, b}. This means that
Apr I(S)AprI(T)={a, b}.
On the other hand, Apr I(F(0) ·G(0)) =
Apr I({0,b})=R, that is, AprI(S·T)=R.
Thus, Apr I(S)AprI(T)(Apr I(S·T).
(2) Define two soft sets S=(F, A) and T=
(G, B) over R, where A={0},B={0},byF(0) =
{0,a,c},G(a)={0,a,b,c}.
By calculations, FI(0) ={0,c},G
I(0) =R.
Hence, FI(0) ∩GI(0) ={0,c}. This means that
Apr I(S)AprI(T)={0,c}.
On the other hand, Apr I(F(0) ·G(0)) =
Apr I({0,a,b,c})=R, that is, Apr I(S·T)=R.
Thus, Apr I(S)AprI(T)(Apr I(S·T).

1486 X. Ma et al. / Rough soft hyperrings
Theorem 4.17. Let Rbe an idempotent hyperring
and Ia normal hyperideal of R.IfS=(F, A)and
T=(G, B)are a soft right hyperideal and a soft left
hyperideal over R, respectively, then
(1) Apr I(S)AprI(T)=Apr I(S·T);
(2) Apr I(S)AprI(T)=Apr I(S·T).
Proof. (1) From the hypothesis, we have F(x)G(y)⊆
F(x) and F(x)G(y)⊆G(y), ∀x∈A, y ∈B. Let
cbe any element of Apr I(F(x)G(y)). Then
c∈I+F(x)G(y), and so, c∈I+F(x) and
c∈I+G(y). That is, c∈FI(x)∩GI(y). Therefore,
Apr I(S·T)⊆AprI(S)Apr I(T). Combining
Theorem 4.15, this completes the proof.
(2) It is similar to (1). 
Note that for any two normal hyperideals Iand J
of R. Then so are I∩Jand I+J.
Lemma 4.18. Let Iand Jbe two normal hyperideals
of Rsuch that I⊆J,S=(F, A)a non-null soft set
over R. Then
(1) Apr I(S)⊇AprJ(S);
(2) Apr I(S)⊆AprJ(S).
Proof. (1) For all x∈A, let a∈AprJ(F(x)) =
FJ(x), then a+J⊆F(x), we have a+I⊆a+J⊆
F(x). Hence a∈Apr I(F(x)), that is, AprJ(F(x)) ⊆
Apr I(F(x)), ∀x∈A. This completes the
proof.
(2) By Lemma 4.3, we have AprI(S)=I+S⊆
J+S=Apr J(S). 
Corollary 4.19. Let Iand Jbe two normal hyperide-
als of Rand S=(F, A)a non-null soft set over R.
Then
(1) Apr (I∩J)(S)⊇Apr(I+J)(S);
(2) Apr (I∩J)(S)⊆Apr(I+J)(S).
Theorem 4.20. Let Ibe a normal hyperideal of R
and S=(F, A)a soft hyperring over R. Then S
is an upper rough soft hyperideal with respect to I
over R.
Proof. By the assumption and Lemma 4.3,
we have AprI(S)+Apr I(S)=I+S+I+S=
I+S=Apr I(S). For any x∈Supp(F, A), we
have RFI(x)=R(I+F(x)) =RI +RF (x)⊆I+
F(x)=FI(x). Similarly, FI(x)R⊆FI(x). This
means that Apr I(S) is an upper rough soft hyperideal
over R.
Theorem 4.21. Let Ibe a normal hyperideal of R
and S=(F, A)any soft hyperideal over R. Then
Apr I(S)/=∅⇔AprI(S)=S.
Proof. If AprI(S)/=∅, then for all x∈Supp(F, A ),
there exists a∈FI(x)⊆F(x), that is, a+I⊆F(x).
Hence I=−a+a+I⊆−a+F(x)⊆F(x). For
any b∈F(x), then b+I⊆b+F(x)⊆F(x), this
implies that b∈FI(x), so F(x)⊆FI(x), hence
FI(x)=F(x), ∀x∈Supp(F, A). This means that
Apr I(S)=S.
Conversely, if Ap rI(S)=S, it is clear that
Apr I(S)/=∅. This completes the proof. 
Definition 4.22. A non-null soft set S=(F, A) over
Ris said to be an addition closed soft set (a mul-
tiplication closed soft set) if for all x, y ∈A, there
exists z∈A(z∈A) such that F(x)+F(y)⊆F(z)
(F(x)F(y)⊆F(z)).
Theorem 4.23. Let I,Jbe two normal hyperideals
of Rand S=(F, A)an addition closed soft set over
R. Then
(1) Apr I(S)+AprJ(S)⊆Apr (I+J)(S).
(2) Apr I(S)/=∅ and AprJ(S)/=∅, then
Apr I(S)+AprJ(S)⊆Apr (I+J)(S).
Proof. (1) For all x, y ∈A, by Lemma 4.3, we have
Apr I(F(x))AprJ(F(y)) =(I+F(x))(J+F(y)) =
IJ +IF (y)+F(x)J+F(x)F(y)⊆I+J+F(x)
F(y). Since Sis a multiplication closed soft set, then
there exist z∈A, such that F(x)F(y)⊆F(z). Hence
Apr I(F(x))AprJ(F(y)) ⊆I+J+F(x)F(y)⊆I+
J+F(z)=Apr (I+J)(F(z)).
(2) For all x, y ∈A, let a∈FI(x)+FJ(y). Then
a∈b+cfor some b∈FI(x) and c∈FJ(y), so
b+I⊆F(x) and c+J⊆F(y). Then a+I+J
⊆b+c+I+J=b+I+c+J⊆F(x)+F(y).
Since Sis an addition closed soft set, then
there exists z∈Asuch that F(x)+F(y)⊆F(z).
Hence a+I+J⊆F(x)+F(y)⊆F(z), which
implies a∈F(I+J)(z)=Apr (I+J)(F(z)). That
is, Apr I(F(x)) +AprJ(F(y)) ⊆Apr (I+J)(F(z)),
∀x∈A. This completes the proof. 
Similarly, we can obtain the following result.
Corollary 4.24. Let I,Jbe two normal hyper-
ideals of Rand S=(F, A)a multiplication
closed soft set over R. Then Apr I(S)·AprJ(S)⊆
Apr (I+J)(S).

X. Ma et al. / Rough soft hyperrings 1487
5. Applications of rough soft hyperrings
in decision making
In this section, we illustrate a novel kind of deci-
sion making methods for rough soft hyperrings, and
provide some relevant algebraic and applied exam-
ples, respectively. Maybe it would be served as a
foundation of rough soft set theory and other decision
making methods in different areas, such as theoretical
computer sciences, information sciences and intelli-
gent systems, and so on.
Decision making method:
Let Rbe a hyperring and Ea set of related
parameters. Let A={e1,e
2,··· ,e
m}⊆E, denote
by withe weight value of ei(i=1,2··· ,m), where
m
i=1ei=1, and S=(F, A) be an original descrip-
tion soft set over R. Let Ibe a normal hyperideal
of Rand (R, I) be a Pawlak approximation space.
Then we present the decision algorithm for rough
soft hyperrings as follows:
Step 1. Input the original description hyperring R,
soft set Sand Pawlak approximation space (R, I),
where Iis a normal hyperideal of R.
Step 2. Compute the lower and upper rough soft
approximation operators Apr I(S) and AprI(S)on
S, respectively.
Step 3. Compute the different values of F(ei),
where F(ei)= |FI(ei)|−|FI(ei)|
|F(ei)|×wi.
Step 4. Find the minimum value F(ek)of F(ei),
where F(ek)=min
iF(ei).
Step 5. The decision is F(ek).
Example 5.1. Assume that we want to find the near-
est accurate hyperring on a soft set S. Consider the
hyperring Ras in Example 4.2. Let I={0,c}be
a normal hyperideal of R. Define a soft set S=
(F, A) over R, where A={e1,e
2,e
3,e
4}, where wi
is the weight value of ei(i=1,2,3,4). The tab-
ular representation of the soft set Sis given in
Table 1.
Table 1
Table for soft set S
0abc
e1,w
1=0.21101
e2,w
2=0.21100
e3,w
3=0.50111
e4,w
4=0.1010 1
Table 2
Table for soft set AprF(S)
0abc
e1,w
1=0.21001
e2,w
2=0.20000
e3,w
3=0.50110
e4,w
4=0.1010 1
Table 3
Table for soft set AprF(S)
0abc
e1,w
1=0.21111
e2,w
2=0.21111
e3,w
3=0.51111
e4,w
4=0.11111
Now, the tabular representations of two soft sets
Apr F(S) and AprF(S) over Rare given by Tables
2 and 3, respectively.
Then, we can calculate F(e1)=0.134,
F(e2)=0.4, F(e3)=0.165, F(e4)=0.2.
This means the minimum value for F(ei)is
F(e1)=0.134. That is, F(e1)={0,a,b}is the
closest accurate hyperring on S.
6. Discussion
Maji et al. [31] first applied soft sets to solve the
decision making problem with the help of rough
approach. The problem of decision making in an
imprecise environment has found importance in
recent years. However, the decision making based on
rough soft sets has not been studied to date, so far as
we are aware. In particular, the decision making with
respect to algebraic structures is even less.
Recently, by means of Feng’s idea in [20], Zhan
[46] firstly applied rough soft sets to algebraic
structures–hemirings, and described some character-
izations of rough soft hemirings. In the present paper,
we put forth a novel rough soft algebraic structure–
hyperring by another kind of way. Let Ibe a normal
hyperideal of a hyperring R, then we define the rela-
tion ≡Iby x≡Iyif and only if x−y∩I/=∅.It
is clear that the relation ≡Iis an equivalence rela-
tion on R. By this idea, we propose the concept of
rough soft hyperrings based on a normal hyperideal
of a hyperring, which is different from Zhan’s idea
in [46].
Based on the above conditions, we first try to put
forth a kind of decision making approaches based
on rough soft hyperrings. It is pointed out that the

1488 X. Ma et al. / Rough soft hyperrings
primary motivation for decision Algorithm is to found
which is the best parameter of a given soft set. In other
words, we focus on finding which is the nearest accu-
rate hyperring on a soft set w.r.t. a normal hyperideal
of hyperrings.
Acknowledgments
The authors are very thankful for the reviewers to
give some valuable comments to improve this paper.
This research is partially supported by a grant of
NNSFC (11561023).
References
[1] H. Aktas¸ and N. C¸aˇ
gman, Soft sets and soft groups, Inform
Sci 177(13) (2007), 2726–2735.
[2] M.I. Ali, Another view on reduction of parameters in soft
sets, Appl Soft Comput 12 (2012), 1814–1821.
[3] M.I. Ali, F. Feng, X.Y. Liu, W.K. Min and M. Shabir, On
some new operations in soft set theory, Comput Math Appl
57(9) (2009), 1547–1553.
[4] R. Ameri and M. Norouzi, New fundamental relation of
hyperrings, European J Combin 334 (2013), 884–891.
[5] N. C¸aˇ
gman and S. Enginoglu, Soft matrix theory and its
decision making, Comput Math Appl 59 (2010), 3308–3314.
[6] N. C¸aˇ
gman and S. Enginoglu, Soft set theory and uni-int
decision making, Eur J Oper Res 207 (2010), 848–855.
[7] D. Chen, E.C. Tsang, D.S. Yeung and X. Wang, The
parametrization reduction of soft sets and its applications,
Comput Math Appl 49 (2005), 757–763.
[8] P. Corsini, Prolegomenaof Hypergroup Theory, Aviani Edi-
tore, 1993.
[9] P. Corsini and V. Leoreanu, Applications of hyperstructure
theory, Advance in Mathematics, Kluwer Academic Pub-
lishers, 2003.
[10] I. Cristea and M. Stefanescu, Hypergroups and n-ary rela-
tions, European J Combin 31 (2010), 780–789.
[11] B. Davvaz, Roughness in rings, Inform Sci 164(1) (2004),
147–163.
[12] B. Davvaz, Isomorphism theorems of hyperrings, Indian J
Pure Appl Math 35 (2004), 321–331.
[13] B. Davvaz, Approximations in hyperrings, J Mult-Valued
Logic & Soft Comput 15 (2009), 471–488.
[14] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory snd
Applications, Academic Press, 2007.
[15] B. Davvaz and A. Salasi, A realization of hyperrings, Comm
Algebra 34 (2006), 4389–4400.
[16] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough
sets, Int J Gen Syst 17 (1990), 191–209.
[17] F. Feng, M. Akram, B. Davvaz and V. Leoreanu-Fotea,
Attribute analysis of information systems based on elemen-
tary soft implications, Knowledge-Based Syst 70 (2014),
281–292.
[18] F. Feng, Y.B. Jun and X.Z. Zhao, Soft semirings, Comput
Math Appl 56(10) (2008), 2621–2628.
[19] F. Feng, X.Y. Liu, V. Leoreanu-Fotea and Y.B. Jun, Soft sets
and soft rough sets, Inform Sci 181(6) (2011), 1125–1137.
[20] F. Feng, C. Li, B. Davvaz and M.I. Ali, Soft sets combined
with fuzzy sets and rough sets: A tentative approach, Soft
Comput 14(9) (2010), 899–911.
[21] M. Jafarpour, H. Aghabozorgi and B. Davvaz, A class of
hyperrings connected to preordered generalized -rings,
European J Combin 44 (2015), 236–241.
[22] M. Krasner, A class of hyperrings and hyperfield, Int J Math
Math Sci 2(1983), 307–312.
[23] V. Leoreanu-Fotea, The lower and upper approximations in
a hypergroup, Inform Sci 178 (2008), 3605–3615.
[24] V. Leoreanu-Fotea and B. Davvaz, Roughness in n-ary
hypergroups, Inform Sci 178 (2008), 4114–4124.
[25] V. Leoreanu-Fotea and B. Davvaz, Fuzzy hyperrings, Fuzzy
Sets and Systems 160 (2009), 2366–2378.
[26] Z. Li, N. Xie and G. Wen,Soft coverings and their parameter
reductions, Appl Soft Comput 31 (2015), 48–60.
[27] Z. Li and T.Xie, The relationship among soft sets, soft rough
sets and topologies, Soft Comput 18 (2014), 717–728.
[28] X. Ma, J. Zhan and V. Leoreanu-Fotea, On (fuzzy) iso-
morphism theorems of -hyperrings, Comput Math Appl
60 (2010), 2594–2600.
[29] P.K. Maji, R. Biswas and A.R. Roy, Soft set theory, Comput
Math Appl 45(4) (2003), 555–562.
[30] P.K. Maji, R. Biswas and A.R. Roy, Fuzzy soft sets, J Fuzzy
Math 9(2001), 589–602.
[31] P.K. Maji, A.R. Roy and R. Biswas, An application of soft
sets in a decision making problem, Comput Math Appl 44
(2002), 1077–1083.
[32] F. Marty, Sur une generalization de la notion de group 8th
Congress Math Scandenaves, Stockholm, 1934, pp. 45–49.
[33] C.G. Massouros, On the theory of hyperrings and hyper-
fields, Algebrai Logika 24 (1985), 728–742.
[34] D. Meng, X. Zhang and K. Qin, Soft rough fuzzy sets and
soft fuzzy rough sets, Comput Math Appl 62(12) (2011),
4635–4645.
[35] S. Mirvakili and B. Davvaz, Applications of the α-relation
to Krasner hyperrings, J Algebra 362 (2012), 145–156.
[36] D. Molodtsov, Soft set theory-first results, Comput Math
Appl 37(4) (1999), 19–31.
[37] Z. Pawlak, Rough sets, Internat J Comput Inform Sci 11(5)
(1982), 341–356.
[38] Y.H. She, On the rough consistency measures of logic theo-
ries and approximate reasoning in rough logic, Int J Approx
Reason 55 (2014), 486–499.
[39] B. Sun and W. Ma, Soft fuzzy rough sets and its application
in decision making, Artif Intell Rev 41(1) (2014), 67–80.
[40] T. Vougiouklis, Hyperstructures and their Representations,
Hadronic Press, Inc 115, Palm Harber, USA, 1994.
[41] Y.Y. Yao, Constructive and algebraic methods of theory of
rough sets, Inform Sci 109 (1998), 21–47.
[42] Y.Y. Yao and B. Yao, Covering based rough set approxima-
tions, Inform Sci 200 (2012), 91–107.
[43] J. Zhan, B. Davvaz and K.P. Shum, On fuzzy isomor-
phism theorems of hypermodules, Soft Comput 11 (2007),
1053–1057.
[44] J. Zhan, B. Davvaz and K.P. Shum, Generalized fuzzy
hyperideals of hyperrings, Comput Math Appl 56 (2008),
1732–1740.
[45] J. Zhan and Y.B. Jun, Soft BL-algebras based on fuzzy sets,
Comput Math Appl 59 (2010), 2037–2046.
[46] J. Zhan, Q. Liu and B. Davvaz, A new rough set theory:
Rough soft hemirings, J Intell Fuzzy Systems 28 (2015),
1687–1697.
[47] J. Zhan and Z. Tan, Approximations in hypergroups, J Appl
Math Comput 21 (2006), 485–494.

X. Ma et al. / Rough soft hyperrings 1489
[48] X.H. Zhang, J. Dai and Y. Yu, On the union and intersection
operations of rough sets based on various approximation
spaces, Inform Sci 292 (2015), 214–229.
[49] W. Zhu, Generalized rough sets based on realtions, Inform
Sci 177(22) (2007), 4997–5011.
[50] Y. Zou and Z. Xiao, Data analysis approaches of soft sets
under incomplete information, Knowledge-Based Syst 21
(2008), 941–945.