U.P.B. Sci. Bull., Series A, Vol. 7, Iss. , 201 ISSN 1223-7027 [628224]

U.P.B. Sci. Bull., Series A, Vol. 7*, Iss. *, 201* ISSN 1223-7027
A NOVEL STUDY OF SOFT SETS OVER n-ARY SEMIGROUPS
Xiaowu Zhou1, Dajing Xiang2and Jianming Zhan2
In this paper, we show that the regular n-ary semigroups can be de-
scribed by using idealistic soft n-ary semigroups. The relationships between regular
n-ary semigroups and soft regular n-ary semigroups are also discussed. Finally,
we introduce quotient n-ary semigroups via soft congruence relations and estab-
lish some homomorphisms and related properties with respect to soft congruence
relations.
Keywords: n-ary semigroups; Soft n-ary semigroups; Idealistic soft n-ary semi-
groups ; Soft congruence relations; Soft homomorphisms;
MSC2010: 03G10; 03G25; 06F05.
1.Introduction
The generalization of classical algebraic structures to n-ary structures was first
initiated by Kasner [17] in 1904. In the following decades and nowadays, a number
of different n-ary systems have been studied in depth in different contexts. Sioson
[20] introduced regular n-ary semigroups and investigated their related properties.
Since then, the nature of regular n-ary semigroups were discussed in detail by Dudek
[11]. In [6, 7, 8], Dudek proved some results and presented many examples of n-ary
groups. Earlier Crombez et al. [2, 3] gave the generalized rings and named it as
(m,n)-rings and introduced their quotient structure. Up till now, the theory of n-
ary systems has many applications, for example, application in physics [21, 22] and
in automata theory [15], fuzzy sets and rough set theory(see[4, 5, 9, 24]) and so on.
In dealing with uncertainties, many theories have been recently developed,
including the theory of probability, theory of fuzzy sets, theory of intuitionistic
fuzzy sets and theory of rough sets and so on. Although many new techniques have
been developed as a result of theories, yet difficulties are still. The major difficulties
posed by these theories are probably due to the inadequacy of parameters. In 1999,
Molodtsov[19] initiated the concept of soft set theory, which was a completely new
approach for modeling uncertainty and had a rich potential for applications in several
directions. Later on, Maji et al. [18] introduced several operations in soft set theory
1Department of Mathematics, Hubei University for Nationalities,Enshi, Hubei Province, 445000,
P. R. China, E-mail: [anonimizat]
2Department of Mathematics, Hubei University for Nationalities,Enshi, Hubei Province, 445000,
P. R. China, E-mail: [anonimizat]
3Department of Mathematics, Hubei University for Nationalities,Enshi, Hubei Province, 445000,
P. R. China, E-mail: [anonimizat]
1

2 Xiaowu Zhou, Dajing Xiang, Jianming Zhan
and carried out a detailed theoretical study on soft sets. The algebraic structure
of soft sets has been studied by several authors. For examples, Akta¸ s and C ¸ aˇ gman
[1] introduced the notion of soft groups and discussed their basic properties. Feng
et al. [12] defined the notions of soft semirings, idealistic soft semirings, soft ideals
and introduced the algebraic properties of semirings. Other applications of soft set
theory in different algebraic structure can be found in [16, 23] and so on. Feng et al.
[13] initiated the soft binary relations, some interesting properties of soft equivalence
and soft congruence relations are discussed.
In this paper, we first recall some concepts and results on n-ary semigroups and
soft sets. In section 3, we define the notion of soft n-ary semigroups and idealistic
softn-ary semigroups over an n-ary semigroup. Some basic related properties with
softn-ary semigroups and idealistic soft n-ary semigroups are proposed. In section
4, we show that the regular n-ary semigroups can be described by using idealistic
softn-ary semigroups. Moreover, we discuss relationships between regular n-ary
semigroups and soft regular n-ary semigroups. In section 5, we give the concept
of soft congruence relations over an n-ary semigroup and introduce quotient n-
ary semigroups via soft congruence relations. Some homomorphisms and related
properties with respect to soft congruence relations are proposed.
2.Preliminaries
A non-empty set Stogether with one n-ary operation f:Sn→S, where
n≥2, is called an n-ary groupoid and is denoted by ( S,f). According to the
general convention used in the theory of n-ary groupoids, the sequence of elements
xi,xi+1,· · ·,xjis denoted by xj
i. In the case j < i , it is the empty symbol. If
xi+1=xi+2=…=xi+t=x, then we write(t)xinstead ofxi+t
i+1. In this convention,
f(x1,x2,…,x n) =f(xn
1),
and
f(x1,…,x i,x,…,x|{z}
t,xi+t+1,…,x n) =f(xi
1,(t)x,xn
i+t+1).
Ann-ary groupoid ( S,f) is called ( i,j)-associative if
f(xi−1
1,f(xn+i−1
i ),x2n−1
n+i) =f(xj−1
1,f(xn+j−1
j ),x2n−1
n+j)
hold for all x1,x2,…,x 2n−1∈S. If this identity holds for all 1 ≤i≤j≤n,then
we say that the operation fis associative, and ( S,f) is called an n-ary semigroup.
Ann-ary semigroup ( S,f) is called idempotent if f(x,…,x ) =xfor allx∈S.
A non-empty subset Hof ann-ary semigroup ( S,f) is ann-ary subsemigroup
if (H,f) is ann-ary subsemigroup, i.e., if it is closed under the operation f. Through-
out this paper, unless otherwise mentioned, Swill denote an n-ary semigroup.
Denition 2.1. [11, 20] A non-empty subset IofSis called an i-ideal ofSif for
everyx1,…,x i−1,xi+1,…,x n∈Switha∈I, thenf(xi−1
1,a,xn
i+1)∈I.Iis called
an ideal of SifIis ani-ideal for every 1≤i≤n.

A novel study of soft sets over n-ary semigroups 3
Denition 2.2. [4]LetRbe an equivalence relation of S.Ris called a con-
gruence of Sif(xi,yi)∈Rimplies (f(xn
1),f(yn
1))∈Rfor all 1≤i≤nand
x1,x2,…,x n,y1,y2,…,y n∈S.
Denition 2.3. [4]A mapping φ:S→TfromSintoTis called a homomorphism
ifφ(f(xn
1)) =g(φ(x1),φ(x2),…,φ (xn))for allx1,x2,…,x n∈S.
Denition 2.4. [19] A pair (F,A)is called a soft set over U, whereA⊆Eand
F:A→P(U)is a set-valued mapping.
For a soft set ( F,A), the set Supp( F,A) ={x∈A|F(x)̸=∅}is called a soft
support of ( F,A). Thus a null soft set is indeed a soft set with an empty support,
and we say that a soft set ( F,A) is non-null if Supp( F,A)̸=∅.
Denition 2.5. [19]A soft set (F,A)overSis called an absolute soft set if F(a) =S
for alla∈A.
Denition 2.6. [14]A soft set (F,A)overSis called a full soft set if∪
x∈AF(x) =S.
Denition 2.7. [12]Let(F,A)and(G,B)be two soft sets over a common universe
U. The inclusion symbol \ e⊆" of (F,A)and(G,B), denoted by (F,A)e⊆(G,B), is
dened as
(1)A⊆B;
(2)F(x)⊆G(x)for allx∈A.
If(F,A)e⊆(G,B)and(G,B)e⊆(F,A), then we denote (F,A) = (G,B).
3.Softn-ary semigroups and idealistic soft n-ary semigroups
In this section, we define the notion of soft n-ary semigroups and idealistic soft
n-ary semigroups over S. Some basic related properties with soft n-ary semigroups
and idealistic soft n-ary semigroups are proposed.
Denition 3.1. Let(F1,A1),(F2,A2),…, (Fn,An)be soft sets over S. Then the
ef-product of them, denoted by ef((F1,A1),(F2,A2),…, (Fn,An)), is dened as a soft
set(G,B) =ef((F1,A1),(F2,A2),…, (Fn,An)), whereB=∩{Ai|i= 1,2,…,n } ̸=
∅andG:B→P(S)dened byG(a) =f(F1(a),F2(a),…,F n(a))for alla∈B.
Denition 3.2. Let(F,A)be a non-null soft set over S. Then (F,A)is called a soft
n-ary semigroup over SifF(a)is ann-ary subsemigroup of Sfor alla∈Supp(F,A).
Example 3.1. LetS={−i,0,i}be a set with a ternary operation fas the usual
multiplication of complex numbers. Then (S,f)is a ternary semigroup. Let (F,A)be
a soft set over S, whereA={a,b,c}andF:A→P(S)be a set-valued function de-
ned byF(x) ={y∈S|(x,y)∈R}for allx∈A, whereR={(a,0),(c,−i),(c,0),(c,i)}.
ThenF(a) ={0},F(b) =∅,F(c) ={−i,0,i}. Therefore (F,A)is a soft ternary
semigroup over S.
Proposition 3.1. A non-null soft set (F,A)overSis a softn-ary semigroup if and
only if ef((F,A),…, (F,A))e⊆(F,A).

4 Xiaowu Zhou, Dajing Xiang, Jianming Zhan
Proof. Let (F,A) be a softn-ary semigroup over S, then for all a∈Supp(F,A),F(a)
is ann-ary subsemigroup of S. By Definition 3.1, we denote ef((F,A),…, (F,A)) =
(G,A), whereG:A→P(S) defined by G(a) =f(F(a),…,F (a)) for alla∈
Supp(F,A).F(a) is ann-ary subsemigroup of S, it follows that f(F(a),…,F (a))⊆
F(a), that isG(a)⊆F(a) for alla∈A. Hence ef((F,A),· · ·,(F,A))e⊆(F,A).
Conversely, if ef((F,A),…, (F,A))e⊆(F,A), it follows that f(F(a),…,F (a))⊆
F(a) for alla∈Supp(F,A) . This means F(a) is ann-ary subsemigroup of S. By
Definition 3.2, ( F,A) is a softn-ary semigroup over S. 
Denition 3.3. Let(F,A)be a non-null soft set over S. Then (F,A)is called aj-
idealistic soft n-ary semigroup over S, ifF(x)is aj-ideal ofSfor allx∈Supp(F,A).
Moreover, if (F,A)is aj-idealistic soft n-ary semigroup of Sfor eachj= 1,2,…,n ,
then (F,A)is called an idealistic soft n-ary semigroup.
Example 3.2. Consider the natural numbers Nwith usual multiplication. Let S=
2N. We dene the 4-ary operation f,f(a,b,c,d ) =abcd
2for alla,b,c,d ∈S. Then
(S,f)is a 4-ary semigroup. Let (F,A)be a soft set over S, whereA=Nand
F:A→P(S)is a set-valued function dened by F(x) ={4xn|n∈N}for all
x∈N. Since for all x1,x2,x3∈Sandr∈F(x),f(xi−1
1,r,x4
i+1)∈F(x)for
i= 1,2,3,4. HenceF(x)is an ideal of (S,f). Therefore (F,A)is an idealistic soft
4-ary semigroup of S.
Remark 3.1. If an idealistic soft n-ary semigroup (F,A)satisfying ef((F,A),…, (F,
A)) = (F,A), then (F,A)is called a soft idempotent idealistic soft n-ary semigroup.
Proposition 3.2. Let(H,E )be an absolute soft set over S. Then, a non-null soft
set(F,A)overSis aj-idealistic soft n-ary semigroup if and only if ef((j−1)
(H,E ),(F,A),
(n−j)
(H,E ))e⊆(F,A),wherej= 1,2,…,n .
Proof. Let (F,A) be aj-idealistic soft n-ary semigroup, then F(a) is aj-ideal ofSfor
alla∈Supp(F,A), it follows that f(xk−1
1,r,xn
k+1)∈F(a) for allx1,x2,…,x n∈S
and allr∈F(a). By Definition 3.1, we denote ef((k−1)
(H,E ),(F,A),(n−k)
(H,E )) = (G,A),
whereG:A→P(S) defined by G(a) =f(j−1
H(a),F(a),n−j
H(a)) for alla∈Supp(F,A).
Since (H,E ) is an absolute soft set over S, soH(a) =Sfor alla∈Supp(F,A). Hence
G(a) =f(j−1
S ,F (a),n−j
S)⊆F(a). Therefore, ef((j−1)
(H,E ),(F,A),(n−j)
(H,E ))e⊆(F,A).
Conversely, if ef((j−1)
(H,E ),(F,A),(n−j)
(H,E ))e⊆(F,A), thenf(j−1
H(a),F(a),n−j
H(a))⊆
F(a) for alla∈Supp(F,A). Since (H,E ) is an absolute soft set over S, soH(a) =S
for alla∈Supp(F,A). Hencef(j−1
S ,F (a),n−j
S)⊆F(a). This means F(a) is aj-ideal
ofS. By Definition 3.3, ( F,A) is aj-idealistic soft n-ary semigroup over S. 
Proposition 3.3. Let(H,E )be an absolute soft set over S. Then, a non-null soft
set(F,A)overSis an idealistic soft n-ary semigroup if (F,A)is aj-idealistic soft
n-ary semigroup for all 1≤j≤n.

A novel study of soft sets over n-ary semigroups 5
Proof. It is straightforward. 
4.The characterizations of regular n-ary semigroups
In this section, we show that the regular n-ary semigroups can be described by
using idealistic soft n-ary semigroups. Moreover, we discuss relationships between
regularn-ary semigroups and soft regular n-ary semigroups.
Denition 4.1. [11]An element a∈Sis called regular if there exist x2,x3,…,x n−1∈
Ssuch thatf(a,xn−1
2,a) =a.Sis called regular if every element of Sis regular.
Denition 4.2. [20]Sis called regular if for all a∈S, there exist xij∈S(i,j=
1,2,…,n )such that
a=f(f(a,x1n
12),f(x21,a,x2n
23),…,f (xnn−1
n1,a))
.
Remark 4.1. In[11], Dudek proved that Denition 4.1 and Denition 4.2 are equiv-
alent.
Example 4.1. LetS={(0 0
0 0)
,(1 0
0 0)
,(0 1
0 0)
,(0 0
1 0)
,(0 0
0 1)}
,
where the ternary operation fis the usual matrix multiplication. One can easily
show that (S,f)is a regular ternary semigroup.
Lemma 4.1. [20]The following conditions are equivalent:
(1)Sis regular,
(2)n∩
i=1Bi=f(B1,B2,…,B n)for alli-dealsBi,
(3) every ideal is idempotent.
Theorem 4.1. Sis regular if and only if
(F1,A1)e(F2,A2)…e(Fn,An) =ef((F1,A1),(F2,A2),…, (Fn,An))
for everyj-idealistic soft n-ary semigroup (Fj,Aj)(j= 1,2,…,n ).
Proof. LetSbe a regular n-ary semigroup and ( Fj,Aj) be anj-idealistic soft n-ary
semigroup over S, respectively. Then ( G,B) =ef((F1,A1),(F2,A2),…, (Fn,An)),
whereB=n∩
j=1AjandGis defined as G(a) =f(F1(a),F2(a),…,F n(a)) for all
a∈B. Also, (K,C ) = (F1,A1)e(F2,A2)…e(Fn,An), whereC=n∩
j=1AjandKis
defined asK(a) =F1(a)∩F2(a)…∩Fn(a) for alla∈C. By Proposition 3.2, we have
ef((F1,A1),(F2,A2),…, (Fn,An))e⊆ef((F1,A1),(H,E ),…, (H,E ))e⊆(F1,A1),
ef((F1,A1),(F2,A2),…, (Fn,An))e⊆ef((H,E ),(F2,A2),…, (H,E ))e⊆(F2,A2),
…
ef((F1,A1),(F2,A2),…, (Fn,An))e⊆ef((H,E ),(H,E ),…, (Fn,An))e⊆(Fn,An),

6 Xiaowu Zhou, Dajing Xiang, Jianming Zhan
where (H,E ) is an absolute soft set over S.
Thus, ef((F1,A1),(F2,A2),…, (Fn,An))e⊆(F1,A1)e(F2,A2)…e(Fn,An).
Letb∈F1(a)∩F2(a)…∩Fn(a), thenb∈Fj(a). SinceSis regular, there exist
xij∈S(i,j= 1,2,…,n ) such that b=f(f(b,x1n
12),f(x21,b,x2n
23),…,f (xnn−1
n1,b)).
By (Fj,Aj) is anj-idealistic soft n-ary semigroup over S, we have Fj(a) is an
j-ideal ofS, sof(b,x1n
12)∈F1(a),f(x21,b,x2n
23)∈F2(a),…,f (xnn−1
n1,b)∈Fn(a).
Henceb∈f(F1(a),F2(a),…,F n(a)). This means F1(a)∩F2(a)…∩Fn(a)⊆
f(F1(a),F2(a),…,F n(a)), for alla∈n∩
j=1Aj. It implies ( F1,A1)e(F2,A2)…e
(Fn,An)e⊆ef((F1,A1),(F2,A2),…, (Fn,An)).
Hence (F1,A1)e(F2,A2)…e(Fn,An) =ef((F1,A1),(F2,A2),…, (Fn,An))
for everyj-idealistic soft n-ary semigroup( Fj,Aj)(j= 1,2,…,n ).
Conversely, suppose that A1=A2=…=An=SandFjis a function
fromAjtoP(S). For alla∈S, we define F1(a) ={a} ∪f(a,S,…,S ),F2(a) =
{a} ∪f(S,a,S,…,S ),· · ·,Fn(a) ={a} ∪f(S,S,…,S,a ). Since
f(F1(a),S,…,S ) =f({a} ∪f(a,S,…,S ),S,…,S )
⊆f(a,S,…,S )∪f(f(a,S,…,S ),S…,S )
=f(a,S,…,S )∪f(a,S,…,S,f (S,…,S ))
⊆f(a,S,…,S )∪f(a,S,…,S )
=f(a,S,…,S ).
Hencea∈F1(a)∩F2(a)…∩Fn(a) =f(F1(a),F2(a),…,F n(a))⊆f(F1(a),S,…,S )⊆
f(a,S,…,S ). In similar way, we have a∈f(S,a,S,…,S ),…,a ∈f(S,…,S,a ).
So
f(f(a,S,…,S ),f(S,a,S,…,S ),…,f (S,…,S,a ))
=f(a,f(S,…,S ),f(a,S,…,S ),…,f (S,…,S ),a)
⊆f(a,S,…,S,a ).
Therefore,
f(F1(a),F2(a),…,F n(a))
=f({a} ∪f(a,S,…,S ),…,{a} ∪f(S,S,…,S,a ))
=f(a,…,a ),…,∪f(f(a,S,…,S ),f(S,a,S,…,S ),…,f (S,…,S,a ))
=f(f(a,S,…,S ),f(S,a,S,…,S ),…,f (S,…,S,a ))
⊆f(a,S,…,S,a ).
This means a∈f(a,S,…,S,a ). HenceSis regular. 
Theorem 4.2. Sis regular if and only if every idealistic soft n-ary semigroup over
Sis soft idempotent.
Proof. LetSbe a regular n-ary semigroup and ( F,A) an idealistic soft n-ary semi-
group. Putting ( F,A) = (F1,A1) = (F2,A2) =…= (Fn,An), then by Theorem
4.1, (F,A) = (F,A)e(F,A)…e(F,A) =ef((F,A),…, (F,A)). Thus (F,A) is soft
idempotent.

A novel study of soft sets over n-ary semigroups 7
Conversely, if all ( Fj,Aj) arej-idealistic soft n-ary semigroups over S, where
j= 1,2,…,n . Then (F1,A1)e(F2,A2)…e(Fn,An) is an idealistic soft n-ary semi-
group over Sand (F1,A1)e(F2,A2)…e(Fn,An)e⊆(Fj,Aj) for eachj= 1,2,· · ·,n.
This implies that
(F1,A1)e(F2,A2)…e(Fn,An)
=ef((F1,A1)e(F2,A2)…e(Fn,An),…, (F1,A1)e(F2,A2)…e(Fn,An))
e⊆ef((F1,A1),(F2,A2),…, (Fn,An)).
Butef((F1,A1),(F2,A2),…, (Fn,An))e⊆(F1,A1)e(F2,A2)…e(Fn,An) always holds.
Thus, (F1,A1)e(F2,A2)…e(Fn,An) =ef((F1,A1),(F2,A2),…, (Fn,An)). Hence
by Theorem 4.1, Sis regular. 
By Theorems 4.1 and 4.2, we have the following Corollary.
Corollary 4.1. Then the following conditions are equivalent:
(1)Sis regular,
(2)(F1,A1)e(F2,A2)…e(Fn,An) =ef((F1,A1),(F2,A2),…, (Fn,An))for
everyj-idealistic soft n-ary semigroup (Fj,Aj)(j= 1,2,…,n ),
(3) every idealistic soft n-ary semigroup is soft idempotent.
Denition 4.3. A softn-ary semigroup (F,A)overSis called a soft regular n-ary
semigroup if for each a∈A,F(a)is regular.
The following examples show that if Sis a regular n-ary semigroup then soft n-
ary semigroup ( F,A) overSmay not be soft regular and if the soft n-ary semigroup
(F,A) overSis soft regular then Smay not be regular.
Example 4.2. Consider the regular ternary semigroup in Example 4.1. Let (F,A)be
a soft set over S, whereA=SandF:A→P(S)be a set-valued function dened by
F((0 0
0 0)
) =S,F((1 0
0 0)
) =F((0 1
0 0)
) ={(0 0
0 0)
,(1 0
0 0)
,(0 1
0 0)}
,
F((0 0
1 0)
) =F((0 0
0 1)
) ={(0 0
0 0)
,(0 0
1 0)
,(0 0
0 1)}
. Then (F,A)
is soft ternary semigroup over S. But it is not soft regular, because F((0 0
1 0)
) =
F((0 0
0 1)
) ={(0 0
0 0)
,(0 0
1 0)
,(0 0
0 1)}
is not regular ternary subsemi-
group ofS.
Example 4.3. Let(S,f)be a 4-ary semigroup derived from the semigroup (S,·),
where the 4-ary operation dened f(d,d,d,d ) =dandf(x,y,z,u ) =awherex,y,z,u
∈ {a,b,c}. Clearly, (S,f)is not a regular 4-ary semigroup. Let A={α,β}be a set
of parameters such that F(α) ={a},F(β) ={a,d}. Then (F,A)is a soft regular
4-ary semigroup over (S,f)becauseF(α)andF(β)are regular 4-ary subsemigroups
of(S,f).

8 Xiaowu Zhou, Dajing Xiang, Jianming Zhan
Remark 4.2. In the above examples, we have demonstrated that the regularity of a
n-ary semigroup Sdoes not imply the regularity of a soft n-ary semigroup over S.
Also, the regularity of a soft n-ary semigroup over a given n-ary semigroup Sdoes
not imply the regularity of the n-ary semigroup. However, we still have the following
proposition.
Proposition 4.1. Let(F,A)be a full soft regular n-ary semigroup over S. ThenS
is a regular n-ary semigroup.
Proof. Let (F,A) be a soft regular n-ary semigroup over S. ThenF(α) is regular for
eachα∈A. Now leta∈S, because (F,A) is a full soft set, we have S=∪
α∈AF(α),
then there exists β∈Asuch thata∈F(β). SinceF(β) is regular, then exist
x2,x3,…,x n−1∈F(β) such that f(a,xn−1
2,a) =a. Sincex2,x3,…,x n−1∈F(β)⊆∪
α∈AF(α) =S, henceSis regular. 
Proposition 4.2. IfSis an idempotent n-ary semigroup, then every soft n-ary
semigroup (F,A)overSis soft regular.
Proof. It is straightforward. 
5.Soft congruence relations over n-ary semigroups and homomor-
phisms
In this section, we give the concept of soft congruence relations over Sand
introduce quotient n-ary semigroups via soft congruence relations. Some homomor-
phisms and related properties with respect to soft congruence relations are proposed.
Denition 5.1. A non-null soft set (ρ,A)overS×Sis called a soft congruence
relation over Sifρ(α)is a congruence relation on Sfor allα∈Supp(ρ,A).
If Supp(ρ,A) =∅, then (ρ,A) is called a null soft congruence relation over S,
denoted ∅2
A.
Example 5.1. Let(S,f)be ann-ary semigroup in Example 3.2 and A=N+.
Consider the set-valued function ρ:A→P(S×S)given byρ(α) ={(x,y)∈
S×S|x≡y(mod α )}for allα∈A. Thenρ(α)is a congruence relation on (S,f).
Hence (ρ,A)is a soft congruence relation on (S,f).
Theorem 5.1. Let(ρ,A)be a soft congruence relation over an n-ary semigroup
(S,f)andS/(ρ,A) ={[x]ρ(α)|x∈S}where [x]ρ(α)={y∈S|(x,y)∈ρ(α),α∈A}.
Then for any α∈A,S/(ρ,A)is ann-ary semigroup under the n-ary operation
dened by
F([x1]ρ(α),[x2]ρ(α),…, [xn]ρ(α)) = [f(xn
1)]ρ(α)
for allx1,x2,…,x n∈S.
Proof. We shall first show that Fis well defined. Let x1,x2,…,x n,y1,y2,…,y n∈S
be such that
[x1]ρ(α)= [y1]ρ(α),[x2]ρ(α)= [y2]ρ(α),…, [xn]ρ(α)= [yn]ρ(α)

A novel study of soft sets over n-ary semigroups 9
for allα∈A. It follows that ( x1,y1)∈ρ(α),(x2,y2)∈ρ(α),…, (xn,yn)∈ρ(α).
Since (ρ,A) is a soft congruence relation over ( S,f), by Definition 4.1, for any α∈A,
ρ(α) is a congruence relation on ( S,f). Hence we have ( f(xn
1),f(yn
1))∈ρ(α). This
means [f(xn
1)]ρ(α)= [f(yn
1)]ρ(α). HenceFis well defined. S/(ρ,A) is closed under
the operation FandFis (i,j)-associative is obvious. Therefore ( S/(ρ,A),F) is an
n-ary semigroup for all α∈A. 
Theorem 5.2. Let(ρ,A)and(σ,B)be two soft congruence relations over an n-ary
semigroupSwith (ρ,A)e⊆(σ,B). Then the soft binary relation (σ,B)/(ρ,A)over
S/(ρ,A), dened by (δ,C) = (σ,B)/(ρ,A)whereC=A∩Band
δ(α) =σ(α)/ρ(α) ={([x]ρ(α),[y]ρ(α))∈S/(ρ,A)×S/(ρ,A),(x,y)∈σ(α)}
for allα∈C, is a soft congruence relation over S/(ρ,A)and
(S/(ρ,A))/((σ,B)/(ρ,A))∼=S/(σ,B).
Proof. SinceC=A∩B=A, then for any α∈C,ρ(α) andσ(α) are congruence
relations on S, soδ(α) is an equivalence relation on S/(ρ,A).
Leta1,a2,…,a n,b1,b2,…,b n∈S, if (a1,b1)∈σ(α),…, (an,bn)∈σ(α), then
we have (f(an
1),f(bn
1))∈σ(α) and ([a1]ρ(α),[b1]ρ(α))∈δ(α),…, ([an]ρ(α),[bn]ρ(α))∈
δ(α). According to Theorem 5.1, ( F([a1]ρ(α),…, [an]ρ(α)),F([b1]ρ(α),…, [bn]ρ(α)) =
([f(an
1)]ρ(α),[f(bn
1)]ρ(α))∈δ(α).This means δ(α) is a congruence relation on S/(ρ,A)
for allα∈C. Hence (σ,B)/(ρ,A) is a soft congruence relation over S/(ρ,A).
From Theorem 5.1, we know ( S/(ρ,A))/((σ,B)/(ρ,A)) andS/(σ,B) are two
n-ary semigroups. Define a mapping:
h: (S/(ρ,A))/((σ,B)/(ρ,A))→S/(σ,B)
byh([[x]ρ(α)]δ(α)) = [x]σ(α)for allx∈Sandα∈A. If [[x]ρ(α)]δ(α)= [[y]ρ(α)]δ(α), then
([x]ρ(α),[x]ρ(α))∈δ(α), it follows that ( x,y)∈σ(α), this means [ x]σ(α)= [y]σ(α).
Hencehis well defined.
LetF∗be ann-ary operation of ( S/(ρ,A))/((σ,B)/(ρ,A)). Then we have
h(F∗([[x1]ρ(α)]δ(α),…, [[xn]ρ(α)]δ(α))
=h([F([x1]ρ(α),…, [xn]ρ(α))]δ(α))
=h([[f(xn
1)]ρ(α)]δ(α))
= [f(xn
1)]σ(α)
=F([x1]σ(α),…, [xn]σ(α))
=F(h([[x1]ρ(α)]δ(α)),…,h ([[xn]ρ(α)]δ(α))).
This means his a homomorphism. If [ x]σ(α)= [y]σ(α), then (x,y)∈σ(α), so
([x]ρ(α),[y]ρ(α))∈δ(α). It follows that [[ x]ρ(α)]δ(α)= [[x]ρ(α)]δ(α), andhis injective.
Furthermore, for any [ y]σ(α)∈S/(σ,B), there exists ρ(α) =σ(α) such that
h([[y]σ(α)]δ(α)) =h([[y]ρ(α)]δ(α)) = [y]σ(α)for allα∈A. Hencehis surjective. This
completes the proof. 
Lemma 5.1. Letφ: (S1,f)→(S2,g)be ann-ary semigroup epimorphism.
(1) Ifγis a congruence relation on S1, deneφ(γ) ={(φ(x),φ(y))∈S2×S2|(x,y)∈

10 Xiaowu Zhou, Dajing Xiang, Jianming Zhan
γ}, thenφ(γ)is a congruence relation on S2.
(2) Ifθis a congruence relation on S2such thatφ−1(θ)̸=∅, whereφ−1(θ) =
{(x,y)∈S1×S1|(φ(x),φ(y))∈θ}, thenφ−1(θ)is a congruence relation on S1.
Proposition 5.1. Letφ: (S1,f)→(S2,g)be ann-ary semigroup epimorphism.
(1) If (ρ,A)is a soft congruence relation over S1, the image φ(ρ,A)of(ρ,A)is
denoted by (φ(ρ),A), then (φ(ρ),A)is a soft congruence relation over S2, where
φ(ρ)(α) ={(φ(x),φ(y))∈S2×S2|(x,y)∈ρ(α)}
for allα∈A.
(2) If (σ,B)is a soft congruence relation over S2such thatφ−1(σ,B)̸=∅2
B, where
φ−1(σ,B)is the inverse image of (σ,B)is denoted by (φ−1(σ),B)and
φ−1(σ)(β) ={(x,y)∈S1×S1|(φ(x),φ(y))∈σ(β)}
for allβ∈B. Thenφ−1(σ,B)is a soft congruence relation over S1.
Theorem 5.3. Letφ: (S1,f)→(S2,g)be ann-ary semigroup epimorphism. If
(ρ,A)is a soft congruence relation over S1, thenS1/(ρ,A)∼=S2/φ(ρ,A).
Proof. By Theorem 5.1 and Proposition 5.1, S1/(ρ,A) andS2/φ(ρ,A) aren-ary
semigroups.
Define a mapping
ψ:S1/(ρ,A)→S2/φ(ρ,A) byψ([x]ρ(α)) = [φ(x)]φ(ρ)(α)
for allx∈S1andα∈A.
We first show that ψis well defined. In fact, let x,x′∈S1, if [x]ρ(α)= [x′]ρ(α),
then (x.x′)∈ρ(α),α∈A. From Proposition 5.1, we have ( φ(x),φ(x′))∈φ(ρ)(α) for
allα∈A, which implies that [ φ(x)]φ(ρ)(α)= [φ(x′)]φ(ρ)(α). Henceψis well defined.
Moreover,ψis a homomorphism. Let x1,x2,…,x n∈S1,α∈A,FandF′be
twon-ary operations of S1/(ρ,A) andS2/φ(ρ,A), respectively. Then we have
ψ(F([x1]ρ(α),…, [xn]ρ(α))
=ψ([f(xn
1)]ρ(α))
=φ(f(xn
1))φ(ρ)(α)
= [g(φ(x1),φ(x2),…,φ (xn))]φ(ρ)(α)
=F′([φ(x1)]φ(ρ)(α),[φ(x2)]φ(ρ)(α),…, [φ(xn)]φ(ρ)(α)).
Henceψis a homomorphism.
For anyx,x′∈S1,α∈A, if [φ(x)]φ(ρ)(α)= [φ(x′)]φ(ρ)(α), then (φ(x),φ(x′))∈
φ(ρ)(α). By Proposition 5.1, we have ( x,x′)∈ρ(α), which implies [ x]ρ(α)= [y]ρ(α).
This means ψis injective.
Sinceφis surjective, then for any [ y]φ(ρ)(α)∈S2/φ(ρ,A),y∈S2,α∈A, there
existsx1∈S1such thatφ(x) =y. Soψ([x]ρ(α)) = [φ(x)]φ(ρ)(α)= [y]φ(ρ)(α), this
means that ψis surjective. Hence ψis a isomorphism and S1/(ρ,A)∼=S2/φ(ρ,A).


A novel study of soft sets over n-ary semigroups 11
Theorem 5.4. Letφ: (S1,f)→(S2,g)be ann-ary semigroup epimorphism. If
(σ,B)is a soft congruence relation over S2andφ−1(σ,B)̸=∅2
B, then
S1/φ−1(σ,B)∼=S2/(σ,B)
Proof. It is similar to the proof of Theorem 5.3 and we omit it. 
6.Conclusion
It is well known that ideals and congruence relations always play an important
role in the study of algebraic structures. In this paper we introduce some basic con-
cepts, examples and related properties with soft n-ary semigroups and idealistic soft
n-ary semigroups over an n-ary semigroups. Furthermore, we give the characteriza-
tion of regular n-ary semigroups by idealistic soft n-ary semigroups, and investigated
its related properties. Moreover, we discuss the quotient n-ary semigroups in terms
of soft congruence relations, and establish isomorphism (soft homomorphism) theo-
rems about n-ary semigroups (soft n-ary semigroups).
In the future study of n-ary semigroups, we can apply idealistic soft n-ary al-
gebraics and soft congruence relations of other n-ary algebras, such as, n-ary groups,
and (m,n)-ary semirings, and so on. We hope this theory can be served as a founda-
tion of some applied fields, such as decision making, data analysis, and forecasting.
Acknowledgements
The authors are very grateful to referees for their valuable comments and
suggestions for improving this paper.
This research is partially supported by a grant of National Natural Science
Foundation of China (11561023; 11461025).
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