Prime idealistic soft BCK=BCI-algebras R. Ameriy , H. Hedayati? , E. Ghasemiany y
Department of Mathematics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran E-mail :
[email protected],
[email protected] [email protected],
[email protected]
?
Department of Mathematics, Faculty of Basic Science, Babol University of Technology, Babol, Iran E-mail :
[email protected],
[email protected]
Abstract The notions of soft prime ideal and prime idealistic soft BCK=BCIalgebras are introduced, and several examples are given. Connections between soft BCK=BCI-algebras and prime idealistic soft BCK=BCI-algebras are provided. The intersection, union, “AND” operation of soft prime ideals and prime idealistic soft BCK=BCI algebras are investigated.
1
Introduction
To solve complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theorise have their own di¢ culties. Uncertainties cannot be handled using traditional mathematical tools but may be 1
dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theorise have their own di¢ culties which are pointed out in [21]. Maji et al. [20] and Molodtsov [21] suggested that one reason for these di¢ culties may be due to the inadequacy of the parametrization tool of the theory. To overcome these di¢ culties, Molodtsov [21] introduced the concept of soft set as a new mathematical tool for dealinng with uncertainties that is free from the di¢ culties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [20] described the application of soft set theory to a decision making problem. Maji et al. [19] also studied several operations of soft sets. Chen et al. [4] presented a new di…nition of soft set parametrization reduction, and compared this de…nition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theorise dealing with uncertainties has been studied by some authors. The study of structures of fuzzy sets in algebraic structures are catrried out by several authors (see [1,5,6,10,11,12,13,14,26]). Akta¸s and Ça¼ gman [1] studied the basic concepts of soft set theory, and compared sofe sets to fuzzy and rough sets, providing examples to clarify their di¤erences. They also discussed the notion of soft groups. In this paper, we deal with the algebraic structure of BCK=BCI -algebras by applying soft set theory. We introduce the notion of soft prime ideals and prime idealistic soft BCK=BCI -algebras, and give several examples. We investigate relations between soft BCK=BCK-algebras and prime idealistic soft BCK=BCI -algebras. We establishe the intersection, union, “AND” operation of soft prime ideals and prime idealistic soft
BCK=BCI -algebras.
2
Basic results on BCK-algebras
An algebra (X;
; 0) of type (2; 0) is called a BCI-algebra if it satis…es the following
conditions:
2
(1) (8x; y; z 2 X)(((x y) (x z)) (z y) = 0); (2) (8x; y 2 X)((x (x y)) y = 0); (3) (8x 2 X)(x x = 0); (4) (8x; y 2 X)(x y = 0; y x = 0 =) x = y): If a BCI-algebra X satis…es the following condition:
(5) (8x 2 X)(0 x = 0); then X is called a BCK -algebra. Any BCK -algebra X
satis…es the following
axioms:
(a1) (8x 2 X)(x 0 = x);
(a2) (8 x; y; z 2 X)(x 6 y =) x z 6 y z; z y 6 z x); (a3) (8x; y; z 2 X)((x y) z = (x z) y);
(a4) (8x; y; z 2 X)((x z) (y z) 6 x y);
where x 6 y if and only if x y = 0. A BCK -algebra X is said to be commutative
if x ^ y = y ^ x for all x; y 2 X , where x ^ y = y
(y x): A nonempty subset S of
BCK=BCI -algebra X is called a subalgebra of X if x y 2 S for all x; y 2 S: A subset A of a BCK -algebra X is called an ideal of X if 0 2 A and if whenever
x y; y 2 A then x 2 A for every x; y 2 X: An ideal A of a commutative BCK -algebra X is said to be prime and denoted by A CP X; if x ^ y 2 A implies x 2 A or y 2 A.
Example 2.1. Let X = f0; a; b; c; dg be a BCK -algebra X with the following cayley table: ^
0
a
b
c
d
0
0
0
0
0
0
a
0
a
0
a
0
b
0
0
b
0
b
c
0
a
0
c
0
d
0
0
b
0
d
and I = f0; a; b; dg : Then I is a prime ideal of X , since x ^ y 2 I implies x 2 I or
y 2 I:
3
Example 2.2. Let X = f0; a; b; cg be a BCK -algebra with the following cayley table: ^
0
a
b
c
0
0
0
0
0
a
0
a
0
a
b
0
0
b
0
c
0
a
0
c
and I = f0; ag. Then I is not a prime ideal of X , since b ^ c = 0 2 I but b 62 I and
c 62 I:
3
Soft sets
Molodtsov [21] de…ned the notion of soft set in the following way: Let U be an initial universe set and E be a set of parameters. Let P (U ) denotes the power set of U and
A
E:
De…nition 3.1. [21] A pair (F; A) is called a sof t set over U , where F is a mapping given by F : A ! P (U ): In other words, a soft set over U is a parameterized family of subsets of the universe
U . For " 2 A, F (" ) may be considered as the set of "-approximate elements of the soft set (F; A). Cleary, a soft set is not a set. For illustration, Molodtsov considered several examples in [21].
De…nition 3.2. [19] Let (F; A) and (G; B) be two soft sets over a common universe U . The intersection of (F; A) and (G; B) is de…ned to be the soft set (H; C) satisfying the following conditions :
(i) C = A \ B; (ii) (8e 2 C)(H(e) = F (e) or G(e)), (as both are same set)).
In this case, we write (F; A) \e (G; B) = (H; C):
De…nition 3.3. [19] Let (F; A) and (G; B) be two soft sets over a common universe U . The union of (H; C) satisfying the following conditions: 4
(i) C = A [ B; (ii) for all e 2 C, 8 > > F (e) if e 2 AnB; > < H(e) = G(e) if e 2 BnA; > > > : F (e) [ G(e) if e 2 A \ B: e (G; B) = (H; C): In this case, we write (F; A) [
De…nition 3.4. [19] If (F; A) and (G; B) are two soft sets over a common universe e (G; B) is de…ned by U , then “ (F; A) AND (G; B)”denoted by (F; A) ^ where H(
4
e (G; B) = (H; A (F; A) ^
B);
; ) = F ( ) \ G( ) for all ( ; ) 2 A
B:
Soft prime ideals
In what follows let X and A be a BCK=BCI-algebra and a nonempty set, respectively, and R will refer to an arbitrary binary relation between an element of X, that is, R is a subset of A De…nition 4.1. A set
X without otherwise speci…ed. valued f unction F : A
! P (X) can be de…ned as
F (x) = fy 2 X j (x; y) 2 Rg for all X 2 A. The pair (F; A) is then a soft set over X. De…nition 4.2. [9] Let (F; A) be a soft set over X . Then (F; A) is called a sof t BCK=BCI -algebra over X if F (x) is a subalgebra of X for all x 2 A: De…nition 4.3. [17] Let (F; A) be a soft BCK=BCI -algebras over X . A soft set (G; I) e (F; A), if it sattis…es: over Xis called a soft ideal of (F; A), denoted by (G; I) C (i) I
A
(ii)(8x 2 I) (G (x) C F (x)):
De…nition 4.4. Let (F; A) be a soft BCK=BCI -algebra over X . A soft ideal (G; I) e p (F; A), if G(x) over X is called a sof t prime ideal of (F; A), denoted by (G; I) C is a prime ideal of F (x) for all x 2 I:
5
Example 4.5. Let X = f0; a; b; c; dg be a BCK -algebra with the following cayley table: 0
a
b
c
d
0
0
0
0
0
0
a
a
0
a
a
a
b
b
b
0
b
b
c
c
c
c
0
c
d
0
d
d
d
0
Let (F; A) be a soft set over X , where A = X and F : A ! P (X) a set-valued function de…ned by
F (x) = fy 2 X j y (y x) 2 f0; agg for all x 2 A. Then (F; A) is a soft BCK -algebra over X . Let (G; I) be a soft set over X , where I = fa; b; cg and G : I ! P (X) a set-valued function de…ned by
G(x) = fy 2 X j y (y x) 2 f0; dgg for all x 2 I .
Then
G(a) = f0; b; c; dg Cp X = F (a); G(b) = f0; a; c; dg Cp
f0; a; c; dg = F (b), and G(c) = f0; a; b; dg Cp f0; a; b; dg = F (c). Hence (G; I) is a soft prime ideal of (F; A):
Theorem 4.6. Let (F; A) be a soft BCK=BCI -algebra over X . For any soft prime ideal (G1 ; I 1 ) and (G2 ; I 2 ) over X where I1 \ I2 6= ?, we have
e (G2 ; I2 ) C e p (F; A); (G2 ; I2 ) C e p (F; A) =) (G1 ; I1 ) \ e p (F; A): (G1 ; I1 )C
Proof. Using De…nition 3.2 we can write
(G1 ; I1 ) \e (G2 ; I2 ) = (G; I),
where I = I 1 \ I2 and G(x) = G1 (x) or G2 (x) for all x 2 I . Obviously I
A and
e p (F; A) G : I ! P (X) is a mapping hence (G; I) is a soft set over X . Since (G1 ; I1 )C
e p (F; A), we know that G(x) = G1 (x) Cp F (x) or G(x) = G2 (x) Cp F (x) and (G2 ; I2 )C
e (G2 ; I2 ) = (G; I)C e p (F; A): for all x 2 I . Hence (G1 ; I1 )\
Corollary 4.7. Let (F; A) be a soft BCK=BCI -algebra over X . For any soft prime ideal (G; I) and (H; I) over X , we have
6
e p (F; A); (H; I) C e p (F; A) =) (G; I) \e (H; I) C e p (F; A): (G; I) C
Proof. Straightforward.
Theorem 4.8. Let (F; A) be a soft BCK=BCI -algebra over X For any soft prime ideal (G; I) and (H; J) over X in which I and J are disjoint, we have
e (H; J) C e p (F; A) , (H; J) C e p (F; A) =) (G; I) [ e p (F; A): (G; I) C
e p (F; A) and (H; J)C e p (F; A). By means of De…nition 3.3, Proof. Assume that (G; I)C we can write (G; I) [e (H; J) = (K; U ) where U = I [ J and for every x 2 U ,
8 > > G(x) > < K(x) = H(x) > > > : G(x) [ H(x)
if x 2 I nJ;
if x 2 J nI; if x 2 I \ J:
Since I \ J = ?, either x 2 I nJ or x 2 J nI for all x 2 U . If x 2 I nJ , then
e P (F; A). If x 2 J nI , then K(x) = H(x)Cp F (x) K(x) = G(x) CP F (x) since (G; I)C
e p (F; A). Thus K(x) Cp F (x) for all x 2 U ,and so since (G; I)C e p (F; A): (G; I) [e (H; J) = (K; U )C
If I and J are not disjoint in Theorem 4.8, this Theorem is not true in general as seen in the following example .
Example 4.9. Let X = f0; a; b; c; dg be a BCK -algebra with the following cayley table: 0
a
b
c
0
0
0
0 0 0
a
a
0
0 0 0
b
b
b
o
b
c
c
c
c
0 0
d
d
d
c
b
7
d
0
0
Let (F; A) be a soft set over X , where A = X and F : A ! P (X) is a set-valued function de…ned by
F (x) = fy 2 X j y (y x) 2 f0; bgg for all x 2 A. Then F (0) = X; F (a) = F (b) = f0; b; c; dg and F (c) = F (d) = f0; bg which are subalgebras of X . Hence (F; A) is a soft BCK -algebra over X . Let (A; I) be a soft set over X , where I = fb; c; dg and A : I ! P (X) is a set-valued function de…ned by
A(x) = fy 2 X j y x = 0g
for all x 2 A. Then A(b) = f0; a; bg Cp f0; b; c; dg = F (b) ; A(c) = f0; a; cg Cp
f0; bg = F (c), and A(d) = X Cp F (d), and so (A; I) is a soft prime ideal (F; A). Let (R; J) be a soft set over X , where J = f0; bg and R : J ! P (X) is a set-valued function de…ned by
R(x) = fy 2 X j y (y x) = 0g
for all x 2 J . Then R(0) = X Cp F (0); R(b) = f0; cg Cp f0; a; b; cg = F (b) , and so
(R; J) is a soft prime ideal of (F; A): Then (G; U ) = (A; I) [e (R; J) is not a soft prime
ideal of (F; A) since G(b) = A(b) [ R(b) = f0; a; b; cg is not an ideal of F (b) because
d
c = b 2 f0; a; b; cg and d 62 f0; a; b; cg. Therefore G (b) is not a prime ideal of
F (b) :
5
Prime idealistic soft BCK/BCI-algebras
De…nition 5.1. Let (F; A) be a soft set over X . Then (F; A) is called a prime idealistic sof t BCK=BCI -algebra over X if F (x) is a prime ideal of X for all x 2 A: Example 5.2. The soft set (F; A) in Example 4.5 is a soft prime ideal of X , since F (0) = F (a) = F (d) = X Cp X , F (b) = f0; a; c; dg Cp X and F (c) = f0; a; b; dg Cp X. Then (F; A) is a prime idealistic soft BCK -algebra over X . Example 5.3. Consider a BCI -algebra X = f0; a; b; cg with the following cayley table:
8
0
a
b
c
0
0
a
b
c
a
a
0
c
b
b
b
c
0
a
c
c
b
a
0
Let A = X and F : A ! P (X) be a set-valued function de…ned by
F (x) = f0; xg for all x 2 A. Then F (0) = f0g ; F (a) = f0; ag ; F (b) = f0; bg and F (c) = f0; cg which are prime ideals of X . Hence (F; A) is a prime idealistic soft BCI -algebra over X .
Proposition 5.4. Let (F; A) and (F; B) be soft sets over X where B
A
X . If
(F; A) is a prime idealistic soft BCK=BCI -algebra over X , then so is (F; B): Proof. Straightforward. The converse of Proposition 5.5 is not ture in general as seen in the following example. For any element x of a BCI -algebra X , we de…ne the order of x, denoted by o(x), as
o(x) = min fn 2 N j 0 xn = 0g Example 5.5. Let X = f0; a; b; c; d; e; f; gg and consider the following cayley table:
Then (X;
0
a
b
c
d
e
f
g
0
0
0
0
0
d
d
d
d
a
a
0
0
0
e
d
d
d
b
b
b
0
0
f
f
d
d
c
c
b
a
0
g
f
e
d
d
d
d
d
d
0
0
0
0
e
e
d
d
d
a
0
0
0
f
f
f
d
d
b
b
0
0
g
g
f
e
d
c
b
a
0
; 0) is a BCI -algebra (see[3]). Let (F; A) be a soft set over X , where
A = X and F : A ! P (X) is a set-valued function de…ned as follows: 9
F (x) = f0g [ fy 2 X j o(x) = o(y)g for all x 2 A. Then F (0) = F (a) = F (b) = F (c) = f0; a; b; cg is a prime ideal of X , but F (d) = F (e) = F (f ) = F (g) = f0; d; e; f; gg is not an ideal of X since b f = d 2
f0; d; e; f; gg and b 62 f0; d; e; f; gg. Therefore (F; A) is not a prime ideal of X. Hence (F; A) is not a prime idealistic soft BCI -algebra over X . If we take B = fa; b; cg
X
and de…ne a set-valued function G : B ! P (X) by
G(X) = fy 2 x j o(x) = o(y)g
for all x 2 B; G (a) = G (b) = G (c) = f0; a; b; cg Cp X; then (G; B) is a prime ideal of X. Hence (G; B) is a prime idealistic soft BCI -algebra over X . Since every prime ideal of a BCK -algebra is a subalgebra, we know that every prime idealistic soft BCK -algebra over a BCK -algebra X is a soft BCK -algebra over X , but the converse is not true as seen in the following example.
Example 5.6. Let X = f0; a; b; c; dg be a BCK -algebra with the following cayley table :
0
a
b
c
d
0
0
0
0 0
0
a
a
0
0 0
0
b
b
b
0 0
0
c
c
c
c
0
0
d
d
c
c
a
0
Let (F; A) be soft set over X , where A = X and F : A ! P (X) is a set-valued function de…ned by
F (x) = fy 2 X j y x = 0g for all x 2 A. Then F (0) = f0g ; F (a) = f0; ag ; F (b) = f0; a; bg ; F (c) = f0; a; b; cg, and F (d) = X which are subalgebras of X . Hence (F; A) is a soft BCK -algebra over
X . But F (c) = f0; a; b; cg is not an ideal of X since d b = c 2 F (c) and d 62 F (c), then F (c) is not a prime ideal of X. Therfore (F; A) is not a prime idealistic soft
BCK -algebra over X . Theorem 5.7. Let (F; A) and (G; B) be two prime idealistic soft BCK=BCI -algebras 10
e (G; B) is a prime idealistic over X . If A \ B 6= ?, then the intersection (F; A) \ sof t BCK=BCI -algebra over X .
Proof. Using De…nition 3.2, we can write (F; A) \e (G; B) = (H; C), where C = A \ B
and H(x) = F (x) or G(x) for all x 2 C . Note that H : C ! P (X) is a mapping, and therefore (H; C) is a soft set over X . Since (F; A) and (G; B) are prime idealistic soft BCK=BCI -algebras over X , it follows that H(x) = F (x) is a prime ideal of X , or
H(x) = G(x) is a prime ideal of X for all x 2 C . Hence (H; C) = (F; A) \e (G; B) is
a prime idealistic soft BCK=BCI -algebra over X .
Corollary 5.8. Let (F; A) and (G; A) be two prime idealistic soft BCK=BCI -algebras e (G; A) is a prime idealistic sof t BCK=BCI over X . Then their intersction (F; A) \
algebra over X .
Proof. Straightforward . Theorem 5.9. Let (F; A) and (G; B) be two prime idealistic soft BCK=BCI -algebras e (G; B) is a prime idealistic over X . If A and B are disjoint, then the union (F; A) [ sof t BCK=BCI -algebra over X .
e (G; B) = (H; C), where C = A [ B Proof. Using De…nition 3.3, we can write (F; A)[ and for every e 2 C ,
8 > > F (e) > < H(e)= G(e) > > > : F (e) [ G(e)
if e 2 A n B; if e 2 B n A; if e 2 A \ B:
Since A\B = ?, either x 2 A n B or x 2 B n A for all x 2 C . If x 2 A n B , then
H(x) = F (x) is a prime ideal of X; since (F; A) is a prime idealistic soft BCK=BCI algebra over X . If x 2 B n A, then H(x) = G(x) is a prime ideal of X; since (G; B) is
a prime idealistic soft BCK=BCI -algebra over X . Hence (H; C) = (F; A) [e (G; B) is a prime idealistic soft BCK=BCI -algebra over X .
Theorem 5.10. If (F; A) and (G; B) are prime idealistic soft BCK=BCI -algebras e (G; B) is a prime idealistic soft BCK=BCI-algebra over X: over X , then (F; A)^ 11
e (G; B) = (H; A Proof. By means of De…nition 3.4, we know that (F; A) ^ H(x; y) = F (x) \ G(y) for all (x; y) 2 A
B) where
B . Since F (x) and G(y) are prime ideals
of X , the intersection F (x) \ G(y) is also a prime ideal of X . Hence H(x; y) is a prime ideal of X for all (x; y) 2 A
B , and therefore (F; A) ^e (G; B) = (H; A
B) is a
prime idealistic soft BCK=BCI -algebra over X .
De…nition 5.11. A prime idealistic soft BCK=BCI-algebra (F; A) over X is said to be whole if F (x) = X for all x 2 A.
Example 5.12. Consider the BCI -algebra X = f0; a; b; c; d; e; f; gg in example 5.5. For C = fb; c; f; gg
X, let H : C ! P (X) be a set-valued function de…ned by H(x) = fy 2 X j y x 2 f0; a; d; egg
for all x 2 C . Then H(x) = X for all x 2 C; and so (H; C) is a whole prime idealistic soft BCI -algebra over X .
Example 5.13. Consider the BCK -algebra X = f0; a; b; c; dg in example 5.6. For D = fb; c; dg
X , let K : D ! P (X) be a set-valued function de…ned by K(x) = fy 2 X j y x 2 f0; a; cgg
for all x 2 D. Then K(x) = X for all x 2 D, and so (K; D) is a whole prime idealistic soft BCK -algebra overX .
De…nition 5.14. Suppose (X1 ;
1 ; 01 )
and (X2 ;
2 ; 02 )
are two BCK-algebras. A
mapping f : X1 ! X2 is called a homomorphism frome X1 into X2 if, for any x; y 2 X1 , f (x
1
y) = f (x)
2
f (y) :
Let f : X ! Y be a mapping of BCK=BCI -algebras. For a set (F; A) over X,
(f (F ); A) is a soft set over Y where f (F ) : A ! P (Y ) is de…ned by f (F )(x) = f (F (x)) for all x 2 A:
Lemma 5.15. Let f : X ! Y be an onto homomorohism of BCK=BCI -algebras. If (F; A) is a prime idealistic soft BCK=BCI -algebra over X , then (f (F ); A) is a prime idealistic soft BCK=BCI -algebra over Y .
12
Proof. For every x 2 A we have f (F )(x) = f (F (x)) is a prime ideal of Y since F (x) is a prime ideal of X and its onto homomorphic image is also a prime ideal of Y . Hence
(f (F ); A) is a prime idealistic soft BCK=BCI -algebra overY . Theorem 5.16. Let f : X ! Y be an onto homomorphism of BCK=BCI -algebras . If (F; A) is whole, then (f (F ); A) is the whole prime idealistic soft BCK=BCI -algebra over Y .
Proof. Suppos that (F; A) is whole. Then F (x) = X for all x 2 A, and so f (F )(x) = f (F (x)) = f (X) = Y
for all x 2 A. It follows from Lemma 5.14 and
De…nition 5.11 that (f (F ); A) is a whole prime idealistic soft BCK=BCI -algebra over
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