Soft intersection near-rings with its applications
Neural Comput & Applic (2012) 21 (Suppl 1):S221–S229 DOI 10.1007/s00521-011-0782-4
ORIGINAL ARTICLE
Soft intersection near-rings with its applications Aslıhan Sezgin • Akın Osman Atagu¨n Naim C ¸ ag˘man
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Received: 18 October 2011 / Accepted: 7 December 2011 / Published online: 24 December 2011 Ó Springer-Verlag London Limited 2011
Abstract In this paper, we first define soft intersection near-ring (soft int near-ring) by using intersection operation of sets. This new notion can be regarded as a bridge among soft set theory, set theory and near-ring theory, since it shows how a soft set effects on a near-ring structure by means of intersection and inclusion of sets. We then derive its basic properties with illustrative examples. Moreover, we obtain some analog of classical near-ring theoretic concepts for soft int near-ring and give the applications of soft int near-ring to near-ring theory. Keywords Soft set Soft int near-ring Soft int subnear-ring Soft int-ideal Soft pre-image Soft image a-Inclusion
1 Introduction The notion of soft set was introduced in 1999 by Molodtsov [1] as a new mathematical tool for dealing with uncertainties. Since its inception, it has received much attention
A. Sezgin (&) Department of Mathematics, Faculty of Arts and Science, Amasya University, 05100 Amasya, Turkey e-mail: [email protected] A. O. Atagu¨n Department of Mathematics, Faculty of Arts and Science, Bozok University, 66100 Yozgat, Turkey e-mail: [email protected] N. C ¸ ag˘man Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpas¸ a University, 60250 Tokat, Turkey e-mail: [email protected]
in the mean of algebraic structures such as groups [2], semirings [3], rings [4], BCK/BCI-algebras [5–7], normalistic soft groups [8], BL-algebras [9], BCH-algebras [10] and near-rings [11]. Atagu¨n and Sezgin [12] defined the concepts of soft subrings and ideals of a ring, soft subfields of a field and soft submodules of a module and studied their related properties with respect to soft set operations also union soft substructues of near-rings and near-ring modules are studied in [13]. Cag˘man et al. defined two new type of group action on a soft set, called group SI-action [14] and group SU-action [14], which are based on the inclusion relation and the intersection of sets and union of sets, respectively. Algebraic structures of soft sets have been studied by some authors. Maji et al. [15] presented some definitions on soft sets and based on the analysis of several operations on soft sets Ali et al. [16] introduced several operations of soft sets and Sezgin and Atagu¨n [17] studied on soft set operations as well. Soft set relations and functions [18] and soft mappings [19] were proposed and many related concepts were discussed, too. Moreover, the theory of soft set has gone through remarkably rapid strides with a wideranging applications especially in soft decision making as in the following studies: [20–22] and some other fields such as [23–26]. Cag˘man and Enginog˘lu [21] redefined the operations of soft sets to develop the soft set theory. By using their definitions, in this paper, we define ‘‘soft int near-ring’’. The structure of soft int near-ring is based on the inclusion relation and intersection of sets, and since this new concept brings the soft set theory, set theory and near-ring theory together, it is very functional by means of improving the soft set theory with respect to near-ring structure. From this view, it functions as a bridge among soft set theory, set theory and near-ring theory. Based on the soft int near-ring
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definition, we introduce the concepts of soft int subnearring and soft int-ideal. We moreover investigate these notions with respect to soft image, soft pre-image and a-inclusion of soft sets. Finally, we give some applications of soft int near-ring to near-ring theory.
2 Preliminary In this section, we recall some basic notions relevant to near-rings and soft sets. (N, ? , .) is called a near-ring if (N1) (N2) (N3)
(N, ?) forms a group (not necessarily abelian) (N, .) forms a semi-group and (a ? b)c = ac ? bc for all a, b, c [ N (i.e. we study on right near-rings).
Throughout this paper, N will always denote a right nearring with zero element 0. A subgroup M of N with MM M is called a subnear-ring of N. A normal subgroup I of N is called a right ideal if IN I and denoted by I C r N: It is called a left ideal if n(s ? i) - ns [ I for all n, s [ N and i [ I and denoted by I C ‘ N: If such a normal subgroup I is both left and right ideal in N, then it is called an ideal of N and denoted by I C N: By a near-field, we shall mean an algebraic system (N, ? , .), where ðN n f0g; :Þ forms a group. If N is a near-field, then its identity element will be denoted by 1N in what follows. For all undefined concepts and notions we refer to [28]. From now on, U refers to an initial universe, E is a set of parameters, P(U) is the power set of U and A; B; C E. Definition 1 by
[1, 21] A soft set fA over U is a set defined
fA : E ! PðUÞ such that fA ðxÞ ¼ ; if x 62 A:
Definition 3 [21] Let fA and fB be soft sets over U. Then, e fB ; is defined as fA [ e fB ¼ union of fA and fB, denoted by fA [ fAe[ B ; where fAe[ B ðxÞ ¼ fA ðxÞ [ fB ðxÞ for all x [ E. e fB ; is defined as Intersection of fA and fB, denoted by fA \ e fB ¼ f ; where f ðxÞ ¼ fA ðxÞ \ fB ðxÞ for all x [ E. fA \ Ae \B Ae \B Definition 4 [21] Let fA and fB be soft sets over U. Then, _-product of fA and fB, denoted by fA_ fB, is defined as fA _ fB = fA_B, where fA_B(x, y) = fA(x) [ fB(y) for all (x, y) [ E 9 E. ^-product of fA and fB, denoted by fA ^ fB, is defined as fA ^ fB = fA^B, where fA^B(x, y) = fA(x)\ fB(y) for all (x, y) [ E 9 E. Definition 5 [14] Let fA and fB be soft sets over the common universe U and W be a function from A to B. Then, soft image of fA under W; denoted by WðfA Þ; is a soft set over U by ðWðfA ÞÞðbÞ (S ffA ðaÞ j a 2 A and WðaÞ ¼ bg; if W1 ðbÞ 6¼ ;; ¼ ;; otherwise for all b [ B. And soft pre-image (or soft inverse image) of fB under W; denoted by W1 ðfB Þ; is a soft set over U by ðW1 ðfB ÞÞðaÞ ¼ fB ðWðaÞÞ for all a [ A. Definition 6 [27] Let fA be a soft set over U and a be a subset of U. Then, upper a-inclusion of fA, denoted by fAa ; and lower a-inclusion of fA, denoted by fAa ; are defined as fAa ¼ fx 2 A j fA ðxÞ ag and fAa ¼ fx 2 A j fA ðxÞ ag;
Here fA is also called approximate function. A soft set over U can be represented by the set of ordered pairs
respectively.
fA ¼ fðx; fA ðxÞÞ : x 2 E; fA ðxÞ 2 PðUÞg:
3 Soft int near-ring
It is clear to see that a soft set is a parametrized family of subsets of the set U. It is worth noting that the sets fA(x) may be arbitrary. Some of them may be empty, some may have nonempty intersection. We refer to [1, 15, 21] for further details. If we define more than one soft set in a subset A of the set of parameters E, then the soft sets will be denoted by fA, gA, hA etc. If we define more then one soft set in some subsets A, B, C etc. of parameters E, then the soft sets will be denoted by fA, fB, fC etc., respectively. Definition 2 [21] Let fA and fB be soft sets over U. Then, fA is a soft subset of fB, denoted by fA ~ fB ; if fA ðxÞ fB ðxÞ for all x [ E.
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In this section, we first define soft intersection near-ring which is abbreviated as soft int near-ring. We then define soft int subnear-ring, soft int-ideal of a near-ring with illustrative examples and study their basic properties with respect to soft set operations. Definition 7 Let N be a near-ring and fN be a soft set over U. Then, fN is called a soft int near-ring over U if it satisfies the following properties: (i) fN ðx þ yÞ fN ðxÞ \ fN ðyÞ; (ii) fN(- x) = fN(x), (iii) fN ðxyÞ fN ðxÞ \ fN ðyÞ for all x, y [ N.
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Example 1 Let N = {0, 1, 2, 3} be the (right) near-ring due to [28] (Near-rings of low order (D-10)) defined by the following tables:
Proof Assume that fN is a soft int near-ring over U. Then, for all x 2 N; fN ð0Þ ¼ fN ðx xÞ fN ðxÞ \ fN ðxÞ ¼ fN ðxÞ \ fN ðxÞ ¼ fN ðxÞ. h
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Theorem 1 Let N be a near-ring and fN be a soft set over U. Then, fN is a soft int near-ring over U if and only if
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Assume that N is the set of parameters and U ¼ x x j x; y 2 Z4 ; 2 2 matrices with Z4 terms, is y y the universal set. We construct a soft set fN over U by 0 0 1 1 3 3 0 0 ; ; ; fN ð0Þ ¼ 1 1 0 0 0 0 3 3 3 3 0 0 fN ð1Þ ¼ ; ; 1 1 0 0 0 0 3 3 1 1 fN ð2Þ ¼ ; ; ; 1 1 0 0 0 0 3 3 0 0 ; : fN ð3Þ ¼ 1 1 0 0 Then, one can easily show that the soft set fN is a soft int near-ring over U. Example 2 In Example 1, assume that N = {0, 1, 2, 3} is again the set of parameters and U = S3, symmetric group, is the universal set. We define a soft set fN by fN ð0Þ ¼ fð12Þg fN ð1Þ ¼ fð12Þ; ð13Þg fN ð2Þ ¼ fð12Þ; ð123Þg fN ð3Þ ¼ fð12Þ; ð13Þg Since fN ð2 2Þ ¼ fN ð0Þ ¼ fð12Þg + fN ð2Þ \ fN ð2Þ ¼ fð12Þ; ð123Þg; fN is not a soft int near-ring over U. It is easy to see that if we take the near-ring as N = {0}, then fN is a soft int near-ring over U no matter how fN is defined and no matter U is. Lemma 1 Let fN be a soft int near-ring over U. Then, fN ð0Þ fN ðxÞ for all x [ N.
(i) fN ðx yÞ fN ðxÞ \ fN ðyÞ (ii) fN ðxyÞ fN ðxÞ \ fN ðyÞ for all x, y [ N. Proof Suppose that fN is a soft int near-ring over. Then, by Definition 7, fN ðxyÞ fN ðxÞ \ fN ðyÞ and fN ðx yÞ fN ðxÞ \ fN ðyÞ ¼ fN ðxÞ \ fN ðyÞ for all x, y [ N. Conversely, assume that fN ðxyÞ fN ðxÞ \ fN ðyÞ and fN ðx yÞ fN ðxÞ \ fN ðyÞ for all x, y [ N. If we choose x = 0, then fN ð0 yÞ ¼ fN ðyÞ fN ð0Þ \ fN ðyÞ ¼ fN ðyÞ by Lemma 1. Similarly, fN ðyÞ ¼ fN ððyÞÞ fN ðyÞ; thus, fN(- y) = fN(y) for all y [ N. Also, by assumption fN ðx þ yÞ fN ðxÞ \ fN ðyÞ ¼ fN ðxÞ \ fN ðyÞ: This completes the proof. h Theorem 2 Let fN be a soft int near-ring over U. If fN (x -y) = fN(0) for any x, y [ N, then fN(x) = fN(y). Proof Assume that fN(x - y) = fN(0) for any x, y [ N. Then, fN ðxÞ ¼ fN ðx y þ yÞ fN ðx yÞ \ fN ðyÞ ¼ fN ð0Þ \ fN ðyÞ ¼ fN ðyÞ and similarly fN ðyÞ ¼ fN ððy xÞ þ xÞ fN ðy xÞ \ fN ðxÞ ¼ fN ððy xÞÞ \ fN ðxÞ ¼ fN ð0Þ \ fN ðxÞ ¼ fN ðxÞ: Thus, fN(x) = fN(y). This completes the proof.
h
Corollary 1 Let fN be a soft int near-ring over U. If fN(x ? y) = fN(0) for any x, y [ N, then fN(x) = fN(y). It is known that if (N, ? , .) is a near-ring, then (N, ?) is a group but not necessarily abelian. That is, for any x, y [ N, x ? y needs not be equal to y ? x. However, we have the following: Theorem 3 Let fN be a soft int near-ring over U and x [ N. Then,
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fN(0). Since fN ð0Þ fN ðxÞ; for all x 2 N; fN ðxyÞ ¼ fN ð0Þ fN ðxÞ and fN ðxyÞ ¼ fN ð0Þ fN ðyÞ; implying that fN ðxyÞ fN ðxÞ \ fN ðyÞ:
fN ðxÞ ¼ fN ð0Þ , fN ðx þ yÞ ¼ fN ðy þ xÞ ¼ fN ðyÞ for all y [ N. Proof Suppose that fN(x ? y) = fN(y ? x) = fN(y) for all y [ N. Then by choosing y = 0, we obtain that fN(x) = fN(0). Conversely, assume that fN(x) = fN(0). Then, by Lemma 1, we have
Case 2 Suppose x = 0 and y = 0. It follows that xy = 0. Then, fN(xy) = fN(1N) = fN(x) and fN(xy) = fN(1N) = fN(y). Thus, fN ðxyÞ fN ðxÞ \ fN ðyÞ:
fN ð0Þ ¼ fN ðxÞ fN ðyÞ;
Case 3 Suppose x = 0 and y = 0. It follows that xy = 0. Then, as in case 1, fN ðxyÞ fN ðxÞ \ fN ðyÞ:
8y 2 N:
ð1Þ
Since fN is a soft int near-ring over U, then fN ðx þ yÞ fN ðxÞ \ fN ðyÞ ¼ fN ðyÞ;
8y 2 N:
Moreover, for all y [ N fN ðyÞ ¼ fN ððxÞ þ xÞ þ yÞ ¼ fN ðx þ ðx þ yÞÞ fN ðxÞ \ fN ðx þ yÞ ¼ fN ðxÞ \ fN ðx þ yÞ ¼ fN ðx þ yÞ since for all y 2 N; fN ðxÞ fN ðyÞ by (1) and x, y [ N implies that x ? y [ N. Therefore, fN ðxÞ fN ðx þ yÞ and so fN(x ? y) = fN(y) for all y [ N. Now, let x [ N. Then, for all y [ N fN ðy þ xÞ ¼ fN ðy þ x þ ðy yÞÞ ¼ fN ðy þ ðx þ yÞ yÞ fN ðyÞ \ fN ðx þ yÞ \ fN ðyÞ ¼ fN ðyÞ \ fN ðx þ yÞ ¼ fN ðyÞ; since fN(x ? y) = fN(y). Furthermore, for all y [ N, fN ðyÞ ¼ fN ðy þ ðx xÞÞ
Now, let x; y 2 N: Then x - y = 0 or x - y = 0. It follows with two cases: Case a Assume that x - y = 0. Then, x = y = 0 or x = 0, y = 0 and x = y. But, since fN(x - y) = fN(0) and fN ð0Þ fN ðxÞ for all x 2 N; it immediately follows that fN ðx yÞ ¼ fN ð0Þ fN ðxÞ \ fN ðyÞ: Case b Assume that x - y = 0. Then, x = 0, y = 0 and x = y or x = 0 and y = 0 or x = 0 and y = 0. Suppose that x = 0, y = 0 and x = y. It follows fN ðx yÞ ¼ fN ð1N Þ ¼ fN ðxÞ fN ðxÞ \ fN ðyÞ: Now let x = 0 and y = 0. Then fN ðx yÞ ¼ fN ð1N Þ ¼ fN ðxÞ fN ðxÞ \ fN ðyÞ: Finally, let x = 0 and y = 0. Then, fN ðx yÞ ¼ fN ð1N Þ ¼ fN ðyÞ fN ðxÞ\ fN ðyÞ: Thus, fN is a soft int near-ring over U. Theorem 5 If fN and fM are soft int near-rings over U, then so is fN ^ fM over U. Proof By Definition 4, let fN ^ fM = fN^M, where fN^M(x, y) = fN(x) \ fM(y) for all ðx; yÞ 2 E E. Since N and M are near-rings, then so is N 9 M. So, let ðx1 ; y1 Þ; ðx2 ; y2 Þ 2 N M: Then, fN^M ððx1 ; y1 Þ ðx2 ; y2 ÞÞ ¼ fN^M ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ \ fM ðy1 y2 Þ
¼ fN ððy þ xÞ xÞ fN ðy þ xÞ \ fN ðxÞ
ðfN ðx1 Þ \ fN ðx2 ÞÞ \ ðfM ðy1 Þ \ fM ðy2 ÞÞ ¼ ðfN ðx1 Þ \ fM ðy1 ÞÞ \ ðfN ðx2 Þ \ fM ðy2 ÞÞ
¼ fN ðy þ xÞ
¼ fN^M ðx1 ; y1 Þ \ fN^M ðx2 ; y2 Þ
by (1). It follows that fN(y ? x) = fN(y) and so fN(x ? y) = fN(y ? x) = fN(y) for all y [ N. This completes the proof. h
fN^M ððx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fN^M ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ \ fM ðy1 y2 Þ ðfN ðx1 Þ \ fN ðx2 ÞÞ \ ðfM ðy1 Þ \ fM ðy2 ÞÞ ¼ ðfN ðx1 Þ \ fM ðy1 ÞÞ \ ðfN ðx2 Þ \ fM ðy2 ÞÞ
Theorem 4 Let N be a near-field and fN be a soft set over U. If fN ð0Þ fN ð1N Þ ¼ fN ðxÞ for all 0 6¼ x 2 N; then fN is a soft int near-ring over U.
¼ fN^M ðx1 ; y1 Þ \ fN^M ðx2 ; y2 Þ
Thus, fN ^ fM is a soft int near-ring over U.
h
Proof Assume that fN ð0Þ fN ð1N Þ ¼ fN ðxÞ for all 0 6¼ x 2 N. We need to show that fN ðx yÞ fN ðxÞ \ fN ðyÞ and fN ðxyÞ fN ðxÞ \ fN ðyÞ for all x; y 2 N. Let x; y 2 N: Then we have the following cases: h
Definition 8 Let fN, gM be soft int near-rings over U. Then, the product of soft int near-rings fN and gM is defined as fN 9 gM = hN9M, where hN9M(x, y) = fN(x) 9 gM(y) for all ðx; yÞ 2 N M:
Case 1 Suppose x = 0 and y = 0 or x = 0 and y = 0. Since N is a near-field, it follows that xy = 0 and fN(xy) =
Theorem 6 If fN and gM are soft int near-rings over U, then so is fN 9 gM over U 9 U.
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Proof By Definition 8, let fN 9 gM = hN9M, where hN9M(x, y) = fN(x) 9 gM(y) for all ðx; yÞ 2 N M: Then, for all ðx1 ; y1 Þ; ðx2 ; y2 Þ 2 N M; hNM ððx1 ; y1 Þ ðx2 ; y2 ÞÞ
Let N ¼ Z6 be the set of parameters and U = D3 = {\x, y[: x3 = y2 = e, xy = yx2} = {e, x, x2, y, yx, yx2}, dihedral group, be the universal set. We define a soft set fN over U by
¼ fN ðx1 x2 Þ gM ðy1 y2 Þ ðfN ðx1 Þ \ fN ðx2 ÞÞ ðgM ðy1 Þ \ gM ðy2 ÞÞ
fN ð0Þ ¼ D3 ;
¼ ðfN ðx1 Þ gM ðy1 ÞÞ \ ðfN ðx2 Þ gM ðy2 ÞÞ ¼ hNM ðx1 ; y1 Þ \ hNM ðx2 ; y2 Þ
fN ð3Þ ¼ fe; y; yx2 ; yxg;
fN ð1Þ ¼ fN ð5Þ ¼ fe; y; yx2 g; fN ð2Þ ¼ fN ð4Þ ¼ fe; x; y; yx2 g:
hNM ððx1 ; y1 Þðx2 ; y2 ÞÞ
Then, fN is a soft int near-ring over U. Now, M = {0, 2, 4}, subnear-ring of N, be the set of parameters. We define a soft set fM over U = D3 by
¼ fN ðx1 x2 Þ gM ðy1 y2 Þ ðfN ðx1 Þ \ fN ðx2 ÞÞ ðgM ðy1 Þ \ gM ðy2 ÞÞ ¼ ðfN ðx1 Þ gM ðy1 ÞÞ \ ðfN ðx2 Þ gM ðy2 ÞÞ ¼ hNM ðx1 ; y1 Þ \ hNM ðx2 ; y2 Þ:
fM ð0Þ ¼ fe; x; y; yx2 g;
Hence, fN 9 gM = hN9M is a soft int near-ring over U 9 U. h Theorem 7 If fN and hN are two soft int near-rings e hN over U. over U, then so is fN \ Proof
It is clear that fM is a soft int subnear-ring of fN over U. Theorem 8 If fN is a soft int near-ring over U; fM f i fN e fK i fN over U. and fK f i fN over U, then fM \ Proof
Let x; y 2 N: Then
Let x; y 2 N. Then
fMe\ K ðx yÞ ¼ fM ðx yÞ \ fK ðx yÞ ðfM ðxÞ \ fM ðyÞÞ \ ðfK ðxÞ \ fK ðyÞÞ
e hN Þðx yÞ ¼ fN ðx yÞ \ hN ðx yÞ ðfN \ ðfN ðxÞ \ fN ðyÞÞ \ ðhN ðxÞ \ hN ðyÞÞ ¼ ðfN ðxÞ \ hN ðxÞÞ \ ðfN ðyÞ \ hN ðyÞÞ e hN ÞðxÞ \ ðfN \ e hN ÞðyÞ; ¼ ðfN \
¼ ðfM ðxÞ \ fK ðxÞÞ \ ðfM ðyÞ \ fK ðyÞÞ ¼ fMe\ K ðxÞ \ fMe\ K ðyÞ
e hN ÞðxyÞ ¼ fN ðxyÞ \ hN ðxyÞ ðfN \ ðfN ðxÞ \ fN ðyÞÞ \ ðhN ðxÞ \ hN ðyÞÞ
and fMe\ K ðxyÞ ¼ fM ðxyÞ \ fK ðxyÞ
¼ ðfN ðxÞ \ hN ðxÞÞ \ ðfN ðyÞ \ hN ðyÞÞ e hN ÞðxÞ \ ðfN \ e hN ÞðyÞ ¼ ðfN \ e hN is a soft int near-ring over U. Therefore, fN \
fM ð2Þ ¼ fM ð4Þ ¼ fe; yx2 g
ðfM ðxÞ \ fM ðyÞÞ \ ðfK ðxÞ \ fK ðyÞÞ ¼ ðfM ðxÞ \ fK ðxÞÞ \ ðfM ðyÞ \ fK ðyÞÞ h
Definition 9 Let M be a subnear-ring of N, fN be a soft int near-ring over U and fM be a nonempty soft subset of fN over U. If fM is itself a soft int near-ring over U, then fM is called a soft int subnear-ring of fN over U and denoted by fM f i fN : Example 3 Consider the additive group (Z6, ?). Under a multiplication given in the following table, (Z6, ? , .) is a (right) near-ring.
.
0
1
2
3
4
5
0
0
0
0
0
0
0
1
3
1
5
3
1
5
2
0
2
4
0
2
4
3
3
3
3
3
3
3
4
0
4
2
0
4
2
5
3
5
1
3
5
1
¼ fMe\ K ðxÞ \ fMe\ K ðyÞ e fK i fN over U. Thus, fM \
h
In Theorem 8, if x 2 N n M or x 2 N n K; then theorem still holds, since empty set is a subset of any set. Definition 10 Let N be a near-ring and fN be a soft int near-ring over U. Then, fN is called an soft int-ideal of N over U, if the following conditions are satisfied: (i) fN ðx þ y xÞ fN ðyÞ; (ii) fN ðxyÞ fN ðxÞ; (iii) fN ðxðy þ zÞ xyÞ fN ðzÞ for all x; y; z 2 N: If fN is a soft int near-ring over U and the conditions (i) and (ii) are satisfied, then fN is called a right soft int-ideal of N over U and if the conditions (i) and (iii) are satisfied, then fN is called a left soft int-ideal of N over U. Example 4 Let N = {0, 1, 2, 3} be the (right) near-ring due to [28] (Near-rings of low order (D-5)) with the following tables:
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3
0
Proposition 1 Let N = N0 and fN be a soft set over U. Then, fN ðxðy þ zÞ xyÞ fN ðzÞ implies that fN ðxyÞ fN ðyÞ for all x; y; z 2 N: Proof Let N be a zero-symmetric near-ring, fN be a soft set over U and fN ðxðy þ zÞ xyÞ fN ðzÞ for all x; y; z 2 N: Choosing y = 0 yields fN ðxð0 þ zÞ x0Þ ¼ fN ðxð0 þ zÞ 0Þ ¼ fN ðxzÞ fN ðzÞ. h Theorem 10 Let fN be a soft int-ideal of N and fM be a soft int-ideal of M over U. Then, fN ^ fM is a soft intideal of N 9 M over U. Proof In Theorem 5, it is shown that if fN and fM are soft int near-rings over U, then so is fN ^ fM. Let ðx1 ; y1 Þ; ðx2 ; y2 Þ; ðx3 ; y3 Þ 2 N M: Then
Let N be the set of parameters and U = D2 = {\x, y[: x2 = y2 = e, xy = yx} = {e, x, y, yx}, dihedral group, be the universal set. We define a soft set fN over U by
fN^M ððx1 ; y1 Þ þ ðx2 ; y2 Þ ðx1 ; y1 ÞÞ
fN ð0Þ ¼ D2 ; fN ð1Þ ¼ fN ð2Þ ¼ fN ð3Þ ¼ fe; x; yxg:
¼ fN^M ðx2 ; y2 Þ; fN^M ððx1 ; y1 Þðx2 ; y2 ÞÞ
Then, one can show that fN is a soft int-ideal of N over U. Now, we define a soft set hN over U by hN ð0Þ ¼ D2 ; hN ð1Þ ¼ hN ð3Þ ¼ fe; x2 g; hN ð2Þ ¼ fe; x2 ; yxg: One can show that hN is a right soft int-ideal of N over U. However, since hN ð3 ð2 þ 2Þ 3 2Þ ¼ hN ð3 0 3 2Þ ¼ hN ð0 3Þ ¼ hN ð0 þ 1Þ ¼ hN ð1Þ+hN ð2Þ; hN is not a left soft int-ideal of N over U. Hence, hN is not a soft int-ideal of N over U.
¼ fN^M ðx1 þ x2 x1 ; y1 þ y2 y1 Þ ¼ fN ðx1 þ x2 x1 Þ \ fM ðy1 þ y2 y1 Þ fN ðx2 Þ \ fM ðy2 Þ
¼ fN^M ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ \ fM ðy1 y2 Þ fN ðx1 Þ \ fM ðy1 Þ ¼ fN^M ðx1 ; y1 Þ and fN^M ððx1 ; y1 Þððx2 ; y2 Þ þ ðx3 ; y3 ÞÞ ðx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fN^M ðx1 ðx2 þ x3 Þ x1 x2 ; y1 ðy2 þ y3 Þ y1 y2 Þ ¼ fN ðx1 ðx2 þ x3 Þ x1 x2 Þ \ fM ðy1 ðy2 þ y3 Þ y1 y2 Þ
Theorem 9 Let N be a near-field and fN be a soft intideal of N over U. Then, fN ð0Þ fN ð1N Þ ¼ fN ðxÞ for all 0 6¼ x 2 N:
fN ðx3 Þ \ fM ðy3 Þ ¼ fN^M ðx3 ; y3 Þ: Therefore, fN ^ fM is a soft int-ideal of N 9 M over U.
Proof Assume that fN is a soft int-ideal of N over U, then fN is a soft int near-ring over U. Since fN ð0Þ fN ðxÞ for all x 2 N; then in particular fN ð0Þ fN ð1N Þ. Now, let 0 6¼ x 2 N: Then fN ðxÞ ¼ fN ð1N xÞ fN ð1N Þ and fN ð1N Þ ¼ fN ðx x1 Þ fN ðxÞ; implying that fN(1N) = fN(x) for all 0 6¼ x 2 N.
h
For a near-ring N, the zero-symmetric part of N denoted by N0 is defined by N0 ¼ fn 2 N j n0 ¼ 0g: It is known that if N is a zero-symmetric near-ring and ICl N; then NI N [28]. Here, we have an analog for this case:
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h Definition 11 Let fN be a soft int-ideal of N and gM be a soft int-ideal of M over U. Then, the product of soft intideals fN and gM is defined as fN 9 gM = hN9M, where hN9M(x, y) = fN(x) 9 gM(y) for all ðx; yÞ 2 N M: Theorem 11 If fN is a soft int-ideal of N and gM is a soft int-ideal of M over U, then fN 9 gM is a soft intideal of N 9 M over U 9 U. Proof In Theorem 6, it is shown that if fN and gM are soft int near-rings over U 9 U, then so is fN 9 gM. Let ðx1 ; y1 Þ; ðx2 ; y2 Þ; ðx3 ; y3 Þ 2 N M;
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Proof It is obvious that 0 2 Nf N. We need to show that ðiÞ x y 2 Nf ; ðiiÞ n þ x n 2 Nf ; ðiiiÞ xn 2 Nf and ðivÞnðs þ xÞ ns 2 Nf for all x; y 2 Nf and n; s 2 N: If x; y 2 Nf ; then fN(x) = fN(y) = fN(0). By Lemma 1, fN ð0Þ fN ðx yÞ; fN ð0Þ fN ðn þ x nÞ; fN ð0Þ fN ðxnÞ and fN ð0Þ fN ðnðs þ xÞ nsÞ for all x; y 2 Nf and n; s 2 N: Since fN is a soft int-ideal of N over U, so for all x; y 2 Nf and n; s 2 N; ðiÞ fN ðx yÞ fN ðxÞ \ fN ðyÞ ¼ fN ð0Þ; ðiiÞ fN ðn þ x nÞ fN ðxÞ ¼ fN ð0Þ; ðiiiÞ fN ðxnÞ fN ðxÞ ¼ fN ð0Þ and ðivÞfN ðnðs þ xÞ nsÞ fN ðxÞ ¼ fN ð0Þ. Hence fN(x - y) = fN(0), fN(n ? x - n) = fN(0), fN(xn) = fN(0) and fN (n(s ? x) - ns) = fN(0) for all x; y 2 Nf and n; s 2 N: h Therefore, Nf is an ideal of N.
fNM ððx1 ; y1 Þ þ ðx2 ; y2 Þ ðx1 ; y1 ÞÞ ¼ fNM ðx1 þ x2 x1 ; y1 þ y2 y1 Þ ¼ fN ðx1 þ x2 x1 Þ fM ðy1 þ y2 y1 Þ fN ðx2 Þ fM ðy2 Þ ¼ fNM ðx2 ; y2 Þ; fNM ððx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fNM ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ fM ðy1 y2 Þ fN ðx1 Þ fM ðy1 Þ ¼ fNM ðx1 ; y1 Þ and fNM ððx1 ; y1 Þððx2 ; y2 Þ þ ðx3 ; y3 ÞÞ ðx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fNM ðx1 ðx2 þ x3 Þ x1 x2 ; y1 ðy2 þ y3 Þ y1 y2 Þ
Theorem 14 Let fN be a soft set over U and a be a subset of U such that ; a fN ð0Þ. If fN is a soft intideal of N over U, then fNa is an ideal of N.
¼ fN ðx1 ðx2 þ x3 Þ x1 x2 Þ fM ðy1 ðy2 þ y3 Þ y1 y2 Þ fN ðx3 Þ fM ðy3 Þ ¼ fNM ðx3 ; y3 Þ Hence, fN 9 gM is a soft int-ideal of N 9 M over U 9 U. h Theorem 12 Let fN and hN be two soft int-ideals of N e hN is a soft int-ideal of N over U. over U. Then, fN \ Proof In Theorem, it is shown that if fN and hN are soft int e hN . Let x; y; z 2 N. Then near-rings over U, then so is fN \ e hN ÞðxþyxÞ ¼ fN ðxþyxÞ\hN ðxþyxÞ ðfN \
Theorem 15 Let fN and fM be soft sets over U and W be a near-ring isomorphism from N to M. If fN is a soft int-ideal of N over U, then WðfN Þ is a soft int-ideal of M over U.
fN ðyÞ\hN ðyÞ e hN ÞðyÞ ¼ ðfN \ e hN ÞðxyÞ ¼ fN ðxyÞ\hN ðxyÞ ðfN \
Proof Let m1 ; m2 ; m3 2 M: Since W is surjective, then there exists n1 ; n2 ; n3 2 N such that Wðn1 Þ ¼ m1 ; Wðn2 Þ ¼ m2 andWðn3 Þ ¼ m3 : Then,
fN ðxÞ\hN ðxÞ e hN ÞðxÞ ¼ ðfN \ e hN ÞðxðyþzÞxyÞ ¼ fN ðxðyþzÞxyÞ ðfN \ \hN ðxðyþzÞxyÞ fN ðzÞ\hN ðzÞ e hN ÞðzÞ ¼ ðfN \ e hN is a soft int-ideal of N over U. Therefore, fN \
Proof Since fN ð0Þ a; then 0 2 fNa and ; 6¼ fNa N: If x; y 2 fNa ; then fN ðxÞ a and fN ðyÞ a. We need to show that ðiÞ x y 2 fNa ; ðiiÞ n þ x n 2 fNa ; ðiiiÞ xn 2 fNa and (iv) nðs þ xÞ ns 2 fNa for all x; y 2 fNa and n; s 2 N: Since fN is a soft int-ideal of N over U, so fN ðx yÞ fN ðxÞ \ fN ðyÞ a \ a ¼ a; fN ðn þ x nÞ fN ðxÞ a; fN ðxnÞ fN ðxÞ a and fN ðnðs þ xÞ nsÞ fN ðxÞ a: Thus, the proof is completed. h
h
4 Applications of soft int near-rings and soft int-ideals In this section, we give the applications of soft image, soft pre-image and upper-a-inclusion of sets to near-ring theory with respect to soft int near-rings and soft int-ideals of a near-ring. Theorem 13 If fN is a soft int-ideal of N over U, then Nf ¼ fx 2 N : fN ðxÞ ¼ fN ð0Þg is an ideal of N.
ðWðfN ÞÞðm1 m2 Þ [ ¼ ffN ðnÞ : n 2 N; WðnÞ ¼ m1 m2 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 m2 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðWðn1 n2 ÞÞ ¼ n1 n2 g [ ¼ ffN ðn1 n2 Þ : ni 2 N; Wðni Þ ¼ mi ; i ¼ 1; 2g [ ffN ðn1 Þ \ fN ðn2 Þ : ni 2 N; Wðni Þ ¼ mi ; i ¼ 1; 2g [ ¼ ffN ðn1 Þ : n1 2 N; Wðn1 Þ ¼ m1 g [ \ ffN ðn2 Þ : n2 2 N; Wðn2 Þ ¼ m2 g ¼ ðWðfN ÞÞðm1 Þ \ ðWðfN ÞÞðm2 Þ Similarly, one can show that ðWðfN ÞÞðm1 m2 Þ ðWðfN ÞÞ ðm1 Þ \ ðWðfN ÞÞðm2 Þ. Also,
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ðW1 ðfM ÞÞðn1 n2 Þ ðW1 ðfM ÞÞðn1 Þ \ ðW1 ðfM ÞÞðn2 Þ:
ðWðfN ÞÞðm1 þ m2 m1 Þ [ ¼ ffN ðnÞ : n 2 N; WðnÞ ¼ m1 þ m2 m1 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 þ m2 m1 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðWðn1 þ n2 n1 ÞÞ
Also, ðW1 ðfM ÞÞðn1 þ n2 n1 Þ ¼ fM ðWðn1 þ n2 n1 ÞÞ ¼ fM ðWðn1 Þ þ Wðn2 Þ Wðn1 ÞÞ fM ðWðn2 ÞÞ
¼ n1 þ n2 n1 g [ ¼ ffN ðn1 þ n2 n1 Þ : ni 2 N; Wðni Þ ¼ mi ; i ¼ 1; 2g [ ffN ðn2 Þ : n2 2 N; Wðn2 Þ ¼ m2 g
¼ ðW1 ðfM ÞÞðn2 Þ Moreover,
¼ ðWðfN ÞÞðm2 Þ
ðW1 ðfM ÞÞðn1 n2 Þ ¼ fM ðWðn1 n2 ÞÞ
Moreover, ðWðfN ÞÞðm1 m2 Þ ¼
¼ fM ðWðn1 ÞWðn2 ÞÞ
[
fM ðWðn1 ÞÞ
ffN ðnÞ : n 2 N; WðnÞ ¼ m1 m2 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 m2 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðWðn1 n2 ÞÞ
Furthermore,
¼ n1 n2 g [ ¼ ffN ðn1 n2 Þ : ni 2 N; Wðni Þ
ðW1 ðfM ÞÞðn1 ðn2 þ n3 Þ n1 n2 Þ ¼ fM ðWððn1 ðn2 þ n3 Þ n1 n2 ÞÞÞ
¼ mi ; i ¼ 1; 2g [ ffN ðn1 Þ : n1 2 N; Wðn1 Þ ¼ m1 g
¼ fM ðWðn1 ÞððWðn2 Þ þ Wðn3 ÞÞ Wðn1 ÞWðn2 ÞÞ
¼ ðW1 ðfM ÞÞðn1 Þ
fM ðWðn3 Þ ¼ ðW1 ðfM ÞÞðn3 Þ
¼ ðWðfN ÞÞðm1 Þ
Hence, W1 ðfM Þ is a soft int-ideal of N over U.
h
Furthermore, ðWðfN ÞÞðm1 ðm2 þ m3 Þ m1 m2 Þ [ ¼ ffN ðnÞ : n 2 N; WðnÞ ¼ m1 ðm2 þ m3 Þ m1 m2 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 ðm2 þ m3 Þ m1 m2 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ n1 ðn2 þ n3 Þ n1 n2 g [ ¼ ffN ðn1 ðn2 þ n3 Þ n1 n2 Þ : ni 2 N; Wðni Þ
5 Conclusion
¼ mi ; i ¼ 1; 2; 3g [ ffN ðn3 Þ : n3 2 N; Wðn3 Þ ¼ m3 g ¼ ðWðfN ÞÞðm3 Þ Hence, WðfN Þ is a soft int-ideal of M over U.
h
Theorem 16 Let fN and fM be soft sets over U and W be a near-ring homomorphism from N to M. If fM is a soft int-ideal of M over U, then W1 ðfM Þ is a soft int-ideal of N over U. Proof
In this paper, by using soft sets and intersection operation of sets, we have defined a new concept, called soft int nearring. This new notion brings the soft set theory, set theory and near-ring theory together and therefore is very functional for obtaining results by means of near-ring structure. Based on the definition, we have introduced the concepts of soft int subnear-rings and soft int-ideals of a near-ring with illustrative examples. We have then investigated these notions with respect to soft image, soft pre-image and a-inclusion of soft sets. Finally, we give some applications of soft int near-rings to near-ring theory. To extend this study, one can further study the other algebraic structures such as different algebras by means of soft intersections.
Let n1 ; n2 ; n3 2 N: Then,
ðW1 ðfM ÞÞðn1 n2 Þ ¼ fM ðWðn1 n2 ÞÞ ¼ fM ðWðn1 Þ Wðn2 ÞÞ fM ðWðn1 ÞÞ \ fM ðWðn2 ÞÞ ¼ ðW1 ðfM ÞÞðn1 Þ \ ðW1 ðfM ÞÞðn2 Þ Similarly, one can show that
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Neural Comput & Applic (2012) 21 (Suppl 1):S221–S229 4. Acar U, Koyuncu F, Tanay B (2010) Soft sets and soft rings. Comput Math Appl 59:3458–3463 5. Jun YB (2008) Soft BCK/BCI-algebras. Comput Math Appl 56:1408–1413 6. Jun YB, Park CH Applications of soft sets in ideal theory of BCK/BCI-algebras. Inform Sci 178:2466–2475 7. Jun YB, Lee KJ, Zhan J (2009) Soft p-ideals of soft BCI-algebras. Comput Math Appl 58:2060–2068 8. Sezgin A, Atagu¨n AO (2011) Soft groups and normalistic soft groups. Comput Math Appl 62(2):685–698 9. Zhan J, Jun YB (2010) Soft BL-algebras based on fuzzy sets. Comput Math Appl 59(6):2037–2046 10. Kazancı O, Yılmaz S¸ , Yamak S (2010) Soft sets and soft BCHalgebras. Hacet J Math Stat 39(2):205–217 11. Sezgin A, Atagu¨n AO, Aygu¨n E (2011) A note on soft near-rings and idealistic soft near-rings. Filomat 25(1):53–68 12. Atagu¨n AO, Sezgin A (2011) Soft substructures of rings, fields and modules. Comput Math Appl 61(3):592–601 13. Sezgin A, Atagu¨n AO, C¸ag˘man N (2011) Union soft substructures of near-rings and N-groups. Neural Comput Appl. doi: 10.1007/s00521-011-0732-1 14. C¸ag˘man N, C¸ıtak F, Aktas¸ H Soft int-groups and its applications to group theory. Neural Comput Appl. doi:10.1007/s00521-0110752-x 15. Maji PK, Biswas R, Roy AR (2003) Soft set theory. Comput Math Appl 45:555–562 16. Ali MI, Feng F, Liu X, Min WK, Shabir M (2009) On some new operations in soft set theory. Comput Math Appl 57:1547–1553
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ORIGINAL ARTICLE
Soft intersection near-rings with its applications Aslıhan Sezgin • Akın Osman Atagu¨n Naim C ¸ ag˘man
•
Received: 18 October 2011 / Accepted: 7 December 2011 / Published online: 24 December 2011 Ó Springer-Verlag London Limited 2011
Abstract In this paper, we first define soft intersection near-ring (soft int near-ring) by using intersection operation of sets. This new notion can be regarded as a bridge among soft set theory, set theory and near-ring theory, since it shows how a soft set effects on a near-ring structure by means of intersection and inclusion of sets. We then derive its basic properties with illustrative examples. Moreover, we obtain some analog of classical near-ring theoretic concepts for soft int near-ring and give the applications of soft int near-ring to near-ring theory. Keywords Soft set Soft int near-ring Soft int subnear-ring Soft int-ideal Soft pre-image Soft image a-Inclusion
1 Introduction The notion of soft set was introduced in 1999 by Molodtsov [1] as a new mathematical tool for dealing with uncertainties. Since its inception, it has received much attention
A. Sezgin (&) Department of Mathematics, Faculty of Arts and Science, Amasya University, 05100 Amasya, Turkey e-mail: [email protected] A. O. Atagu¨n Department of Mathematics, Faculty of Arts and Science, Bozok University, 66100 Yozgat, Turkey e-mail: [email protected] N. C ¸ ag˘man Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpas¸ a University, 60250 Tokat, Turkey e-mail: [email protected]
in the mean of algebraic structures such as groups [2], semirings [3], rings [4], BCK/BCI-algebras [5–7], normalistic soft groups [8], BL-algebras [9], BCH-algebras [10] and near-rings [11]. Atagu¨n and Sezgin [12] defined the concepts of soft subrings and ideals of a ring, soft subfields of a field and soft submodules of a module and studied their related properties with respect to soft set operations also union soft substructues of near-rings and near-ring modules are studied in [13]. Cag˘man et al. defined two new type of group action on a soft set, called group SI-action [14] and group SU-action [14], which are based on the inclusion relation and the intersection of sets and union of sets, respectively. Algebraic structures of soft sets have been studied by some authors. Maji et al. [15] presented some definitions on soft sets and based on the analysis of several operations on soft sets Ali et al. [16] introduced several operations of soft sets and Sezgin and Atagu¨n [17] studied on soft set operations as well. Soft set relations and functions [18] and soft mappings [19] were proposed and many related concepts were discussed, too. Moreover, the theory of soft set has gone through remarkably rapid strides with a wideranging applications especially in soft decision making as in the following studies: [20–22] and some other fields such as [23–26]. Cag˘man and Enginog˘lu [21] redefined the operations of soft sets to develop the soft set theory. By using their definitions, in this paper, we define ‘‘soft int near-ring’’. The structure of soft int near-ring is based on the inclusion relation and intersection of sets, and since this new concept brings the soft set theory, set theory and near-ring theory together, it is very functional by means of improving the soft set theory with respect to near-ring structure. From this view, it functions as a bridge among soft set theory, set theory and near-ring theory. Based on the soft int near-ring
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definition, we introduce the concepts of soft int subnearring and soft int-ideal. We moreover investigate these notions with respect to soft image, soft pre-image and a-inclusion of soft sets. Finally, we give some applications of soft int near-ring to near-ring theory.
2 Preliminary In this section, we recall some basic notions relevant to near-rings and soft sets. (N, ? , .) is called a near-ring if (N1) (N2) (N3)
(N, ?) forms a group (not necessarily abelian) (N, .) forms a semi-group and (a ? b)c = ac ? bc for all a, b, c [ N (i.e. we study on right near-rings).
Throughout this paper, N will always denote a right nearring with zero element 0. A subgroup M of N with MM M is called a subnear-ring of N. A normal subgroup I of N is called a right ideal if IN I and denoted by I C r N: It is called a left ideal if n(s ? i) - ns [ I for all n, s [ N and i [ I and denoted by I C ‘ N: If such a normal subgroup I is both left and right ideal in N, then it is called an ideal of N and denoted by I C N: By a near-field, we shall mean an algebraic system (N, ? , .), where ðN n f0g; :Þ forms a group. If N is a near-field, then its identity element will be denoted by 1N in what follows. For all undefined concepts and notions we refer to [28]. From now on, U refers to an initial universe, E is a set of parameters, P(U) is the power set of U and A; B; C E. Definition 1 by
[1, 21] A soft set fA over U is a set defined
fA : E ! PðUÞ such that fA ðxÞ ¼ ; if x 62 A:
Definition 3 [21] Let fA and fB be soft sets over U. Then, e fB ; is defined as fA [ e fB ¼ union of fA and fB, denoted by fA [ fAe[ B ; where fAe[ B ðxÞ ¼ fA ðxÞ [ fB ðxÞ for all x [ E. e fB ; is defined as Intersection of fA and fB, denoted by fA \ e fB ¼ f ; where f ðxÞ ¼ fA ðxÞ \ fB ðxÞ for all x [ E. fA \ Ae \B Ae \B Definition 4 [21] Let fA and fB be soft sets over U. Then, _-product of fA and fB, denoted by fA_ fB, is defined as fA _ fB = fA_B, where fA_B(x, y) = fA(x) [ fB(y) for all (x, y) [ E 9 E. ^-product of fA and fB, denoted by fA ^ fB, is defined as fA ^ fB = fA^B, where fA^B(x, y) = fA(x)\ fB(y) for all (x, y) [ E 9 E. Definition 5 [14] Let fA and fB be soft sets over the common universe U and W be a function from A to B. Then, soft image of fA under W; denoted by WðfA Þ; is a soft set over U by ðWðfA ÞÞðbÞ (S ffA ðaÞ j a 2 A and WðaÞ ¼ bg; if W1 ðbÞ 6¼ ;; ¼ ;; otherwise for all b [ B. And soft pre-image (or soft inverse image) of fB under W; denoted by W1 ðfB Þ; is a soft set over U by ðW1 ðfB ÞÞðaÞ ¼ fB ðWðaÞÞ for all a [ A. Definition 6 [27] Let fA be a soft set over U and a be a subset of U. Then, upper a-inclusion of fA, denoted by fAa ; and lower a-inclusion of fA, denoted by fAa ; are defined as fAa ¼ fx 2 A j fA ðxÞ ag and fAa ¼ fx 2 A j fA ðxÞ ag;
Here fA is also called approximate function. A soft set over U can be represented by the set of ordered pairs
respectively.
fA ¼ fðx; fA ðxÞÞ : x 2 E; fA ðxÞ 2 PðUÞg:
3 Soft int near-ring
It is clear to see that a soft set is a parametrized family of subsets of the set U. It is worth noting that the sets fA(x) may be arbitrary. Some of them may be empty, some may have nonempty intersection. We refer to [1, 15, 21] for further details. If we define more than one soft set in a subset A of the set of parameters E, then the soft sets will be denoted by fA, gA, hA etc. If we define more then one soft set in some subsets A, B, C etc. of parameters E, then the soft sets will be denoted by fA, fB, fC etc., respectively. Definition 2 [21] Let fA and fB be soft sets over U. Then, fA is a soft subset of fB, denoted by fA ~ fB ; if fA ðxÞ fB ðxÞ for all x [ E.
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In this section, we first define soft intersection near-ring which is abbreviated as soft int near-ring. We then define soft int subnear-ring, soft int-ideal of a near-ring with illustrative examples and study their basic properties with respect to soft set operations. Definition 7 Let N be a near-ring and fN be a soft set over U. Then, fN is called a soft int near-ring over U if it satisfies the following properties: (i) fN ðx þ yÞ fN ðxÞ \ fN ðyÞ; (ii) fN(- x) = fN(x), (iii) fN ðxyÞ fN ðxÞ \ fN ðyÞ for all x, y [ N.
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Example 1 Let N = {0, 1, 2, 3} be the (right) near-ring due to [28] (Near-rings of low order (D-10)) defined by the following tables:
Proof Assume that fN is a soft int near-ring over U. Then, for all x 2 N; fN ð0Þ ¼ fN ðx xÞ fN ðxÞ \ fN ðxÞ ¼ fN ðxÞ \ fN ðxÞ ¼ fN ðxÞ. h
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Theorem 1 Let N be a near-ring and fN be a soft set over U. Then, fN is a soft int near-ring over U if and only if
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Assume that N is the set of parameters and U ¼ x x j x; y 2 Z4 ; 2 2 matrices with Z4 terms, is y y the universal set. We construct a soft set fN over U by 0 0 1 1 3 3 0 0 ; ; ; fN ð0Þ ¼ 1 1 0 0 0 0 3 3 3 3 0 0 fN ð1Þ ¼ ; ; 1 1 0 0 0 0 3 3 1 1 fN ð2Þ ¼ ; ; ; 1 1 0 0 0 0 3 3 0 0 ; : fN ð3Þ ¼ 1 1 0 0 Then, one can easily show that the soft set fN is a soft int near-ring over U. Example 2 In Example 1, assume that N = {0, 1, 2, 3} is again the set of parameters and U = S3, symmetric group, is the universal set. We define a soft set fN by fN ð0Þ ¼ fð12Þg fN ð1Þ ¼ fð12Þ; ð13Þg fN ð2Þ ¼ fð12Þ; ð123Þg fN ð3Þ ¼ fð12Þ; ð13Þg Since fN ð2 2Þ ¼ fN ð0Þ ¼ fð12Þg + fN ð2Þ \ fN ð2Þ ¼ fð12Þ; ð123Þg; fN is not a soft int near-ring over U. It is easy to see that if we take the near-ring as N = {0}, then fN is a soft int near-ring over U no matter how fN is defined and no matter U is. Lemma 1 Let fN be a soft int near-ring over U. Then, fN ð0Þ fN ðxÞ for all x [ N.
(i) fN ðx yÞ fN ðxÞ \ fN ðyÞ (ii) fN ðxyÞ fN ðxÞ \ fN ðyÞ for all x, y [ N. Proof Suppose that fN is a soft int near-ring over. Then, by Definition 7, fN ðxyÞ fN ðxÞ \ fN ðyÞ and fN ðx yÞ fN ðxÞ \ fN ðyÞ ¼ fN ðxÞ \ fN ðyÞ for all x, y [ N. Conversely, assume that fN ðxyÞ fN ðxÞ \ fN ðyÞ and fN ðx yÞ fN ðxÞ \ fN ðyÞ for all x, y [ N. If we choose x = 0, then fN ð0 yÞ ¼ fN ðyÞ fN ð0Þ \ fN ðyÞ ¼ fN ðyÞ by Lemma 1. Similarly, fN ðyÞ ¼ fN ððyÞÞ fN ðyÞ; thus, fN(- y) = fN(y) for all y [ N. Also, by assumption fN ðx þ yÞ fN ðxÞ \ fN ðyÞ ¼ fN ðxÞ \ fN ðyÞ: This completes the proof. h Theorem 2 Let fN be a soft int near-ring over U. If fN (x -y) = fN(0) for any x, y [ N, then fN(x) = fN(y). Proof Assume that fN(x - y) = fN(0) for any x, y [ N. Then, fN ðxÞ ¼ fN ðx y þ yÞ fN ðx yÞ \ fN ðyÞ ¼ fN ð0Þ \ fN ðyÞ ¼ fN ðyÞ and similarly fN ðyÞ ¼ fN ððy xÞ þ xÞ fN ðy xÞ \ fN ðxÞ ¼ fN ððy xÞÞ \ fN ðxÞ ¼ fN ð0Þ \ fN ðxÞ ¼ fN ðxÞ: Thus, fN(x) = fN(y). This completes the proof.
h
Corollary 1 Let fN be a soft int near-ring over U. If fN(x ? y) = fN(0) for any x, y [ N, then fN(x) = fN(y). It is known that if (N, ? , .) is a near-ring, then (N, ?) is a group but not necessarily abelian. That is, for any x, y [ N, x ? y needs not be equal to y ? x. However, we have the following: Theorem 3 Let fN be a soft int near-ring over U and x [ N. Then,
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fN(0). Since fN ð0Þ fN ðxÞ; for all x 2 N; fN ðxyÞ ¼ fN ð0Þ fN ðxÞ and fN ðxyÞ ¼ fN ð0Þ fN ðyÞ; implying that fN ðxyÞ fN ðxÞ \ fN ðyÞ:
fN ðxÞ ¼ fN ð0Þ , fN ðx þ yÞ ¼ fN ðy þ xÞ ¼ fN ðyÞ for all y [ N. Proof Suppose that fN(x ? y) = fN(y ? x) = fN(y) for all y [ N. Then by choosing y = 0, we obtain that fN(x) = fN(0). Conversely, assume that fN(x) = fN(0). Then, by Lemma 1, we have
Case 2 Suppose x = 0 and y = 0. It follows that xy = 0. Then, fN(xy) = fN(1N) = fN(x) and fN(xy) = fN(1N) = fN(y). Thus, fN ðxyÞ fN ðxÞ \ fN ðyÞ:
fN ð0Þ ¼ fN ðxÞ fN ðyÞ;
Case 3 Suppose x = 0 and y = 0. It follows that xy = 0. Then, as in case 1, fN ðxyÞ fN ðxÞ \ fN ðyÞ:
8y 2 N:
ð1Þ
Since fN is a soft int near-ring over U, then fN ðx þ yÞ fN ðxÞ \ fN ðyÞ ¼ fN ðyÞ;
8y 2 N:
Moreover, for all y [ N fN ðyÞ ¼ fN ððxÞ þ xÞ þ yÞ ¼ fN ðx þ ðx þ yÞÞ fN ðxÞ \ fN ðx þ yÞ ¼ fN ðxÞ \ fN ðx þ yÞ ¼ fN ðx þ yÞ since for all y 2 N; fN ðxÞ fN ðyÞ by (1) and x, y [ N implies that x ? y [ N. Therefore, fN ðxÞ fN ðx þ yÞ and so fN(x ? y) = fN(y) for all y [ N. Now, let x [ N. Then, for all y [ N fN ðy þ xÞ ¼ fN ðy þ x þ ðy yÞÞ ¼ fN ðy þ ðx þ yÞ yÞ fN ðyÞ \ fN ðx þ yÞ \ fN ðyÞ ¼ fN ðyÞ \ fN ðx þ yÞ ¼ fN ðyÞ; since fN(x ? y) = fN(y). Furthermore, for all y [ N, fN ðyÞ ¼ fN ðy þ ðx xÞÞ
Now, let x; y 2 N: Then x - y = 0 or x - y = 0. It follows with two cases: Case a Assume that x - y = 0. Then, x = y = 0 or x = 0, y = 0 and x = y. But, since fN(x - y) = fN(0) and fN ð0Þ fN ðxÞ for all x 2 N; it immediately follows that fN ðx yÞ ¼ fN ð0Þ fN ðxÞ \ fN ðyÞ: Case b Assume that x - y = 0. Then, x = 0, y = 0 and x = y or x = 0 and y = 0 or x = 0 and y = 0. Suppose that x = 0, y = 0 and x = y. It follows fN ðx yÞ ¼ fN ð1N Þ ¼ fN ðxÞ fN ðxÞ \ fN ðyÞ: Now let x = 0 and y = 0. Then fN ðx yÞ ¼ fN ð1N Þ ¼ fN ðxÞ fN ðxÞ \ fN ðyÞ: Finally, let x = 0 and y = 0. Then, fN ðx yÞ ¼ fN ð1N Þ ¼ fN ðyÞ fN ðxÞ\ fN ðyÞ: Thus, fN is a soft int near-ring over U. Theorem 5 If fN and fM are soft int near-rings over U, then so is fN ^ fM over U. Proof By Definition 4, let fN ^ fM = fN^M, where fN^M(x, y) = fN(x) \ fM(y) for all ðx; yÞ 2 E E. Since N and M are near-rings, then so is N 9 M. So, let ðx1 ; y1 Þ; ðx2 ; y2 Þ 2 N M: Then, fN^M ððx1 ; y1 Þ ðx2 ; y2 ÞÞ ¼ fN^M ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ \ fM ðy1 y2 Þ
¼ fN ððy þ xÞ xÞ fN ðy þ xÞ \ fN ðxÞ
ðfN ðx1 Þ \ fN ðx2 ÞÞ \ ðfM ðy1 Þ \ fM ðy2 ÞÞ ¼ ðfN ðx1 Þ \ fM ðy1 ÞÞ \ ðfN ðx2 Þ \ fM ðy2 ÞÞ
¼ fN ðy þ xÞ
¼ fN^M ðx1 ; y1 Þ \ fN^M ðx2 ; y2 Þ
by (1). It follows that fN(y ? x) = fN(y) and so fN(x ? y) = fN(y ? x) = fN(y) for all y [ N. This completes the proof. h
fN^M ððx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fN^M ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ \ fM ðy1 y2 Þ ðfN ðx1 Þ \ fN ðx2 ÞÞ \ ðfM ðy1 Þ \ fM ðy2 ÞÞ ¼ ðfN ðx1 Þ \ fM ðy1 ÞÞ \ ðfN ðx2 Þ \ fM ðy2 ÞÞ
Theorem 4 Let N be a near-field and fN be a soft set over U. If fN ð0Þ fN ð1N Þ ¼ fN ðxÞ for all 0 6¼ x 2 N; then fN is a soft int near-ring over U.
¼ fN^M ðx1 ; y1 Þ \ fN^M ðx2 ; y2 Þ
Thus, fN ^ fM is a soft int near-ring over U.
h
Proof Assume that fN ð0Þ fN ð1N Þ ¼ fN ðxÞ for all 0 6¼ x 2 N. We need to show that fN ðx yÞ fN ðxÞ \ fN ðyÞ and fN ðxyÞ fN ðxÞ \ fN ðyÞ for all x; y 2 N. Let x; y 2 N: Then we have the following cases: h
Definition 8 Let fN, gM be soft int near-rings over U. Then, the product of soft int near-rings fN and gM is defined as fN 9 gM = hN9M, where hN9M(x, y) = fN(x) 9 gM(y) for all ðx; yÞ 2 N M:
Case 1 Suppose x = 0 and y = 0 or x = 0 and y = 0. Since N is a near-field, it follows that xy = 0 and fN(xy) =
Theorem 6 If fN and gM are soft int near-rings over U, then so is fN 9 gM over U 9 U.
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Proof By Definition 8, let fN 9 gM = hN9M, where hN9M(x, y) = fN(x) 9 gM(y) for all ðx; yÞ 2 N M: Then, for all ðx1 ; y1 Þ; ðx2 ; y2 Þ 2 N M; hNM ððx1 ; y1 Þ ðx2 ; y2 ÞÞ
Let N ¼ Z6 be the set of parameters and U = D3 = {\x, y[: x3 = y2 = e, xy = yx2} = {e, x, x2, y, yx, yx2}, dihedral group, be the universal set. We define a soft set fN over U by
¼ fN ðx1 x2 Þ gM ðy1 y2 Þ ðfN ðx1 Þ \ fN ðx2 ÞÞ ðgM ðy1 Þ \ gM ðy2 ÞÞ
fN ð0Þ ¼ D3 ;
¼ ðfN ðx1 Þ gM ðy1 ÞÞ \ ðfN ðx2 Þ gM ðy2 ÞÞ ¼ hNM ðx1 ; y1 Þ \ hNM ðx2 ; y2 Þ
fN ð3Þ ¼ fe; y; yx2 ; yxg;
fN ð1Þ ¼ fN ð5Þ ¼ fe; y; yx2 g; fN ð2Þ ¼ fN ð4Þ ¼ fe; x; y; yx2 g:
hNM ððx1 ; y1 Þðx2 ; y2 ÞÞ
Then, fN is a soft int near-ring over U. Now, M = {0, 2, 4}, subnear-ring of N, be the set of parameters. We define a soft set fM over U = D3 by
¼ fN ðx1 x2 Þ gM ðy1 y2 Þ ðfN ðx1 Þ \ fN ðx2 ÞÞ ðgM ðy1 Þ \ gM ðy2 ÞÞ ¼ ðfN ðx1 Þ gM ðy1 ÞÞ \ ðfN ðx2 Þ gM ðy2 ÞÞ ¼ hNM ðx1 ; y1 Þ \ hNM ðx2 ; y2 Þ:
fM ð0Þ ¼ fe; x; y; yx2 g;
Hence, fN 9 gM = hN9M is a soft int near-ring over U 9 U. h Theorem 7 If fN and hN are two soft int near-rings e hN over U. over U, then so is fN \ Proof
It is clear that fM is a soft int subnear-ring of fN over U. Theorem 8 If fN is a soft int near-ring over U; fM f i fN e fK i fN over U. and fK f i fN over U, then fM \ Proof
Let x; y 2 N: Then
Let x; y 2 N. Then
fMe\ K ðx yÞ ¼ fM ðx yÞ \ fK ðx yÞ ðfM ðxÞ \ fM ðyÞÞ \ ðfK ðxÞ \ fK ðyÞÞ
e hN Þðx yÞ ¼ fN ðx yÞ \ hN ðx yÞ ðfN \ ðfN ðxÞ \ fN ðyÞÞ \ ðhN ðxÞ \ hN ðyÞÞ ¼ ðfN ðxÞ \ hN ðxÞÞ \ ðfN ðyÞ \ hN ðyÞÞ e hN ÞðxÞ \ ðfN \ e hN ÞðyÞ; ¼ ðfN \
¼ ðfM ðxÞ \ fK ðxÞÞ \ ðfM ðyÞ \ fK ðyÞÞ ¼ fMe\ K ðxÞ \ fMe\ K ðyÞ
e hN ÞðxyÞ ¼ fN ðxyÞ \ hN ðxyÞ ðfN \ ðfN ðxÞ \ fN ðyÞÞ \ ðhN ðxÞ \ hN ðyÞÞ
and fMe\ K ðxyÞ ¼ fM ðxyÞ \ fK ðxyÞ
¼ ðfN ðxÞ \ hN ðxÞÞ \ ðfN ðyÞ \ hN ðyÞÞ e hN ÞðxÞ \ ðfN \ e hN ÞðyÞ ¼ ðfN \ e hN is a soft int near-ring over U. Therefore, fN \
fM ð2Þ ¼ fM ð4Þ ¼ fe; yx2 g
ðfM ðxÞ \ fM ðyÞÞ \ ðfK ðxÞ \ fK ðyÞÞ ¼ ðfM ðxÞ \ fK ðxÞÞ \ ðfM ðyÞ \ fK ðyÞÞ h
Definition 9 Let M be a subnear-ring of N, fN be a soft int near-ring over U and fM be a nonempty soft subset of fN over U. If fM is itself a soft int near-ring over U, then fM is called a soft int subnear-ring of fN over U and denoted by fM f i fN : Example 3 Consider the additive group (Z6, ?). Under a multiplication given in the following table, (Z6, ? , .) is a (right) near-ring.
.
0
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0
0
0
0
0
0
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1
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1
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3
1
5
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0
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3
3
3
3
3
3
3
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0
4
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4
2
5
3
5
1
3
5
1
¼ fMe\ K ðxÞ \ fMe\ K ðyÞ e fK i fN over U. Thus, fM \
h
In Theorem 8, if x 2 N n M or x 2 N n K; then theorem still holds, since empty set is a subset of any set. Definition 10 Let N be a near-ring and fN be a soft int near-ring over U. Then, fN is called an soft int-ideal of N over U, if the following conditions are satisfied: (i) fN ðx þ y xÞ fN ðyÞ; (ii) fN ðxyÞ fN ðxÞ; (iii) fN ðxðy þ zÞ xyÞ fN ðzÞ for all x; y; z 2 N: If fN is a soft int near-ring over U and the conditions (i) and (ii) are satisfied, then fN is called a right soft int-ideal of N over U and if the conditions (i) and (iii) are satisfied, then fN is called a left soft int-ideal of N over U. Example 4 Let N = {0, 1, 2, 3} be the (right) near-ring due to [28] (Near-rings of low order (D-5)) with the following tables:
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Proposition 1 Let N = N0 and fN be a soft set over U. Then, fN ðxðy þ zÞ xyÞ fN ðzÞ implies that fN ðxyÞ fN ðyÞ for all x; y; z 2 N: Proof Let N be a zero-symmetric near-ring, fN be a soft set over U and fN ðxðy þ zÞ xyÞ fN ðzÞ for all x; y; z 2 N: Choosing y = 0 yields fN ðxð0 þ zÞ x0Þ ¼ fN ðxð0 þ zÞ 0Þ ¼ fN ðxzÞ fN ðzÞ. h Theorem 10 Let fN be a soft int-ideal of N and fM be a soft int-ideal of M over U. Then, fN ^ fM is a soft intideal of N 9 M over U. Proof In Theorem 5, it is shown that if fN and fM are soft int near-rings over U, then so is fN ^ fM. Let ðx1 ; y1 Þ; ðx2 ; y2 Þ; ðx3 ; y3 Þ 2 N M: Then
Let N be the set of parameters and U = D2 = {\x, y[: x2 = y2 = e, xy = yx} = {e, x, y, yx}, dihedral group, be the universal set. We define a soft set fN over U by
fN^M ððx1 ; y1 Þ þ ðx2 ; y2 Þ ðx1 ; y1 ÞÞ
fN ð0Þ ¼ D2 ; fN ð1Þ ¼ fN ð2Þ ¼ fN ð3Þ ¼ fe; x; yxg:
¼ fN^M ðx2 ; y2 Þ; fN^M ððx1 ; y1 Þðx2 ; y2 ÞÞ
Then, one can show that fN is a soft int-ideal of N over U. Now, we define a soft set hN over U by hN ð0Þ ¼ D2 ; hN ð1Þ ¼ hN ð3Þ ¼ fe; x2 g; hN ð2Þ ¼ fe; x2 ; yxg: One can show that hN is a right soft int-ideal of N over U. However, since hN ð3 ð2 þ 2Þ 3 2Þ ¼ hN ð3 0 3 2Þ ¼ hN ð0 3Þ ¼ hN ð0 þ 1Þ ¼ hN ð1Þ+hN ð2Þ; hN is not a left soft int-ideal of N over U. Hence, hN is not a soft int-ideal of N over U.
¼ fN^M ðx1 þ x2 x1 ; y1 þ y2 y1 Þ ¼ fN ðx1 þ x2 x1 Þ \ fM ðy1 þ y2 y1 Þ fN ðx2 Þ \ fM ðy2 Þ
¼ fN^M ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ \ fM ðy1 y2 Þ fN ðx1 Þ \ fM ðy1 Þ ¼ fN^M ðx1 ; y1 Þ and fN^M ððx1 ; y1 Þððx2 ; y2 Þ þ ðx3 ; y3 ÞÞ ðx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fN^M ðx1 ðx2 þ x3 Þ x1 x2 ; y1 ðy2 þ y3 Þ y1 y2 Þ ¼ fN ðx1 ðx2 þ x3 Þ x1 x2 Þ \ fM ðy1 ðy2 þ y3 Þ y1 y2 Þ
Theorem 9 Let N be a near-field and fN be a soft intideal of N over U. Then, fN ð0Þ fN ð1N Þ ¼ fN ðxÞ for all 0 6¼ x 2 N:
fN ðx3 Þ \ fM ðy3 Þ ¼ fN^M ðx3 ; y3 Þ: Therefore, fN ^ fM is a soft int-ideal of N 9 M over U.
Proof Assume that fN is a soft int-ideal of N over U, then fN is a soft int near-ring over U. Since fN ð0Þ fN ðxÞ for all x 2 N; then in particular fN ð0Þ fN ð1N Þ. Now, let 0 6¼ x 2 N: Then fN ðxÞ ¼ fN ð1N xÞ fN ð1N Þ and fN ð1N Þ ¼ fN ðx x1 Þ fN ðxÞ; implying that fN(1N) = fN(x) for all 0 6¼ x 2 N.
h
For a near-ring N, the zero-symmetric part of N denoted by N0 is defined by N0 ¼ fn 2 N j n0 ¼ 0g: It is known that if N is a zero-symmetric near-ring and ICl N; then NI N [28]. Here, we have an analog for this case:
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h Definition 11 Let fN be a soft int-ideal of N and gM be a soft int-ideal of M over U. Then, the product of soft intideals fN and gM is defined as fN 9 gM = hN9M, where hN9M(x, y) = fN(x) 9 gM(y) for all ðx; yÞ 2 N M: Theorem 11 If fN is a soft int-ideal of N and gM is a soft int-ideal of M over U, then fN 9 gM is a soft intideal of N 9 M over U 9 U. Proof In Theorem 6, it is shown that if fN and gM are soft int near-rings over U 9 U, then so is fN 9 gM. Let ðx1 ; y1 Þ; ðx2 ; y2 Þ; ðx3 ; y3 Þ 2 N M;
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Proof It is obvious that 0 2 Nf N. We need to show that ðiÞ x y 2 Nf ; ðiiÞ n þ x n 2 Nf ; ðiiiÞ xn 2 Nf and ðivÞnðs þ xÞ ns 2 Nf for all x; y 2 Nf and n; s 2 N: If x; y 2 Nf ; then fN(x) = fN(y) = fN(0). By Lemma 1, fN ð0Þ fN ðx yÞ; fN ð0Þ fN ðn þ x nÞ; fN ð0Þ fN ðxnÞ and fN ð0Þ fN ðnðs þ xÞ nsÞ for all x; y 2 Nf and n; s 2 N: Since fN is a soft int-ideal of N over U, so for all x; y 2 Nf and n; s 2 N; ðiÞ fN ðx yÞ fN ðxÞ \ fN ðyÞ ¼ fN ð0Þ; ðiiÞ fN ðn þ x nÞ fN ðxÞ ¼ fN ð0Þ; ðiiiÞ fN ðxnÞ fN ðxÞ ¼ fN ð0Þ and ðivÞfN ðnðs þ xÞ nsÞ fN ðxÞ ¼ fN ð0Þ. Hence fN(x - y) = fN(0), fN(n ? x - n) = fN(0), fN(xn) = fN(0) and fN (n(s ? x) - ns) = fN(0) for all x; y 2 Nf and n; s 2 N: h Therefore, Nf is an ideal of N.
fNM ððx1 ; y1 Þ þ ðx2 ; y2 Þ ðx1 ; y1 ÞÞ ¼ fNM ðx1 þ x2 x1 ; y1 þ y2 y1 Þ ¼ fN ðx1 þ x2 x1 Þ fM ðy1 þ y2 y1 Þ fN ðx2 Þ fM ðy2 Þ ¼ fNM ðx2 ; y2 Þ; fNM ððx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fNM ðx1 x2 ; y1 y2 Þ ¼ fN ðx1 x2 Þ fM ðy1 y2 Þ fN ðx1 Þ fM ðy1 Þ ¼ fNM ðx1 ; y1 Þ and fNM ððx1 ; y1 Þððx2 ; y2 Þ þ ðx3 ; y3 ÞÞ ðx1 ; y1 Þðx2 ; y2 ÞÞ ¼ fNM ðx1 ðx2 þ x3 Þ x1 x2 ; y1 ðy2 þ y3 Þ y1 y2 Þ
Theorem 14 Let fN be a soft set over U and a be a subset of U such that ; a fN ð0Þ. If fN is a soft intideal of N over U, then fNa is an ideal of N.
¼ fN ðx1 ðx2 þ x3 Þ x1 x2 Þ fM ðy1 ðy2 þ y3 Þ y1 y2 Þ fN ðx3 Þ fM ðy3 Þ ¼ fNM ðx3 ; y3 Þ Hence, fN 9 gM is a soft int-ideal of N 9 M over U 9 U. h Theorem 12 Let fN and hN be two soft int-ideals of N e hN is a soft int-ideal of N over U. over U. Then, fN \ Proof In Theorem, it is shown that if fN and hN are soft int e hN . Let x; y; z 2 N. Then near-rings over U, then so is fN \ e hN ÞðxþyxÞ ¼ fN ðxþyxÞ\hN ðxþyxÞ ðfN \
Theorem 15 Let fN and fM be soft sets over U and W be a near-ring isomorphism from N to M. If fN is a soft int-ideal of N over U, then WðfN Þ is a soft int-ideal of M over U.
fN ðyÞ\hN ðyÞ e hN ÞðyÞ ¼ ðfN \ e hN ÞðxyÞ ¼ fN ðxyÞ\hN ðxyÞ ðfN \
Proof Let m1 ; m2 ; m3 2 M: Since W is surjective, then there exists n1 ; n2 ; n3 2 N such that Wðn1 Þ ¼ m1 ; Wðn2 Þ ¼ m2 andWðn3 Þ ¼ m3 : Then,
fN ðxÞ\hN ðxÞ e hN ÞðxÞ ¼ ðfN \ e hN ÞðxðyþzÞxyÞ ¼ fN ðxðyþzÞxyÞ ðfN \ \hN ðxðyþzÞxyÞ fN ðzÞ\hN ðzÞ e hN ÞðzÞ ¼ ðfN \ e hN is a soft int-ideal of N over U. Therefore, fN \
Proof Since fN ð0Þ a; then 0 2 fNa and ; 6¼ fNa N: If x; y 2 fNa ; then fN ðxÞ a and fN ðyÞ a. We need to show that ðiÞ x y 2 fNa ; ðiiÞ n þ x n 2 fNa ; ðiiiÞ xn 2 fNa and (iv) nðs þ xÞ ns 2 fNa for all x; y 2 fNa and n; s 2 N: Since fN is a soft int-ideal of N over U, so fN ðx yÞ fN ðxÞ \ fN ðyÞ a \ a ¼ a; fN ðn þ x nÞ fN ðxÞ a; fN ðxnÞ fN ðxÞ a and fN ðnðs þ xÞ nsÞ fN ðxÞ a: Thus, the proof is completed. h
h
4 Applications of soft int near-rings and soft int-ideals In this section, we give the applications of soft image, soft pre-image and upper-a-inclusion of sets to near-ring theory with respect to soft int near-rings and soft int-ideals of a near-ring. Theorem 13 If fN is a soft int-ideal of N over U, then Nf ¼ fx 2 N : fN ðxÞ ¼ fN ð0Þg is an ideal of N.
ðWðfN ÞÞðm1 m2 Þ [ ¼ ffN ðnÞ : n 2 N; WðnÞ ¼ m1 m2 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 m2 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðWðn1 n2 ÞÞ ¼ n1 n2 g [ ¼ ffN ðn1 n2 Þ : ni 2 N; Wðni Þ ¼ mi ; i ¼ 1; 2g [ ffN ðn1 Þ \ fN ðn2 Þ : ni 2 N; Wðni Þ ¼ mi ; i ¼ 1; 2g [ ¼ ffN ðn1 Þ : n1 2 N; Wðn1 Þ ¼ m1 g [ \ ffN ðn2 Þ : n2 2 N; Wðn2 Þ ¼ m2 g ¼ ðWðfN ÞÞðm1 Þ \ ðWðfN ÞÞðm2 Þ Similarly, one can show that ðWðfN ÞÞðm1 m2 Þ ðWðfN ÞÞ ðm1 Þ \ ðWðfN ÞÞðm2 Þ. Also,
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Neural Comput & Applic (2012) 21 (Suppl 1):S221–S229
ðW1 ðfM ÞÞðn1 n2 Þ ðW1 ðfM ÞÞðn1 Þ \ ðW1 ðfM ÞÞðn2 Þ:
ðWðfN ÞÞðm1 þ m2 m1 Þ [ ¼ ffN ðnÞ : n 2 N; WðnÞ ¼ m1 þ m2 m1 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 þ m2 m1 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðWðn1 þ n2 n1 ÞÞ
Also, ðW1 ðfM ÞÞðn1 þ n2 n1 Þ ¼ fM ðWðn1 þ n2 n1 ÞÞ ¼ fM ðWðn1 Þ þ Wðn2 Þ Wðn1 ÞÞ fM ðWðn2 ÞÞ
¼ n1 þ n2 n1 g [ ¼ ffN ðn1 þ n2 n1 Þ : ni 2 N; Wðni Þ ¼ mi ; i ¼ 1; 2g [ ffN ðn2 Þ : n2 2 N; Wðn2 Þ ¼ m2 g
¼ ðW1 ðfM ÞÞðn2 Þ Moreover,
¼ ðWðfN ÞÞðm2 Þ
ðW1 ðfM ÞÞðn1 n2 Þ ¼ fM ðWðn1 n2 ÞÞ
Moreover, ðWðfN ÞÞðm1 m2 Þ ¼
¼ fM ðWðn1 ÞWðn2 ÞÞ
[
fM ðWðn1 ÞÞ
ffN ðnÞ : n 2 N; WðnÞ ¼ m1 m2 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 m2 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðWðn1 n2 ÞÞ
Furthermore,
¼ n1 n2 g [ ¼ ffN ðn1 n2 Þ : ni 2 N; Wðni Þ
ðW1 ðfM ÞÞðn1 ðn2 þ n3 Þ n1 n2 Þ ¼ fM ðWððn1 ðn2 þ n3 Þ n1 n2 ÞÞÞ
¼ mi ; i ¼ 1; 2g [ ffN ðn1 Þ : n1 2 N; Wðn1 Þ ¼ m1 g
¼ fM ðWðn1 ÞððWðn2 Þ þ Wðn3 ÞÞ Wðn1 ÞWðn2 ÞÞ
¼ ðW1 ðfM ÞÞðn1 Þ
fM ðWðn3 Þ ¼ ðW1 ðfM ÞÞðn3 Þ
¼ ðWðfN ÞÞðm1 Þ
Hence, W1 ðfM Þ is a soft int-ideal of N over U.
h
Furthermore, ðWðfN ÞÞðm1 ðm2 þ m3 Þ m1 m2 Þ [ ¼ ffN ðnÞ : n 2 N; WðnÞ ¼ m1 ðm2 þ m3 Þ m1 m2 g [ ¼ ffN ðnÞ : n 2 N; n ¼ W1 ðm1 ðm2 þ m3 Þ m1 m2 Þg [ ¼ ffN ðnÞ : n 2 N; n ¼ n1 ðn2 þ n3 Þ n1 n2 g [ ¼ ffN ðn1 ðn2 þ n3 Þ n1 n2 Þ : ni 2 N; Wðni Þ
5 Conclusion
¼ mi ; i ¼ 1; 2; 3g [ ffN ðn3 Þ : n3 2 N; Wðn3 Þ ¼ m3 g ¼ ðWðfN ÞÞðm3 Þ Hence, WðfN Þ is a soft int-ideal of M over U.
h
Theorem 16 Let fN and fM be soft sets over U and W be a near-ring homomorphism from N to M. If fM is a soft int-ideal of M over U, then W1 ðfM Þ is a soft int-ideal of N over U. Proof
In this paper, by using soft sets and intersection operation of sets, we have defined a new concept, called soft int nearring. This new notion brings the soft set theory, set theory and near-ring theory together and therefore is very functional for obtaining results by means of near-ring structure. Based on the definition, we have introduced the concepts of soft int subnear-rings and soft int-ideals of a near-ring with illustrative examples. We have then investigated these notions with respect to soft image, soft pre-image and a-inclusion of soft sets. Finally, we give some applications of soft int near-rings to near-ring theory. To extend this study, one can further study the other algebraic structures such as different algebras by means of soft intersections.
Let n1 ; n2 ; n3 2 N: Then,
ðW1 ðfM ÞÞðn1 n2 Þ ¼ fM ðWðn1 n2 ÞÞ ¼ fM ðWðn1 Þ Wðn2 ÞÞ fM ðWðn1 ÞÞ \ fM ðWðn2 ÞÞ ¼ ðW1 ðfM ÞÞðn1 Þ \ ðW1 ðfM ÞÞðn2 Þ Similarly, one can show that
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