A NEW APPROACH TO M(G)-GROUP SOFT UNION ACTION AND ITS APPLICATIONS TO M(G)-GROUP THEORY

P.Jeyaraman 1 & R.Nagarajan2

1Assistant Professor, Department of Mathematics, Bharathiar University,Coimbatore-046.

2Associate Professor, Department of Mathematics ,J J College of Engineering & Technology Tiruchirappalli- 620009, Tamilnadu, India

ABSTRACT:

In this paper, we define a new type of M(G)-group action , called M(G)-group soft union(SU) action and M(G)-ideal soft union(SU) action on a soft set. This new concept illustrates how a soft set effects on an M(G)-group in the mean of union and inclusion of sets and its function as bridge among soft set theory, set theory and M(G)-group theory. We also obtain some analog of classical M(G)- group theoretic concepts  for M(G)-group SU-action. Finally, we give the application of SU-actions on M(G)-group to M(G)-group theory.

KEYWORDS:

 soft set, M(G)-group, M(G)-group SU-action, M(G)–ideal SU-action, soft pre-image, soft anti-image, α– inclusion.

 AMS MATHEMATICS SUBJECT CLASSIFICATION: 03E70,08E40,

1.INTRODUCTION:

Soft set theory as in [1, 2, 11, 14, 15, 16, 18, 25, 28] was introduced in 1999 by Molodtsov [22] for dealing with uncertainties and it has gone through remarkably rapid strides in the mean of algebraic structures. Maji et al. [19] presented some definitions on soft sets and based on the analysis of several operations on soft sets Ali et al. [3] introduced several operations Moreover, Atagun and Sezgin [4] defined the concepts of soft sub rings and ideals of a ring, soft subfields of a field and soft sub modules of a module and studied their related properties with respect to soft set operations. Furthermore, soft set relations and functions [5] and soft mappings [21] with many related concepts were discussed. The theory of soft set has also a wide-ranging applications especially in soft decision making as in the following studies: [6, 7, 23, 29] Sezgin et.al [25] introduced a new concept to the literature of N-group called N-group soft intersection action. Operations of soft sets have been studied by some authors, too. of soft sets and Sezgin and Atagun [26] studied on soft set operations as well. In this paper, we define a new type of M(G)- group action on a soft set, which we call M(G)- group soft union action and abbreviate as “M(G)- group SU action “which is based on the inclusion  relation and union of sets. Since M(G)-   group

SU-action gathers soft set theory and set theory and M(G) –group theory, it is useful in improving the soft set theory with respect to M(G)- group structures. Based on this new notion, we then introduce the concepts of M(G)-ideal SU-action and show that if M(G)-group SU-action over U. Moreover, we investigate these notions with respect to soft image, soft pre-image and give their applications to M(G)- group theory.

2.PRELIMINARIES:

In this section, we recall some basic notions relevant to M(G)- groups and soft sets. By a near- ring, we shall mean an algebraic system (M(G),+, • ),

where

(N1) (M(G), +) forms a group (not necessarily abelian)

(N2) (M(G),  •) forms a semi group and

(N3) (x + y)z = xz + yz for all x,y,z ∈ G.

Throughout this paper, M(G) will always denote right near-ring. A normal subgroup H of M(G) is called a left ideal of M(G) if g(f+i)-gf ∈ H for all g,f ∈ M(G) and i ∈ I and  denoted  by  H⊲M(G). For a near-ring M(G), the zero-symmetric part of M(G) denoted by MO(G) is defined by MO(G) ={g∈ S / g0=0}.

Let G be a group, written additively but not necessarily abelian, and let M(G) be the set { f / f : G→ G } of all functions from G to G. An addition operation can be defined on M(G) ; given f,g  in M(G), then the mapping f+g from G to G is given by (f+g)x = f(x) +g(x) for x in G.Then (M(G), + ) is also group, which is abelian if and only if G is abelian. Taking the composition of mappings as the product, M(G) becomes a near-ring.

Let G be a group. Then, under the operation below;

µ : M(G) x G → G (f,a )  → fa

(G, µ ) is called M(G)-group. Let M(G) be a near-ring, G1  and G2  two M(G)-groups. Then ϕ : G1

→ G2   is called M(G)- homomorphism if for all x,y ε G1, for all g ε M(G),

  • Φ (x+y) = ϕ(x) + ϕ(y)
  • Φ (gx) = g ϕ(x). It is denoted by G. Clearly M(G) itself is an M(G)-group by natural operations.

For all undefined concepts and notions we refer to (24). From now on, U refers to on initial universe, E is a set of parameters P(U) is the power set of U and A,B,C⊆ E.

Definition[22]: A pair (F,A) is called a soft 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.

Note that a soft set (F, A) can be denoted by FA. In this case, when we define more than one soft set in some subsets A, B, C of parameters E, the soft sets will be denoted by FA, FB, FC, respectively. On the other case, when we define more than one soft set in a subset A of the set   of

parameters E, the soft sets will be denoted by FA,GA, HA, respectively. For more details, we refer to [11,17,18,26,29,7].

Definition[6] :The relative complement of the soft set FA over U is denoted by FrA, where

A : A → P(U) is a mapping given as F A(a) =U \FA(a), for all a ∈ A.

Definition[6]: Let FA and GB be two soft sets over U such that A∩B ≠ ∅,. The restricted intersection of FA and GB is denoted by FA ⋓ GB, and is defined as FA ⋓ GB =(H,C),

where C = A∩B and for all c ∈ C, H(c) = F(c)∩G(c).

Definition[6]: Let FA and GB be two soft sets over U such that A∩B ≠ ∅,. The restricted union of FA and GB is denoted by FA∪R GB, and is defined as FA∪R GB = (H,C),where C = A∩B and for all c ∈ C, H(c) = F(c)∪G(c).

2.5 Definition[12]: Let FA  and GB  be soft sets over the common universe U and ƒbe a function from A to B. Then we can define the soft set ƒ (FA) over U, where ƒ (FA) : B→P(U) is a set valued function defined by ƒ (FA)(b) =∪{F(a) | a ∈ A and ƒ (a) = b},

ifƒ−1(b) ≠ ∅,  = 0 otherwise for all b ∈ B. Here, ƒ (FA) is called the soft image of FA under ƒ. Moreover  we can define a soft  set ƒ−1(GB)  over  U, where ƒ−1(GB)  :  A → P(U)  is a set-valued function  defined  by ƒ−1(GB)(a)  =  G(ƒ (a))  for  all  a  ∈  A.  Then, ƒ−1(GB)  is  called  the  soft  pre image (or inverse image) of GB under ƒ.

2.6.Definition[13]: Let FA and GB be soft sets over the common universe U and ƒ be a function from A to B. Then we can define the soft set ƒ⋆(FA) over U, where ƒ⋆(FA) : B→P(U) is a set-valued function defined by ƒ⋆(FA)(b)=∩{F(a) | a ∈ A and ƒ (a) = b},  if ƒ−1(b) ≠ ∅,

=0 otherwise for all b ∈ B. Here, ƒ⋆(FA) is called the soft anti image of FA  under ƒ.

2.7 Definition [8]:  Let fA  be a soft set over U and α be a subset of U. Then, lower α-inclusion  of

a soft set fA, denoted by f α   , is defined as f α   = {x   A : f  (x) ⊆ α}

3.M(G) –GROUP SU-ACTION

In this section, we first define M(G)-group soft union action, abbreviated as M(G)-group SU- action with illustrative examples. We then study their basic results with respect to soft set operation.

3.1Definition: Let S be an M(G)- group and fs be a fuzzy soft set over U, then fs is called fuzzy SU-action on M(G)- group over U if it satisfies the following conditions;

(FSUN-1) fs  (x+y) ⊆ fs  (x) U fs  (y)

(FSUN-2) fs  (-x) ⊆ fs  (x)

(FSUN-3) fs    (gx) ⊆ fs    (x) For all x,y ∈ S and g ∈M(G).

3.1 Example: Consider the near-ring module M(G) ={e,f,g,h},be the near-ring under  the  operation defined by the following table:

s1Let G=M(G)  be the set of  functions and

s2.JPG

s3.JPG

s4.JPG

s5

s6

s7

s8.JPG

s9

s10.JPG

s11

s12

3.2Definition : Let  be fuzzy SU-action on M(G)- group over U. Then product of fuzzy

s13

s14.JPG

4.SU-action on  M(G)-ideal structures

s15.JPGs16.JPG

4.1Example : Consider  M(G) =  with the following tables

s17.JPG

s18

s19.JPG

s20

s21

5.Application of fuzzy SU-action on M(G)- group

s22.JPG

s23

s24s25.JPGs26.JPG

s27.JPG

s28.JPG

Conclusion:

In this paper, we have defined a new type of N-module action on a fuzzy soft set, called fuzzy SU-action on M(G)- group by using the soft sets. This new concept picks up the soft set theory, fuzzy theory and M(G)- group theory together and therefore, it is very functional for obtaining results in the mean of M(G)- group structure. Based on this definition, we have introduced the concept of fuzzy SU-action on M(G)-ideal. We have investigated these notions with respect to soft image, soft pre-image and lower -inclusion of soft sets. Finally, we give some application of fuzzy SU-action on M(G)-ideal to M(G)- group theory. To extend this study, one can further study the other algebraic structures such as different algebra in view of their SU-actions.

Acknowledgement:

The authors are highly grateful to the referees for their valuable comments and suggestions for improving papers.

References:

 [1] Acar U., Koyuncu F., Tanay B., Soft sets and soft rings, Comput. Math. Appl., 59(2010), 3458-3463.

 [2] Aktas. H., C. agman N., ˘ Soft sets and soft groups, Inform. Sci., 177(2007), 2726-2735.

[3] Ali M.I.,Feng F., Liu X., Min W.K., Shabir M., On some new operations in soft set theory, Comput. Math. Appl., 57(2009), 1547-1553.

 [4] Atagun A.O., Sezgin A., Soft substructures of rings, fields and modules, Comput. Math. Appl., 61(3)(2011), 592-601.

[5] Babitha K.V., Sunil J.J., Soft set relations and functions, Comput. Math. Appl., 60(7)(2010), 1840-1849.

[6] Cagman N., Enginoglu S.,Soft matrix theory and its decision making, Comput. Math. Appl., 59(2010), 3308-3314.

 [7] Cagman N, Engino ˘ glu S., ˘ Soft set theory and uni-int decision making, Eur. J. Oper. Res., 207(2010), 848-855.

[8] Cagman N., C¸ ıtak F., Aktas¸ H., ˘ Soft int-groups and its applications to group theory, Neural Comput. Appl., DOI: 10.1007/s00521-011-0752-x.

[9] Cagman N., Sezgin A., Atag ˘ un A.O., ¨ Soft uni-groups and its applications to group theory, (submitted).

[10] Cagman N., Sezgin A., Atag ˘ un A.O., ¨ α-inclusions and their applications to group theory, (submitted). ˘

[11] F. Feng, W. Pedrycz, On scalar products and decomposition theorems of fuzzy soft sets, Journal of Multivalued Logic and Soft Computing, 2015, 25(1), 45-80.

[12] F. Feng, J. Cho, W. Pedrycz, H. Fujita, T. Herawan, Soft set based association rule mining, Knowledge-Based Systems, 111 (2016), 268-282

 [13] Feng F., Li C., Davvaz B., Ali M.I., Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14(6)(2010), 899-911.

[14] Jun Y.B., Soft BCK/BCI-algebras, Comput. Math. Appl., 56(2008), 1408 -1413.

[15] Jun Y.B., Park C.H., Applications of soft sets in ideal theory of BCK/ BCI-algebras, Inform. Sci., 178(2008), 2466-2475.

[16] Jun Y.B., Lee K.J., Zhan J., Soft p-ideals of soft BCI-algebras, Comput. Math. Appl., 58(2009), 2060-2068.

[17] Jun Y.B., Lee K.J., Park C.H., Soft set theory applied to ideals in d-algebras, Comput. Math. Appl., 57(3)(2009), 367-378.

[18] Kazancı O., Yılmaz S¸., Yamak S., Soft sets and soft BCH-algebras, Hacet. J. Math. Stat., 39(2)(2010), 205-217.

[19] Maji P.K., Biswas R., Roy A.R., Soft set theory, Comput. Math. Appl., 45(2003), 555-562. [20] Maji P.K., Roy A.R., Biswas R., An application of soft sets in a decision making problem, Comput. Math. Appl., 44(2002), 1077-1083.

[21] Majumdar P., Samanta S.K., On soft mappings, Comput. Math. Appl., 60 (9)(2010), 2666-2672.

[22] Molodtsov D., Soft set theory-first results, Comput. Math. Appl., 37(1999), 19-31.

[23] Molodtsov D.A., Leonov V.Yu., Kovkov D.V., Soft sets technique and its application, Nechetkie Sistemy i Myagkie Vychisleniya, 1(1)(2006), 8-39.

[24] Pilz G., Near-rings, North Holland Publishing Company, Amsterdam-New York-Oxford, 1983.

[25] Sezgin A., Atagun A.O., Ayg ¨ un E., ¨ A note on soft near-rings and idealistic soft near-rings, Filomat., 25(1)(2011), 53-68.

[26] Sezgin A., Atagun A.O., ¨ On operations of soft sets, Comput. Math. Appl., 61(5)(2011), 1457-1467.

 [27] Wendt G., On Zero Divisors in Near-Rings, Int. J. Algebra, 3(1)(2009), 21-32.

[28] Zhan J., Jun Y.B., Soft BL-algebras based on fuzzy sets, Comput. Math. Appl., 59(6)(2010), 2037-2046.

[29] Zou Y., Xiao Z., Data analysis approaches of soft sets under incomplete information, Knowl-Based Syst., 21(2008), 941-945.

 [30] J. Zhan, B. Yu, V. Fotea, Characterizations of two kinds of hemi rings based on probability spaces, Soft Comput, 20 (2016), 637-648.

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