Research Article Int-Soft (Generalized) Bi-Ideals of Semigroups

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1 e cientific World Journal Volume 2015, rticle ID , 6 pages Research rticle Int-oft (Generalized) i-ideals of emigroups Young ae Jun 1 and eok-zun ong 2 1 Department of Mathematics Education, Gyeongsang National University, Jinju , Republic of Korea 2 Department of Mathematics, Jeju National University, Jeju , Republic of Korea Correspondence should be addressed to eok-zun ong; szsong@jejunu.ac.kr Received 14 ugust 2014; ccepted 23 December 2014 cademic Editor: Guilong Liu Copyright 2015 Y.. Jun and.-z. ong. This is an open access article distributed under the Creative Commons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The notions of int-soft semigroups and int-soft left (resp., right) ideals in semigroups are studied in the paper by ong et al. (2014). In this paper, further properties and characterizations of int-soft left (right) ideals are studied, and the notion of int-soft (generalized) bi-ideals is introduced. Relations between int-soft generalized bi-ideals and int-soft semigroups are discussed, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are considered. Given a soft set (α; )overu, int-soft (generalized) bi-ideals generated by (α; )areestablished. 1. Introduction s a new mathematical tool for dealing with uncertainties, the notion of soft sets is introduced by Molodtsov [1]. The author pointed out several directions for the applications of soft sets. t present, works on the soft set theory are progressing rapidly. Maji et al. [2] described the application of soft set theory to a decision making problem. Maji et al. [3] also studied several operations on the theory of soft sets. Çağman and Enginoglu [4] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision making method based on FP-soft set theory and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Feng [5] considered the application of soft rough approximations in multicriteria group decision making problems. ktaş and Çağman [6] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. In general, it is well known that the soft set theory is a good mathematical model to deal with uncertainty. Nevertheless, it is also a new notion worth applying to abstract algebraic structures. o, we can provide the possibility of a new direction of soft sets based on abstractalgebraicstructuresindealingwithuncertainty.in fact, in the aspect of algebraic structures, the soft set theory has been applied to rings, fields, and modules (see [7, 8]), groups (see [6]), semirings (see [9]), (ordered) semigroups (see [10, 11]),andhypervectorspaces(see[12]). ong et al. [11] introduced the notion of int-soft semigroups and int-soft left (resp., right) ideals and investigated several properties. Our aim in this paper is to apply the soft sets to one of abstract algebraic structures, the so-called semigroup. o, we take a semigroup as the parameter set for combining soft sets with semigroups. This paper is a continuation of [11]. We first study further properties and characterizations of int-soft left (right) ideals. We introduce the notion of intsoft (generalized) bi-ideals and provide relations between intsoft generalized bi-ideals and int-soft semigroups. We discuss characterizations of (int-soft) generalized bi-ideals and intsoft bi-ideals. Given a soft set (α, ) over U, we establish intsoft (generalized) bi-ideals generated by (α, ). 2. Preliminaries Let be a semigroup. Let and be subsets of. Thenthe multiplication of and is defined as follows: = {ab a,b }. (1) semigroup issaidtoberegularifforeveryx there exists a such that xax=x. semigroup is said to be left (resp., right) zero if xy = x (resp., xy = y)forallx, y.

2 2 The cientific World Journal semigroup is said to be left (right) simple if it contains no proper left (right) ideal. semigroup is said to be simple if it contains no proper two-sided ideal. nonemptysubsetof is called (i) a subsemigroup of if,thatis,ab for all a, b, (ii) a left (resp., right) ideal of if (resp., ), that is, xa (resp., ax )forallx and a, (iii) a two-sided ideal of if it is both a left and a right ideal of, (iv) a generalized bi-ideal of if, (v) a bi-ideal of if it is both a semigroup and a generalized bi-ideal of. soft set theory is introduced by Molodtsov [1], and Çağman and Enginoğlu [13] provided new definitions and various results on soft set theory. In what follows, let U be an initial universe set and E be a set of parameters. Let P(U) denote the power set of U and,,c,..., E. Definition 1 (see [1, 13]). soft set (α, ) over U is defined to be the set of ordered pairs (α, ) := {(x, α (x)) :x E,α(x) P (U)}, (2) where α:e P(U) such that α(x) = 0 if x. The function α is called approximate function of the soft set (α, ). Thesubscript in the notation α indicates that α is the approximate function of (α, ). For a soft set (α, ) over U and a subset γ of U, theγinclusive set of (α, ), denoted by i (α; γ), isdefinedtobe the set i (α;γ):={x γ α(x)}. (3) For any soft sets (α, ) and (β, ) over U, we define (α, ) (β, ) if α (x) β(x) x. (4) The soft union of (α, ) and (β, ) is defined to be the soft set (α β,)over U in which α βis defined by (α β)(x) =α(x) β(x) x. (5) The soft intersection of (α, ) and (β, ) is defined to be the soft set (α β,)over U in which α βis defined by (α β) (x) =α(x) β(x) x. (6) The int-soft product of (α, ) and (β, ) isdefinedtobethe soft set (α β, ) over U in which α βis a mapping from to P(U) given by (α β)(x) = { {α (y) β (z)} if y, z such that { { 0 otherwise. (7) 3. Int-oft Ideals In what follows, we take E=, as a set of parameters, which is a semigroup unless otherwise specified. Definition 2 (see [11]). soft set (α, ) over U is called an intsoft semigroup over U if it satisfies ( x, y ) (α (x) α(y) α(xy)). (8) Definition 3 (see [11]). soft set (α, ) over U is called an intsoft left (resp., right) ideal over U if it satisfies ( x, y ) (α (xy) α(y)(resp., α (xy) α(x))). (9) If a soft set (α, ) over U is both an int-soft left ideal and an int-soft right ideal over U, wesaythat(α, ) is an int-soft two-sided ideal over U. Obviously, every int-soft (resp., right) ideal over U is an int-soft semigroup over U. uttheconverseisnottruein general (see [11]). Proposition 4. Let (α, ) be an int-soft left ideal over U. IfG is a left zero subsemigroup of, then the restriction of (α, ) to G is constant; that is, α(x) = α(y) for all x, y G. Proof. Let x, y G.Thenxy = x and yx=y.thus α (x) =α(xy) α(y)=α(yx) α(x), (10) and so α(x) = α(y) for all x, y G. imilarly, we have the following proposition. Proposition 5. Let (α, ) be an int-soft right ideal over U.IfG is a right zero subsemigroup of, then the restriction of (α, ) to G is constant; that is, α(x) = α(y) for all x, y G. Proposition 6. Let (α, ) be an int-soft left ideal over U. If the set of all idempotent elements of forms a left zero subsemigroup of, then α(x) = α(y) for all idempotent elements x and y of. Proof. ssume that the set I () :={x xis an idempotent element of } (11) is a left zero subsemigroup of. Foranyu, V I(), wehave uv =uand Vu =V.Hence α (u) =α(uv) α(v) =α(vu) α(u) (12) and thus α(u) = α(v) for all idempotent elements u and V of. imilarly, we have the following proposition. Proposition 7. Let (α, ) be an int-soft right ideal over U. If the set of all idempotent elements of forms a right zero subsemigroup of, then α(x) = α(y) for all idempotent elements x and y of.

3 The cientific World Journal 3 For a nonempty subset of and Φ, Ψ P(U) with Φ Ψ, define a map as follows: Φ if x, : P (U), x { Ψ otherwise. (13) Then,)is a soft set over U,whichiscalledthe(Φ, Ψ)- characteristic soft set. The soft set,) is called the (Φ, Ψ)-identitysoft set over U.The(Φ, Ψ)-characteristic soft set with Φ=Uand Ψ=0is called the characteristic soft set and is denoted by (χ,).the(φ, Ψ)-identity soft set with Φ=Uand Ψ=0is called the identity soft set and is denoted by (χ,). Lemma 8. Let,)and,)be (Φ, Ψ)-characteristic soft sets over U where and are nonempty subsets of. Then the following properties hold: (1) =, (2) =. Proof. (1) Let x.ifx,thenx and x.thus we have ) (x) = (x) =Φ= (x). (14) If x,thenx or x.hencewehave ) (x) = (x) =Ψ= (x). (15) Therefore =. (2) For any x,supposex. Then there exist a and b such that x=ab.thuswehave ) (x) = { (y) (z)} (a) (b) =Φ, (16) and so )(x) = Φ. incex,weget (x) = Φ. upposex.thenx =abfor all a and b.iffor some y, z,theny or z. Hence If x ) (x) = =yzfor all x, y,then { =Ψ= (x). (y) (z)} (17) ) (x) =Ψ= (x). (18) In any case, we have =. Theorem 9. For the (Φ, Ψ)-identity soft set,),let (α, ) be a soft set over U such that α(x) Φ for all x. Then the following assertions are equivalent: (1) (α, ) isanint-softleftidealoveru, (2) α,) (α,). Proof. uppose that (α, ) is an int-soft left ideal over U. Let x.iffor some y, z,then α)(x) = { (y) α (z)} Otherwise, we have α,) (α,). {Φ α (yz)} = α (x). (19) α)(x) = 0 α(x). Therefore Conversely, assume that α,) (α,).forany x, y,wehave α(xy) α)(xy) (x) α(y) =Φ α(y)=α(y). Hence (α, ) is an int-soft left ideal over U. imilarly, we have the following theorem. (20) Theorem 10. For the (Φ, Ψ)-identity soft set,),let (α, ) be a soft set over U such that α(x) Φ for all x. Then the following assertions are equivalent: (1) (α, ) isanint-softrightidealoveru, (2) (α,) (α,). Corollary 11. For the (Φ, Ψ)-identity soft set,),let (α, ) be a soft set over U such that α(x) Φ for all x. Then the following assertions are equivalent: (1) (α, ) is an int-soft two-sided ideal over U, (2) α,) (α,)and (α,) (α,). Note that the soft intersection of int-soft left (right, twosided) ideals over U is an int-soft left (right, two-sided) ideal over U. In fact, the soft intersection of int-soft left (right, twosided) ideals containing a soft set (α, ) over U is the smallest int-soft left (right, two-sided) ideal over U. For any soft set (α, ) over U, the smallest int-soft left (right, two-sided) ideal over U containing (α, ) is called the int-soft left (right, two-sided) ideal over U generated by (α, ) and is denoted by (α, ) l ((α, ) r,(α,) 2 ). Theorem 12. Let be a monoid with identity e.then(α, ) l = (β, ),where for all a. β (a) = {α(y) a=xy,x,y }, (21)

4 4 The cientific World Journal Proof. Let a.incea=ea,wehave β (a) = {α(y) a=xy,x,y } α(a), (22) and so (α, ) (β,).forallx, y,wehave β(xy)= {α (x 2 ) xy=x 1 x 2,x 1,x 2 } {α (z 2 ) xy=(xz 1 )z 2,y=z 1 z 2,z 1,z 2 } {α (z 2 ) y=z 1 z 2,z 1,z 2 } =β(y). (23) Thus (β, ) is an int-soft left ideal over U.Nowlet(γ, ) be an int-soft left ideal over U such that (α, ) (γ,).thenα(a) γ(a) for all a and β (a) = {α (x 2 ) a=x 1 x 2,x 1,x 2 } {γ (x 2 ) a=x 1 x 2,x 1,x 2 } {γ (x 1 x 2 ) a=x 1 x 2,x 1,x 2 } =γ(a). This implies that (β, ) (γ,). Therefore (α, ) l =(β,). imilarly, we have the following theorem. (24) Theorem 13. Let be a monoid with identity e.then(α, ) r = (β, ),where for all a. β (a) = {α (x) a=xy,x,y }, (25) 4. Int-oft (Generalized) i-ideals Definition 14. softset(α, ) over U is called an int-soft generalized bi-ideal over U if it satisfies ( x, y ) (α (xyz) α(x) α(z)). (26) We know that any int-soft generalized bi-ideal may not be an int-soft semigroup by the following example. Example 15. Let = {a, b, c, d} be a semigroup with the following Cayley table: a b c d a a a a a b a a a a c a a b a d a a b b Let (α, ) be a soft set over U=Z defined as follows: α: P (U), 2Z { if x=a, x 4N { if x {b, d}, { 4Z if x=c. (27) (28) Then (α, ) is an int-soft generalized bi-ideal over U=Z.ut it is not an int-soft semigroup over U=Zsince α(c) α(c) = 4Z 4N = α(b) = α(cc). If a soft set (α, ) over U is both an int-soft semigroup and an int-soft generalized bi-ideal over U,then we say that(α, ) is an int-soft bi-ideal over U. We provide a condition for an int-soft generalized bi-ideal to be an int-soft semigroup. Theorem 16. In a regular semigroup, every int-soft generalized bi-ideal is an int-soft semigroup, that is, an int-soft biideal. Proof. Let (α, ) be an int-soft generalized bi-ideal over U and let x and y be any elements of. Then there exists a such that y=yay,andso α(xy) =α(x(yay)) =α(x(ya)y) α(x) α(y). (29) Therefore (α, ) is an int-soft semigroup over U. We consider characterizations of an int-soft (generalized) bi-ideal. Lemma 17. For any nonempty subset of,thefollowingare equivalent: (1) is a generalized bi-ideal of, (2) the (Φ, Ψ)-characteristic soft set,) over U is an int-soft generalized bi-ideal over U for any Φ, Ψ P(U) with Φ Ψ. Proof. ssume that is a generalized bi-ideal of.letφ, Ψ P(U) with Φ Ψand x, y, z.ifx, z,then (x) = (z) and xyz.hence Φ= If x or z,then (xyz) = Φ = (z). (30) (x) = Ψ or (z) = Ψ.Hence (xyz) Ψ = (z). (31) Therefore,)is an int-soft generalized bi-ideal over U for any Φ, Ψ P(U) with Φ Ψ. Conversely, suppose thatthe (Φ, Ψ)-characteristic soft set,)over U is an int-soft generalized bi-ideal over U for any Φ, Ψ P(U) with Φ Ψ.Letabe any element of. Then a=xyzfor some x, z and y.then (a) = =Φ Φ=Φ, (xyz) (z) (32) and so (a) = Φ.Thusa,whichshowsthat. Therefore is a generalized bi-ideal of. Theorem 18. softset(α, ) over U is an int-soft generalized bi-ideal over U if and only if the nonempty γ-inclusive set of (α, ) is a generalized bi-ideal of for all γ U.

5 The cientific World Journal 5 Proof. ssume that (α, ) is an int-soft semigroup over U.Let γ Ube such that i (α; γ) =0.Leta and x, y i (α; γ). Then α(x) γ and α(y) γ.itfollowsfrom(26) that α(xay) α(x) α(y) γ (33) and that xay i (α; γ).thusi (α; γ) is a generalized bi-ideal of. Conversely, suppose that the nonemptyγ-inclusive set of (α, ) is a generalized bi-ideal of for all γ U.Letx, y, z be such that α(x) = γ x and α(z) = γ z.takingγ=γ x γ z implies that x, z i (α; γ).hencexyz i (α; γ),andso α (xyz) γ=γ x γ z =α(x) α(z). (34) Therefore (α, ) is an int-soft generalized bi-ideal over U. Lemma 19 (see [11]). softset(α, ) over U is an int-soft semigroup over U if and only if the nonempty γ-inclusive set of (α, ) is a subsemigroup of for all γ U. Combining Theorem 18 and Lemma 19, wehavethe following characterization of an int-soft bi-ideal. Theorem 20. softset(α, ) over U isanint-softbi-idealover U if and only if the nonempty γ-inclusive set of (α, ) is a biideal of for all γ U. Theorem 21. For the identity soft set (χ,)and a soft set (α, ) over U,thefollowingareequivalent: (1) (α, ) is an int-soft generalized bi-ideal over U, (2) (α χ α,) (α,). Proof. ssume that (α, ) is an int-soft generalized bi-ideal over U. Leta be any element of. If(α χ α)(a) = 0, then it is clear that (α χ α,) (α,). Otherwise, there exist x, y, u, V such that a=xyand x=uv. ince(α, ) is an int-soft generalized bi-ideal over U, itfollowsfrom(26) that α (uvy) α(u) α(y) (35) and that (α χ α)(a) = ((α χ ) (x) α (y)) a=xy = (( (α (u) χ (V))) α (y)) a=xy x=uv Conversely, suppose that (α χ α,) (α,).forany x, y, z,leta=xyz.then α(xyz)=α(a) (α χ α)(a) = ((α χ ) (b) α(c)) a=bc (α χ )(xy) α(z) =( (α (u) χ (V))) α (z) xy=uv (α(x) χ (y)) α (z) = (α (x) U) α(z) =α(x) α(z). (37) Therefore (α, ) is an int-soft generalized bi-ideal over U. Theorem 22. Let (α, ) be an int-soft semigroup over U. Then (α, ) is an int-soft bi-ideal over U if and only if (α χ α,) (α,),where(χ,)is the identity soft set over U. Proof. ItisthesameastheproofofTheorem21. Theorem 23. If (α, ) and (β, ) are int-soft generalized biideals over U, then so is the soft intersection (α β,). Proof. For any a, b, x,wehave (α β) (axb) =α(axb) β(axb) (α (a) α(b)) (β(a) β(b)) (α(a) β(a)) (α(b) β(b)) =(α β) (a) (α β)(b). (38) Thus (α β,)is an int-soft generalized bi-ideals over U. Theorem 24. If (α, ) and (β, ) are int-soft bi-ideals over U, then so is the soft intersection (α β,). Proof. ItisthesameastheproofofTheorem23. = (( (α (u) U)) α(y)) a=xy x=uv = (α (u) α (y)) a=uvy α(uvy) a=uvy =α(a). Therefore (α χ α,) (α,). (36) Theorem 25. If is a group, then every int-soft generalized biideal is a constant function. Proof. Let be a group with identity e and let (α, ) be an intsoft generalized bi-ideal over U.Foranyx,wehave α (x) =α(exe) α(e) α(e) =α(e) =α(ee) =α((xx 1 )(x 1 x)) = α (x (x 1 x 1 )x) α(x) α(x) =α(x), and so α(x) = α(e). Therefore α is a constant function. (39)

6 6 The cientific World Journal For any soft set (α, ) over U, the smallest int-soft (generalized) bi-ideal over U containing (α, ) is called an intsoft (generalized) bi-ideal over U generated by (α, ) and is denoted by (α, ) b. Theorem 26. Let be a monoid with identity e and let (α, ) be a soft set over U such that α(x) α(e) for all x.then (α, ) b =(β,),where β (a) = {α (x 1 ) α(x 3 ) a=x 1 x 2 x 3,x 1,x 2,x 3 }, (40) for all a. Proof. Let a.incea = eea it follows from hypothesis that β (a) = {α (x 1 ) α(x 3 ) a=x 1 x 2 x 3,x 1,x 2,x 3 } α(e) α(a) =α(a) and that (α, ) (β,).foranyx, y, z,weget β (x) = {α (x 1 ) α(x 3 ) x=x 1 x 2 x 3,x 1,x 2,x 3 }, ince β (z) = {α (z 1 ) α(z 3 ) z=z 1 z 2 z 3,z 1,z 2,z 3 }, β(xyz) = {α (u 1 ) α(u 3 ) xyz=u 1 u 2 u 3,u 1,u 2,u 3 } {α (x 1 ) α(z 3 ) xyz=x 1 (x 2 x 3 yz 1 z 2 )z 3, β (x) β(z) x=x 1 x 2 x 3,z=z 1 z 2 z 3 }. = {(α (x 1 ) α(x 3 )) (α (z 1 ) α(z 3 )) x=x 1 x 2 x 3, z = z 1 z 2 z 3 }, (41) (42) (43) we have β(x) β(z) β(xyz).lety=e.thenβ(x) β(z) β(xz) for all x, z.hence(β, ) is an int-soft (generalized) bi-ideal over U.Let(γ, ) be an int-soft (generalized) bi-ideal over U such that (α, ) (γ,).foranya,wehave β (a) = {α (x 1 ) α(x 3 ) a=x 1 x 2 x 3,x 1,x 2,x 3 } {γ (x 1 ) γ(x 3 ) a=x 1 x 2 x 3,x 1,x 2,x 3 } Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. cknowledgments The authors wish to thank the anonymous reviewer(s) for the valuable suggestions. This research was supported by asic cience Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, cience and Technology (no. 2012R ). References [1] D. Molodtsov, oft set theory First results, Computers and Mathematics with pplications,vol.37,no.4-5,pp.19 31,1999. [2] P.K.Maji,.R.Roy,andR.iswas, napplicationofsoftsets in a decision making problem, Computers & Mathematics with pplications,vol.44,no.8-9,pp ,2002. [3] P.K.Maji,R.iswas,and.R.Roy, oftsettheory, Computers & Mathematics with pplications,vol.45,no.4-5,pp , [4] N. Çağman and. Enginoglu, FP-soft set theory and its applications, nnals of Fuzzy Mathematics and Informatics,vol. 2, no. 2, pp , [5] F. Feng, oft rough sets applied to multicriteria group decision making, nnals of Fuzzy Mathematics and Informatics, vol. 2, no. 1, pp , [6] H. ktaş andn.çağman, oftsetsandsoftgroups, Information ciences,vol.177,no.13,pp ,2007. [7] U. car, F. Koyuncu, and. Tanay, oft sets and soft rings, Computers & Mathematics with pplications,vol.59,no.11,pp , [8].O.tagün and. ezgin, oft substructures of rings, fields and modules, Computers & Mathematics with pplications,vol. 61,no.3,pp ,2011. [9] F. Feng, Y.. Jun, and X. Zhao, oft semirings, Computers & Mathematics with pplications, vol.56,no.10,pp , [10] Y.. Jun, K. J. Lee, and. Khan, oft ordered semigroups, Mathematical Logic Quarterly,vol.56,no.1,pp.42 50,2010. [11]. Z. ong, H.. Kim, and Y.. Jun, Ideal theory in semigroups based on inter sectional soft sets, The cientific World Journal, vol. 2014, rticle ID , 7 pages, [12] Y.. Jun, C. H. Park, and N. O. lshehri, Hypervector spaces based on intersectional soft sets, bstract and pplied nalysis, vol. 2014, rticle ID , 6 pages, [13] N. Çağman and. Enginoğlu, oft set theory and uni-int decision making, European Operational Research, vol. 207, no. 2, pp , {γ (x 1 x 2 x 3 ) a=x 1 x 2 x 3,x 1,x 2,x 3 } =γ(a), (44) and so (β, ) (γ,). Therefore (α, ) b =(β,).

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