Characterizations of Hemi-Rings by their Bipolar-Valued Fuzzy h-ideals

Transcription

1 Inf Sci Lett 4 No ( Information Sciences Letters An International Journal haracterizations of Hemi-Rings by their Bipolar-Valued Fuzzy h-ideals Tahir Mahmood and Khizar Hayat Department of Mathematics and Statistics International Islamic University H-10 Islamabad Pakistan Received: 18 Aug 2014 Revised: 18 Dec 2014 Accepted: 19 Dec 2014 Published online: 1 May 2015 Abstract: In this paper we employed bipolar fuzzy set theory to hemi-rings and we popularized notation of bipolar valued fuzzy interior h-ideal bipolar-valued fuzzy prime h-ideal bipolar-valued fuzzy semiprime h-ideal bipolar-valued fuzzy h-bi-ideal and bipolarvalued fuzzy h-quasi-ideal We also introduced some basic properties and definition of bipolar-valued fuzzy h-ideals of hemi-ring Then we introduced results of biplar-valued fuzzy h-ideal of h-hemi-regular and h-hemi-simple hemi-rings Keywords: Hemi-rings bipolar valued fuzzy h-ideal h-hemi-regular hemi-ring h-hemi-simple hemi-rings 1 Introduction Afterwards Zadeh [12] popularized fuzzy set there have been several generalizations of this essential concept Fuzzy sets are extremely useful to deal many problems in applied mathematics control engineering information sciences expert systems and theory of automata etc Although there are many generalizations of fuzzy sets but non of these deal with the problems related to the contrary characteristics of the members having membership degree 0 Lee [7] handled this problem by introducing the concept of Bipolar-valued fuzzy (BVF sets The BVF sets aggregate a proper information conception structure to solving daily life problems The sweet taste of foodstuffs is a bipolar valued fuzzy set Assuming that sweet taste of foodstuff as a positive membership value then bitter taste of foodstuffs as a negative membership value The remaining foodstuffs of taste like acidic saline chilly etc are extraneous to the sweet and bitter foodstuffs Thus these foodstuffs are accepted as zero membership values There are two types of appearance in bipolar valued fuzzy sets so called approved display and diminished display With the broad concern semirings presented by Vandiver [9] have been explored by many researchers [1 2 3] Ideals of hemi-rings as a class of particular hemi-ring play an essential role in the algebraic structure theories anyhow many properties of hemi-rings are described by ideals On the other hand generally ideals in hemi-rings do not correspond with the ideals in rings Subsequently Henriksen [4] defined k-ideals of hemi-rings Iizuka [5] presented another more restricted ideals of hemi-rings called h-ideals According to new concept of ideals La Torre [6] analyzed exhaustively h-ideals and k-ideals of hemi-rings In 2014 M zhou et al [8] contemplated the applications of bipolar fuzzy theory to hemi-rings In this paper we introduced some basic definition theorem and examples about bipolar-valued fuzzy h-ideals and we haracterized properties of h-hemi-regular and h-hemi-simple hemi-ring by using bipolar-valued fuzzy h-ideals bipolar-valued h-bi-ideals and bipolar-valued h-quasi-ideals 2 Preliminaries In this section for basic definitions of hemi-ring and basic concepts of BVF sets we reffer to [3] and [7] respectively Let R be a universe Expresses + =λ + λ + : R [01] and =λ λ : R [ 10] We symbolize B=z(λ + (zλ (z a BVF set in R where λ + (z + denotes the satisfaction degree of z R about some property generally it is known as a positive membership degree and λ (z denotes the satisfaction degree of z R about some implicit counter-property generally it is known as a negative membership degree For the sake of orresponding author tahirbakhat@yahoocom

2 52 T Mahmood K Hayat: haracterizations of Hemi-Rings by their Bipolar-Valued simplicity we shall use the symbol B=(λ + λ for BVF set B=z(λ + (zλ (z 21 Definition Let R be a universe and M R Then BVF characteristic function is given by M = ( M + M where M + 1 if z M (z= 0 if z / M M 1 if z M (z= 0 if z / M 22 Definition Let R be a universe and t (01] Then a BVF set B=(λ + λ in R of the form λ + t (z= + i f z=x 0 i f z x λ t (z= i f z=x 0 i f z x is called BVF point with value t = (t + t (01] [ 10 and support x It is express as x t = (x t + x t A BVF point x t is said to belong to BVF subset B written as x t B if B(x t ie λ + (x t + and λ (x t Moreover throught this paper R is hem-ring unless else particularized 3 Main Results In this section we introduced some basic definitions of BVF h-ideals and some theorem regarding BVF h-ideals 31 Definition A BVF set B=(λ + λ is called BVF h-subhemi-ring of R if it holds (1 x t B y r B = (x+y min(t r B (2 x t B y r B = (xy min(t r B (3 x+a 1 + z=a 2 + z (a 1 t B (a 2 r B = (x min(t r B xyza 1 a 2 R & t = (t + t r =(r + r (01] [ Definition A BVF set B = (λ + λ is called BVF left (resp right h-ideal of R if it holds(1 (3 and (4 x t B = (yx t B (resp (5 (xy t B x y R&t =(t + t (01] [ 10 B is called a BVF h-ideal if it is both left and right BVF h-ideal of R 33 Definition A BVF set B=(λ + λ is called BVF interior h-ideal of R if it holds(1 (2 (3 and (6 y t B = (xyz t B xyz R & t =(t + t (01] [ Definition A BVF set B=(λ + λ is called BVF h-bi-ideal of R if it holds(1 (2 (3 and (7 x t B y r B = (xzy min(t r B x y z R & t r (01] [ Definition Let B = (λ + λ be a BVF set of a commutative hemi-ring R with unity Then B is called BVF prime h-ideal of R if it holds(1(3(4 (5 and (8 (xy t B = (x t B or y t B x y R & t =(t + t (01] [ Definition Let B = (λ + λ be a BVF subset of a commutative hemi-ring R with unity Then B is called BVF semi-prime h-ideal of R if it holds(1(3(4(5 and (9 (x 2 t B= x t B x R& t (01] [ Remark In rest of the paper we denote set of BVF left h-ideals of R BVF right h-ideals of R BVF h-ideals of R BVF interior h-ideals of R BVF h-bi-ideals of R BVF prime h-ideals of R and BVF semi-prime h-ideals of R by BVFLhI(R BVFRhI(R BVFhI(R BVFIhI(R BVFhbI(R BVFPhI(R and BVFShI(R respectively

3 Inf Sci Lett 4 No (2015 / wwwnaturalspublishingcom/journalsasp Theorem The conditions (1 to (9 are equivelent to (1 to (9 respectively x 1 x 2 yr 1 r 2 where: (1 λ + (x 1 + x 2 minλ + (x 1 λ + (x 2 λ (x 1 + x 2 maxλ (x 1 λ (x 2 (2 λ + (x 1 x 2 minλ + (x 1 λ + (x 2 λ (x 1 x 2 maxλ (x 1 λ (x 2 (3 x 1 + r 1 + y=r 2 + y = λ + (x 1 min λ + (r 1 λ + (r 2 λ (x 1 max λ (r 1 λ (r 2 (4 λ + (x 1 x 2 λ + (x 2 λ (x 1 x 2 λ (x 2 (5 λ + (x 1 x 2 λ + (x 1 λ (x 1 x 2 λ (x 1 (6 λ + (x 1 yx 2 λ + (y λ (x 1 yx 2 λ (y (7 λ + (x 1 zx 2 minλ + (x 1 λ + (x 2 λ (x 1 zx 2 maxλ (x 1 λ (x 2 (8 λ + (x 1 x 2 maxλ + (x 1 λ + (x 2 λ (x 1 x 2 minλ (x 1 λ (x 2 (9 λ +( x 2 1 λ + (x 1 λ ( x 2 1 λ (x 1 Proof Straightforward 39 Theorem A BVF set B = (λ + λ BVFLhI(R(resp BVFRhI(R BVFhI(R BVFIhI(R BVFhbI(R BVFPhI(R BVFShI(R iff it holds following sets of conditions (1 (3 (4 (resp (1 (3 (5 (1 (3 (4 (5 (1 (2 (3 (6 (1 (2 (3 (7 (1 (3 (4 (5 (8 and (1 (3 (4 (5 (9 310 Example We define BVF set B as follows 0 1 p p µ µ learly B BVFhI(R 311 Example onsider hemi-ring N 0 with respect to the usual + and Let t 1 t 2 [01 be such that t 1 t 2 ie (t 1 t 1 (t 2 t 2 Define BVF set B=(λ + λ by λ + t1 if x 3 (x= t 2 if x / 3 and λ t1 if x 3 (x= t 2 if x / 3 x N 0 Where 3 is set of generators of 3 Then B= (λ + λ BVFhbI(N Remark If B = (λ + λ BVFLhI(R (BVFRhI(R BVFhI(R BVFIhI(R BVFhbI(R BVFPhI(R BVFShI(R then λ + (0 λ + (x and λ (0 λ (x x R 313 Theorem Let /0 I R Then I BVFLhI(R (resp BVFRhI(R BVFhI(R BVFIhI(R BVFhbI(R iff I is a left h-ideal (resp right h-ideal h-ideal interior h-ideal h-bi-ideal of R Proof Straightforward 314 Theorem Let /0 I R where R is a commutative hemi-ring with unity Then I = ( I + I BVFPhI(R(resp BV FShI(R iff I is a prime h-ideal(resp semi-prime h-ideals of R respectively Proof Straightforward onsider R=01 p p defined by p p p p p p p p p p p p p p p p 0 1 p p p p Theorem Every BVF h-ideal is a BVF interior h-ideal of R 316 Remark Generally converse of Theorem 315 is not true

4 54 T Mahmood K Hayat: haracterizations of Hemi-Rings by their Bipolar-Valued 317 Example onsider R = 0 p q r defined by the following operations + 0 p q r 0 0 p q r p p 0 r q q q r 0 p r r q p 0 Define B as follows 0 p q r p 0 q 0 q q r 0 q 0 q 0 p q r µ µ Then B=(µ + µ BVFIhI(R but B=(µ + µ / BVFhI(R As B(pqr = B(0 = (04 07 and B(q = (01 03 this shows µ + (pqr > µ + (q and µ (pqr < µ (q On the other hand B(pp = (01 03 and B(p = (04 07 this shows µ + (pp µ + (p and µ (pp µ (p 318 Theorem If B = (µ + µ BVFPhI(R then B=(µ + µ BVFShI(R Proof Suppose B = (λ + λ BVFPhI(R Then by definition λ + (xy maxλ + (xλ + (y and λ (xy minλ (xλ (y For y = x λ + (x λ + (x 2 and λ (x λ (x 2 Hence B BVFShI(R This complete the proof 319 Remark Generally converse of Theorem 318 is not true 320 Example Let N 0 =0 N and p 1 p 2 p 3 be the distinct prime numbers in N 0 If K 0 = N 0 and K l = p 1 p 2 p 3 p l N 0 where l = then K 0 K 1 K 2 K n K n+1 As every non-zero elememt of N 0 has unique prime factorization for l = 2 3 K l is a semi-prime h-ideal but not a prime h-ideal Then by (314 K l = ( +K K BVFShI(R but l l K l = ( +K K / BVFPhI(R l l 321 Theorem Let B i = ( λ i + λi : i Ω is a family BVF subsets of R and B i BVFLhI(R (resp BVFRhI(R BVFhI(R and BVFIhI(R Then B = B i BVFLhI(R (resp i Ω BVFRhI(R BVFIhI(R and BVFIhI(R where B = (λ + λ with λ + = λ i + and λ = λi ( λ + i Ω i Ω λ i + λ λi i Ω 322 Definition Let B 1 = (λ + λ and B 2 = (µ + µ be two BVF subset of R The h-intrinsic product of B 1 =(λ + λ and B 2 = (µ + µ is denoted and described as (B 1 h B 2 (x = ((λ + h µ + (x (λ h µ (x x R if x can be signified as x+σ m a ib i + z=σ n c jd j + z so that (λ + h µ + (x= ( m λ + ( ( m ( ai µ + bi ( ( n ( c j µ + d j x+σ m a ib i +z=σ n c jd j +z ( n λ + (λ h µ (x= ( m ( ( m ( λ ai µ bi ( x+σ m a ib i +z=σ n c n ( ( n ( jd j +z λ c j µ d j And (B 1 h B 2 (x = (00 x R if x cannot be signified as x+σ m a ib i + z=σ n c jd j + z 323 Theorem Let M 1 M 2 be h-ideals of R Then we have (im 1 M 2 iff + M 1 + M 2 M 1 M 2 (ii + M 1 + M 2 = + M 1 M 2 M 1 M 2 = M 1 M 2 (iii + M 1 h + M 2 = + M 1 M 2 M 1 h M 2 = M 1 M 2 Proof(i(ii Straightforwad (iii Let M1 M 2 = ( +M1M2 M1M2 Suppose x R and x M 1 M 2 so + = 1 = 1 Now let x+ M 1 M 2 M 1 M 2 Σ m p iq i + z=σ n r js j + z for some p r A 1 and q s M 2 Then ( M + 1 h M + 2 (x= ( m M + 1 (a i ( m M + 2 (b i x+σ m a ib i +z=σ n c jd j +z ( n + M 1 (c j ( n M + 2 (d j ( m M + 1 (p i ( m M + 2 (q i ( n + M 1 (r j ( n M + 2 (s j

5 Inf Sci Lett 4 No (2015 / wwwnaturalspublishingcom/journalsasp 55 = 1 ( M 1 h M 2 (x= m M 1 (a i ( m M 2 (b i x+σ m a ib i +z=σ n c jd j +z n M 1 (c j n M + 2 (d j m M 1 (p i ( m M 2 (q i n M 1 (r j n M 2 (s j = 1 Hence(iii is proved 324 Theorem A BVF subset B = (λ + λ BVFLhI(R (resp BVFRhI(R iff it holds (1 (3 and ( + R h λ + (x λ + (x( R h λ (x λ (x(resp (λ + h + R (x λ+ (x (λ h R (x λ (x 325 Lemma Let B 1 = (λ + λ BVFRhI(R and B 2 = (µ + µ BVFLhI(R Then λ + h µ + λ + µ + and λ h µ λ µ 326 Definition A BVF set B=(λ + λ is a called BVF h-quasi-ideal of R iff it holds for trlr 1 r 2 R (1 λ + (t+ r minλ + (tλ + (r λ (t+ r maxλ (tλ (r (3 t+ r 1 + l = r 2 + l = λ + (t minλ + (r 1 λ + (r 2 λ (t maxλ (r 1 λ (r 2 (10 (λ + h + R (+ R h λ + λ + (λ h R ( R h λ λ 327 Remark In rest of paper set of BVF h-quasi-ideal of R is denoted by BVFhqI(R 328 Example In Example 311 B=(λ + λ BVFhqI(N Theorem A BVF set B=(λ + λ BVFhqI(R iff all level subsets U(Bt /0 are h-quasi-ideal of R 330 Theorem Let /0 I R Then I BVFhqI(R iff I is a h-quasiideal of R 331 Lemma Let B = (λ + λ BVFRhI(R and B = (µ + µ BVFLhI(R Then B B BVFhqI(R Proof Straightforward 332 Lemma If B = (λ + λ BVFhqI(R then B=(λ + λ BVFhbI(R Proof Let B=(λ + λ BVFhqI(R It is sufficient to prove λ + (xyz minλ + (x λ + (z λ (xyz maxλ (xλ (z and λ + (xy minλ + (xλ + (y λ (xy maxλ (xλ (y x y z R Now we have λ + (xyz ((λ + h R + (+ R h λ + (xyz = min(λ + h R + (xyz(+ R h λ + (xyz m (λ + (a i n ( n (λ + (c j = min m (λ + (b i n ( n (λ + (d j minminλ + (0λ + (xminλ + (0λ + (z = minλ + (xλ + (z Analogously we have λ (xyz ((λ h R ( R h λ (xyz = max(λ h R (xyz( R h λ (xyz m (λ (a i n n (λ (c j = max m (λ (b i n n (λ (d j

6 56 T Mahmood K Hayat: haracterizations of Hemi-Rings by their Bipolar-Valued maxmaxλ (0λ (xmaxλ (0λ (z = maxλ (xλ (z Similarly we can prove λ + (xy minλ + (xλ + (y λ (xy maxλ (xλ (y Therefore B BVFhbI(R 4 haracterization of hemi-rings by their bipolar-valued fuzzy h-ideals In this section we applied concepts of Y Q Yin etal [10 11] on BVF h-ideals of R 41 Theorem Let B 1 =(λ + λ B 2 =(µ + µ be two BVF subsets of R Then we say λ + µ + µ λ iff λ + [ ]µ + µ [ ]λ and µ + [ ]λ + λ [ ]µ 42 Lemma The relation is called equivalance relation on BVF subsets of R 43 Lemma Hemi-ring R is h-hemi-regular iff MRM = M for every h- quasi-ideal M of R 44 Theorem Let B 1 = (λ + λ B 2 = (µ + µ BVFhI(R Then λ + [ ]µ + µ [ ]λ iff λ + µ + µ λ x R 45 Theorem If B 1 = (λ + λ BVFRhI(R and B 2 = (µ + µ BVFLhI(R Then (λ + h µ + (λ + µ + (λ h µ (λ µ iff R is h-hemi-regular Proof Suppose R is h-hemi-regular Let B 1 BVFRhI(R and B 2 BVFLhI(R then s R by the Lemma 325 (λ + h µ + (s (λ + µ + (s (λ h µ (s (λ µ (s and so by the Theorem 44 (λ + h µ + (s[ ](λ + µ + (s (λ µ (s[ ](λ h µ (s Now since R is h-hemi-regular so s R p q z R such that s + sps + z = sqs + z Thus (λ + h µ + (s = [ n λ + (p i ( n µ + (q i s+σ n p iq i +z=σ m r jt j +z ] ( m λ + (r j ( m µ + (t j minλ + (spλ + (sq µ + (s=(λ + µ + (s (λ + h µ + (s (λ + µ + (s and (λ h µ (s= [ n λ (p i n µ (q i s+σ n p iq i +z=σ m r jt j +z ] m λ (r j m µ (t j maxλ (spλ (sq µ (s =(λ µ (s (λ + h µ + (s (λ µ (s Hence(λ + h µ + (λ + µ + (λ h µ (λ µ onversely let P be a right and Q be a left h-ideal of R Then P = ( ( P + P BVFRhI(R and Q = Q + Q BVFLhI(R Now by the Theorem PQ = + P h Q + + PQ = + P h Q + + P + Q and + PQ = + P h Q + + P + Q = + P Q = PQ=P Q Hence R is h-hemi-regular 46 Theorem Let B BVFhbI(R Then λ + (λ + h + R h λ + λ (λ h R h λ iff R is h-hemi-regular ProofSuppose B BVFhbI(R (λ + h R + h λ + (x = ( m (λ + h R + (a i n (λ + h n R + (c j ( m (λ + (b i ( n (λ + (d j ((λ min + h R + (xa ((λ + h R + (xcλ+ (x m (λ + (a i n ( n (λ + (c j = min m (λ + (b i n ( n (λ + (d j λ + (x minminλ + (xaxλ + (xcxminλ + (xaxλ + (xcx λ + (x

7 Inf Sci Lett 4 No (2015 / wwwnaturalspublishingcom/journalsasp 57 Since xa+xaxa+za=xcxa+za and xc+xaxc+zc= xcxc + zc we have that (λ + h R + h λ + (x minλ + (xλ + (xλ + (x λ + (x Similarly (λ h R h λ (x = m (λ h R (a i n (λ h R (c j m (λ (b i n (λ (d j ((λ max h R (xa ((λ h R (xcλ (x = max n c j d j +z m (λ (a i n (λ (c j m (λ (b i n (λ (d j λ (x maxmaxλ (xaxλ (xcxmaxλ (xax λ (xcxλ (x Since xa+xaxa+za=xcxa+za and xc+xaxc+zc= xcxc + zc we have that (λ h R h λ (x maxλ (xλ (xλ (x = λ (x This proves λ + (λ + h R + h λ + λ (λ h R h λ onversely let M be any h-bi-ideal of R Then by the Theorem 313 M BVFhbI(R Since M + + M h R + h M + by the Theorem 323 M + + M h R + h M + = + and M MRM On the MRM other hand since M is h-bi-ideal of R so that MRM M This implies MRM M and MRM M = M therefore MRM = M Hence R is h-hemi-regular 47 Theorem The following conditions for R are equivalent: (i R is h-hemi-regular hemi-ring (ii minλ + µ + λ + h µ + h λ + maxλ µ λ h µ h λ for every B = (λ + λ BVFhbI(R and for every B =(µ + µ BVFhI(R (iii minλ + µ + λ + h µ + h λ + maxλ µ λ h µ h λ for every B = (λ + λ BVFhqI(R and for every B =(µ + µ BVFhI(R Proof(i = (ii Suppose(i holds (λ + h µ + h λ + (x = n ((λ min + h µ + (xa ((λ + h µ + (xcλ + (x = min n n ( m (λ + h µ + (a i ( n (λ + h µ + (c j ( m (λ + (b i ( n (λ + (d j ( m (λ + (a i ( n (λ + (c j ( m (µ + (b i ( n (µ + (d j ( m (λ + (a i ( n (λ + (c j ( m (µ + (b i ( n (µ + (d j λ + (x minminλ + (x µ + (axa µ + (cxa minλ + (x µ + (axcλ + (cxcλ + (x Since xa+xaxa+za=xcxa+za and xc+xaxc+zc= xcxc+zc we have that (λ + h µ + h λ + (x minminλ + (x µ + (xminλ + (x µ + (x = minλ + (x µ + (x Similarly (λ h µ h λ (x = m (λ h µ (a i n (λ h µ (c j m (λ (b i n (λ (d j ((λ max h µ (xa ((λ h µ (xcλ (x

8 58 T Mahmood K Hayat: haracterizations of Hemi-Rings by their Bipolar-Valued = max ( m (λ (a i ( n (λ (c j ( m (µ (b i n (µ (d j m (λ (a i n (λ (c j m (µ (b i n (µ (d j λ (x maxmaxλ (x µ (axa µ (cxa maxλ (xλ (axc µ (cxcλ (x Since xa + xaxa + za = xcxa + za and xc+xaxc+zc= xcxc+zc we have that (λ h µ h λ (x maxmaxλ (x µ (xmaxλ (x µ (x = maxλ (x µ (x This proves(ii (ii = (iii By the Lemma 332 it is straightforward (iii = (i Assume that (iii holds Let M be any h-quasi-ideal of R By the Theorem 313 M BVFhqI(R Since R BVFhI(R Now from (iii M + + M h R + h M + by the Theorem 323 M + + M h R + h M + = + and M MRM On the MRM other hand since M is h-bi-ideal of R so that MRM M This implies MRM M and MRM M = M therefore MRM = M Hence R is h-hemi-regular hemi-ring 48 Theorem Let R be a h-hemi-simple and B = (λ + λ be a BVF subset of R Then B BVFhI(R iff B BVFIhI(R Proof By the Theorem 315 if B=(λ + λ BVFhI(R then B=(λ + λ BVFIhI(R onversely assume B = (λ + λ BVFhI(R Let p q R a i b i c i d i e j f j g j h j R such that p+ Σ m a ipb i c i pd i + z=σ n e j p f j g j ph j + z Which implies pq+σ m a ipb i c i pd i q+zq=σ n e j p f j g j ph j q+zq Thus λ + (pq min λ +( Σ m a ipb i c i pd i q λ (Σ + n e j p f j g j ph j q 05 λ + (p And λ (pq max λ ( Σ m a ipb i c i pd i q λ (Σ n e j p f j g j ph j q 05 λ (p Thus B BVFRhI(R Similarly we can show B = (λ + λ BVFLhI(R Hence proved the theorem 49 Theorem If B 1 = (λ + λ BVFIhI(R and B 2 = (µ + µ BVFIhI(R Then (λ + h µ + (λ + µ + (λ h µ (λ µ iff R is h-semi-simple Proof Suppose for any B 1 = (λ + λ B 2 = (µ + µ BVFLhI(R By the Lemma (325 (λ + h µ + (λ + µ + (λ h µ (λ µ And by the Theorem (44 (λ + h µ + [ ](λ + µ + (λ µ [ ](λ h µ Since R is h-hemi-simple so s R c i d i e i f i c j d j e j f j R such that s+σ m c isd i e i s f i + z=σ n c j sd j e j s f j + z Thus (λ + h µ + (s=[ m λ + (a i ( m µ + (b i x+σ m a ib i +z=σ n a j b j +z ( n λ + (a j µ + (e j s f j ( n µ + (b j minλ + (c i sd i λ (c + j sd j ] µ + (e i s f i minλ + (s µ + (s =(λ + µ + (s and (λ h µ (s= [ m λ (a i m µ (b i x+σ m a ib i +z=σ n a j b j +z ( n λ (a j ( n µ (b j maxλ (c i sd i λ (c j sd j ] µ + (e i s f i µ (e + j s f j maxλ (s µ (s =(λ µ (s This proves(λ + h µ + (λ + µ + (λ h µ (λ µ onversely let I be a h-ideal of R By the Theorem 315 I is an interior h-ideal of R Now by the Theorem 313 I = ( I + I BVFIhI(R We have that I + = I + I + and + I = I + h I + I + I + and by the Theorem 323 I + = + = I = I 2 I2 Therefore R is h-hemi-simple References [1] S Ghosh Matrices over semirings inform vol90 pp

9 Inf Sci Lett 4 No (2015 / wwwnaturalspublishingcom/journalsasp 59 [2] K Glazek A guide to litrature on semirings and their applications in mathematics and in f ormation sciences with complete bibliography Kluwer Acad Publ Nederland 2002 [3] J S Golan Semirings and their applications Kluwer Acad Publ 1999 [4] M Henriksen Ideals in semirings with commutative addition Amer Math Soc Notices vol6 no1 p [5] K Iizuka On the Jacobson radical o f a semiring Tohoku Math J vol11 no3 pp [6] D R La Torre On h-ideals and k-ideals in hemi rings Publ Math Debrecen vol12 no2 pp [7] K M Lee Bipolar-Valued Fuzzy sets and their operations Proc Int onf on Intelligent Technologies Bangkok Thailand pp [8] Min Zhou and Shenggang Applications o f Bipolar Fuzzy theory to hemi rings Int J of innovative comp inf and cont vol10 no2 pp [9] H S Vandiver Note on a simple type o f algebra in which cancellation law o f addition does not hold Bulletin of the American Mathematical Society vol40 no12 pp [10] Y Q Yin H Li T he characterizations o f h-hemiregular hemi rings and h-intra-hemiregular hemi ringsinform Sci vol178 no17 pp [11] Y Q Yin X Huang D Xu and HLi T he characterizations o f h-hemisimple hemi rings Int J of fuzy systems vol11 pp [12] L A Zadeh Fuzzy sets Inform and ontrol vol8 no3 pp Tahir Mahmood is Assistant Professor of Mathematics at Department of Mathematics and Statistics International Islamic University Islamabad Pakistan He received his Ph D degree from Quaid-i-Azam University Islamabad Pakistan in 2012 under the supervision of Professor Dr Muhammad Shabir His areas of interest are Algebraic and Fuzzy Algebraic structures He has 13 international publications to his credit and he has also produced 4 MS students Khizar Hayat is MS mathematics student in International Islamic University Islamabad Pakistan He is working under the supervision of Dr Tahir Mahmood His areas of interest are algebra fuzzy algebra and topology Recently he submitted 3 research paper in reputed international journals of mathematics