Neutrosophic Ideals of Γ-Semirings

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1 ISSN: Number: 6, Year: 014, Pages: Receved: Accepted: Edtors-n-Chef : Nam Çağman Area Edtor: Oktay Muhtaroglu Neutrosophc Ideals of Γ-Semrngs Debabrata Mandal 1 (dmandaljumath@gmal.com) Department of Mathematcs, Raja Peary Mohan College, Uttarpara, Hooghly-7158, Inda Abstract - Neutrosophc deals of a Γ-semrng are ntroduced and studed n the sense of Smarandache[14], along wth some operatons such as ntersecton, composton, cartesan product etc. on them. Among the other results/characterzatons, t s shown that all the operatons are structure preservng. Keywords - Cartesan product, Homomorphsm, Ideal, Intersecton, Neutrosophc. 1 Introducton Uncertantes, whch could be caused by nformaton ncompleteness, data randomness lmtatons of measurng nstruments, etc., are pervasve n many complcated problems n bology, engneerng, economcs, envronment, medcal scence and socal scence. We cannot successfully use the classcal methods for these problems. To solve ths problem, the concept of fuzzy sets was ntroduced by Zadeh [15] n 1965 where each element have a degree of membershp and has been extensvely appled to many scentfc felds. As a generalzaton of fuzzy sets, the ntutonstc fuzzy set was ntroduced by Atanassov [1] n 1986, where besdes the degree of membershp of each element there was consdered a degree of non-membershp wth (membershp value + non-membershp value) 1. There are also several well-known theores, for nstances, rough sets, vague sets, nterval-valued sets etc. whch can be consdered as mathematcal tools for dealng wth uncertantes. In 1995, nspred from the sport games (wnnng/te/defeatng), votes, from (yes/na/no), from decson makng (makng a decson/ hestatng/not makng), from (accepted/pendng/rejected) etc. and guded by the fact that the law of excluded mddle dd not work any longer n the modern logcs, F. Smarandache [14] combned the non-standard analyss [4, 11] wth a tr-component logc/set/probablty theory and wth phlosophy and ntroduced Neutrosophc set whch represents the man dstncton between fuzzy and ntutonstc fuzzy logc/set. Here he ncluded the mddle component..e. the neutral/ ndetermnate/unknown part (besdes the truth/membershp 1 Correspondng Author

2 Journal of New Results n Scence 6 (014) and falsehood/non-membershp components that both appear n fuzzy logc/set) to dstngush between absolute membershp and relatve membershp or absolute nonmembershp and relatve non-membershp (see, [6, 13]). There are also several authors [, 3, 8] who have enrched the theory of neutrosophc sets. Inspred from the above dea and motvated by the fact that semrngs arse naturally n combnatorcs, mathematcal modellng, graph theory, automata theory, parallel computaton system etc., n the paper, I have used that to study the deals, whch play a central role n the structure theory and useful for many purposes, of Γ-semrngs[10] - a generalzaton of semrngs [5, 7] and obtan some of ts characterzatons. Prelmnares We recall the followng results for subsequent use. Defnton.1. Let S and Γ be two addtve commutatve semgroups wth zero. Then S s called a Γ-semrng f there exsts a mappng S Γ S S ( (a,α,b) aαb) satsfyng the followng condtons: () (a + b)αc = aαc + bαc () aα(b + c) = aαb + aαc () a(α + β)b = aαb + aβb (v) aα(bβc) = (aαb)βc (v) 0 S αa = 0 S = aα0 S (v) a0 Γ b = 0 S = b0 Γ a for all a, b, c S and for all α, β Γ. For smplfcaton we wrte 0 nstead of 0 S and 0 Γ. Defnton.. A left deal I of Γ-semrng S s a nonempty subset of S satsfyng the followng condtons: () If a, b I then a + b I () If a I, s S and γ Γ then sγa I () I S. A rght deal of S s defned n an analogous manner and an deal of S s a nonempty subset whch s both a left deal and a rght deal of S. Defnton.3. Let R, S be two Γ-semrngs and a, b R, γ Γ. A functon f : R S s sad to be a homomorphsm f () f(a + b) = f(a) + f(b) () f(aγb) = f(a)γf(b)

3 Journal of New Results n Scence 6 (014) () f(0 R ) = 0 S where 0 R and 0 S are the zeroes of R and S respectvely. Defnton.4. A neutrosophc set A on the unverse of dscourse X s defned as A = {< x, A T (x), A I (x), A F (x) >, x X}, where A T, A I, A F : X ] 0, 1 + [ and 0 A T (x) + A I (x) + A F (x) 3 +. From phlosophcal pont of vew, the neutrosophc set takes the value from real standard or non-standard subsets of ] 0, 1 + [. But n real lfe applcaton n scentfc and engneerng problems t s dffcult to use neutrosophc set wth value from real standard or non-standard subset of ] 0, 1 + [. Hence we consder the neutrosophc set whch takes the value from the subset of [0, 1]. 3 Man Results Throughout ths secton unless otherwse mentoned S denotes a Γ-semrng. Defnton 3.1. Let µ = (µ T, µ I, µ F ) be a non-empty neutrosophc subset of a Γ- semrng S (.e. anyone of µ T (x), µ I (x) or µ F (x) not equal to zero for some x S). Then µ s called a neutrosophc left deal of S f () µ T (x + y) mn{µ T (x), µ T (y)}, µ T (xγy) µ T (y) () µ I (x + y) µi (x)+µ I (y), µ I (xγy) µ I (y) () µ F (x + y) max{µ F (x), µ F (y)}, µ F (xγy) µ F (y). for all x, y S and γ Γ. Smlarly we can defne neutrosophc rght deal of S. Example 3.. Let S be the addtve commutatve semgroup of all non-postve ntegers and Γ be the addtve commutatve semgroup of all non-postve even ntegers. Then S s a Γ-semrng f aγb denotes the usual multplcaton of ntegers a, γ, b where a, b S and γ Γ. Defne a neutrosophc subset µ of S as follows (1, 0, 0) f x = 0 µ(x) = (0.8, 0.3, 0.4) f x s even (0.3,.0, 0.7) f x s odd Then the neutrosophc set µ of S s a neutrosophc deal of S. Theorem 3.3. A neutrosophc set µ of a Γ-semrng S s a neutrosophc left deal of S f and only f any level subsets µ T t := {x S : µ T (x) t, t [0, 1]}, µ I t := {x S : µ I (x) t, t [0, 1]} and µ F t := {x S : µ F (x) t, t [0, 1]} are left deals of S. Proof. Assume that the neutrosophc set µ of S s a neutrosophc left deal of S. Then anyone of µ T, µ I or µ F s not equal to zero for some x S.e., n other words anyone of µ T t, µ I t or µ F t s not equal to zero for all t [0, 1]. So t s suffcent to consder that all of them are not equal to zero. Suppose x, y µ t = (µ T t, µ I t, µ F t ), s S and γ Γ. Then µ T (x + y) mn{µ T (x), µ T (y)} mn{t, t} = t µ I (x + y) µi (x)+µ I (y) t+t = t µ F (x + y) max{µ F (x), µ F (y)} max{t, t} = t

4 Journal of New Results n Scence 6 (014) whch mples x + y µ T t, µ I t, µ F t.e., x + y µ t. Also µ T (sγx) µ T (x) t µ I (sγx) µ I (x) t µ F (sγx) µ F (x) t Hence sγx µ t. Therefore µ t s a left deal of S. Conversely, suppose µ t ( φ) s a left deal of S. If possble µ s not a neutrosophc left deal. Then for x, y S anyone of the followng nequalty s true. µ T (x + y) < mn{µ T (x), µ T (y)} µ I (x + y) < µi (x)+µ I (y) µ F (x + y) > max{µ F (x), µ F (y)} For the frst nequalty, choose t 1 = 1 [µt (x+y)+mn{µ T (x), µ T (y)}]. Then µ T (x+y) < t 1 < mn{µ T (x), µ T (y)} whch mples x, y µ T t 1 but x + y µ T t 1 - a contradcton. For the second nequalty, choose t = 1 [µi (x+y)+mn{µ I (x), µ I (y)}]. Then µ I (x+y) < t < µi (x)+µ I (y) whch mples x, y µ I t but x + y µ I t - a contradcton. For the thrd nequalty, choose t 3 = 1 [µf (x + y) + max{µ F (x), µ F (y)}]. Then µ F (x + y) > t 3 > max{µ F (x), µ F (y)} whch mples x, y µ F t 3 but x + y µ F t 3 - a contradcton. So, n any case we have a contradcton to the fact that µ t s a left deal of S. Hence the result follows. Defnton 3.4. [9] Let µ and ν be two neutrosophc subsets of S. The ntersecton of µ and ν s defned by (µ T ν T )(x) = mn{µ T (x), ν T (x)} for all x S. (µ I ν I )(x) = mn{µ I (x), ν I (x)} (µ F ν F )(x) = max{µ F (x), ν F (x)} Proposton 3.5. Intersecton of a non-empty collecton of neutrosophc left deals s also a neutrosophc left deal of S. Proof. Let {µ : I} be a non-empty famly of neutrosophc left deals of a Γ-semrng S and x, y S, γ Γ. Then µ T )(x + y) = nf µt (x + y) nf {mn{µt (x), µ T (y)}} = mn{nf µt (x), nf µt (y)} = mn{ µ T )(x), µ T )(y)}. µ I )(x + y) = nf µi (x + y) nf = nf µi (x)+nf µi (y) = µi (x)+ µi (y). µ I (x)+µi (y)

5 Journal of New Results n Scence 6 (014) µ F )(x + y) = sup = max{sup µ F (x + y) sup µ F (x), sup {max{µ F (x), µ F (y)}} µ F (y)} = max{ µ F )(x), µ F )(y)}. µ T )(xγy) = nf µt (xγy) nf µt (y) = µ T )(y). µ I )(xγy) = nf µi (xγy) nf µi (y) = µ I )(y). µ F )(xγy) = sup µ F (xγy) sup Hence µ s a neutrosophc left deal of S. µ F (y) = µ F )(y). Proposton 3.6. Let f : R S be a morphsm of Γ-semrngs. Then () If φ s a neutrosophc left deal of S, then f 1 (φ) [1] s a neutrosophc left deal of R. () If f s surjectve morphsm and µ s a neutrosophc left deal of R, then f(µ) [1] s a neutrosophc left deal of S. Proof. Let f : R S be a morphsm of Γ-semrngs. () Let φ be a neutrosophc left deal of S and r, s R, γ Γ. f 1 (φ T )(r + s) = φ T (f(r + s)) = φ T (f(r) + f(s)) mn{φ T (f(r)), φ T (f(s))} = mn{(f 1 (φ T ))(r), (f 1 (φ T ))(s)}. f 1 (φ I )(r + s) = φ I (f(r + s)) = φ I (f(r) + f(s)) φi (f(r))+φ I (f(s)) = (f 1 (φ I ))(r)+(f 1 (φ I ))(s). f 1 (φ F )(r + s) Agan = φ F (f(r + s)) = φ F (f(r) + f(s)) max{φ F (f(r)), φ F (f(s))} = max{(f 1 (φ F ))(r), (f 1 (φ F ))(s)}. (f 1 (φ T ))(rγs) = φ T (f(rγs)) = φ T (f(r)γf(s)) φ T (f(s)) = (f 1 (φ T ))(s). (f 1 (φ I ))(rγs) = φ I (f(rγs)) = φ I (f(r)γf(s)) φ I (f(s)) = (f 1 (φ I ))(s). (f 1 (φ F ))(rγs) = φ F (f(rγs)) = φ F (f(r)γf(s)) φ F (f(s)) = (f 1 (φ F ))(s). Thus f 1 (φ) s a neutrosophc left deal of R.

6 Journal of New Results n Scence 6 (014) () Suppose µ be a neutrosophc left deal of R and x, y S, γ Γ. Then (f(µ T ))(x + y ) = sup µ T (z) sup µ T (x + y) sup{mn{µ T (x), µ T (y)}} z f 1 (x +y ) x f 1 (x ), } = mn{(f(µ T ))(x ), (f(µ T ))(y )}. = mn{sup µ T (x) x f 1 (x ), sup µ T (y) (f(µ I ))(x + y ) = sup µ I (z) sup µ I (x + y) sup µi (x)+µ I (y) z f 1 (x +y ) x f 1 (x ), = 1 [sup µi (x) x f 1 (x ) + sup µ I (y) ] = 1 [(f(µi ))(x ) + (f(µ I ))(y )]. (f(µ F ))(x + y ) = nf µ F (z) z f 1 (x +y ) nf µ F (x + y) x f 1 (x ), nf{max{µ F (x), µ F (y)}} } = max{(f(µ F ))(x ), (f(µ F ))(y )}. = max{nf µ F (x) x f 1 (x ), nf µ F (y) Agan f(µ T )(x γy ) f(µ I )(x γy ) = sup µ T (z) z f 1 (x γy ) sup µ T (y) = sup µ I (z) z f 1 (x γy ) sup µ I (y) sup µ T (xγy) x f 1 (x ), = f(µ T )(y ). sup µ I (xγy) x f 1 (x ), = f(µ I )(y ). f(µ F )(x γy ) = nf µ F (z) z f 1 (x γy ) nf µ F (y) Thus f(µ)s a neutrosophc left deal of S. nf µ F (xγy) x f 1 (x ), = f(µ F )(y ). Defnton 3.7. [9] Let µ and ν be two neutrosophc subsets of S. The cartesan product of µ and ν s defned by (µ T ν T )(x, y) = mn{µ T (x), ν T (y)} for all x, y S. (µ I ν I )(x, y) = µi (x) + ν I (y) (µ F ν F )(x, y) = max{µ F (x), ν F (y)} Theorem 3.8. Let µ and ν be two neutrosophc left deals of S. neutrosophc left deal of S S. Then µ ν s a

7 Journal of New Results n Scence 6 (014) Proof. Let (x 1, x ), (y 1, y ) S S and γ Γ. Then (µ T ν T )((x 1, x ) + (y 1, y )) = (µ T ν T )(x 1 + y 1, x + y ) = mn{µ T (x 1 + y 1 ), ν T (x + y )} mn{mn{µ T (x 1 ), µ T (y 1 )}, mn{ν T (x ), ν T (y )}} = mn{mn{µ T (x 1 ), ν T (x )}, mn{µ T (y 1 ), ν T (y )}} = mn{(µ T ν T )(x 1, x ), (µ T ν T )(y 1, y )}. (µ I ν I )((x 1, x ) + (y 1, y )) = (µ I ν I )(x 1 + y 1, x + y ) = µi (x 1 +y 1 )+ν I (x +y ) 1{ µi (x 1 )+µ I (y 1 ) + νi (x )+ν I (y ) } = 1{ µi (x 1 )+ν I (x ) + µi (y 1 )+ν I (y ) } = 1 {(µi ν I )(x 1, x ) + (µ I ν I )(y 1, y )}. (µ F ν F )((x 1, x ) + (y 1, y )) = (µ F ν F )(x 1 + y 1, x + y ) = max{µ F (x 1 + y 1 ), ν F (x + y )} max{max{µ F (x 1 ), µ F (y 1 )}, max{ν F (x ), ν F (y )}} = max{max{µ F (x 1 ), ν F (x )}, max{µ F (y 1 ), ν F (y )}} = max{(µ F ν F )(x 1, x ), (µ F ν F )(y 1, y )}. (µ T ν T )((x 1, x )γ(y 1, y )) = (µ T ν T )(x 1 γy 1, x γy ) = mn{µ T (x 1 γy 1 ), ν T (x γy )} mn{µ T (y 1 ), ν T (y )} = (µ T ν T )(y 1, y ). (µ I ν I )((x 1, x )γ(y 1, y )) = (µ I ν I )(x 1 γy 1, x γy ) = µi (x 1 γy 1 )+ν I (x γy ) µi (y 1 )+ν I (y ) = (µ I ν I )(y 1, y ). (µ F ν F )((x 1, x )γ(y 1, y )) = (µ F ν F )(x 1 γy 1, x γy ) = max{µ F (x 1 γy 1 ), ν F (x γy )} max{µ F (y 1 ), ν F (y )} = (µ F ν F )(y 1, y ). Hence µ ν s a neutrosophc left deal of S S. Theorem 3.9. Let µ be a neutrosophc subset of S. Then µ s a neutrosophc left deal of S f and only f µ µ s a neutrosophc left deal of S S. Proof. Suppose µ be a neutrosophc subset of S. If µ s a neutrosophc left deal of S then by Theorem 3.8, µ µ s a neutrosophc left deal of S S. Conversely, suppose µ µ s a neutrosophc left deal of S S and x 1, x, y 1, y S, γ Γ. Then mn{µ T (x 1 + y 1 ), µ T (x + y )} = (µ T µ T )(x 1 + y 1, x + y ) = (µ T µ T )((x 1, x ) + (y 1, y )) mn{(µ T µ T )(x 1, x ), (µ T µ T )(y 1, y )} = mn{mn{µ T (x 1 ), µ T (x )}, mn{µ T (y 1 ), µ T (y )}}.

8 Journal of New Results n Scence 6 (014) µ I (x 1 +y 1 )+µ I (x +y ) = (µ I µ I )(x 1 + y 1, x + y ) = (µ I µ I )((x 1, x ) + (y 1, y )) (µi µ I )(x 1,x )+(µ I µ I )(y 1,y ) = 1[ µi (x 1 )+µ I (x ) + µi (y 1 )+µ I (y ) ]. max{µ F (x 1 + y 1 ), µ F (x + y )} = (µ F µ F )(x 1 + y 1, x + y ) = (µ F µ F )((x 1, x ) + (y 1, y )) max{(µ F µ F )(x 1, x ), (µ F µ F )(y 1, y )} = max{max{µ F (x 1 ), µ F (x )}, max{µ F (y 1 ), µ F (y )}}. Now, puttng x 1 = x, x = 0, y 1 = y and y = 0, n the above nequaltes and notng that µ T (0) µ T (x), µ I (0) = 0 and µ F (0) µ F (x) for all x S we obtan Next, we have µ T (x + y) mn{µ T (x), µ T (y)} µ I (x + y) µi (x)+µ I (y) µ F (x + y) max{µ F (x), µ F (y)}. mn{µ T (x 1 γy 1 ), µ T (x γy )} = (µ T µ T )(x 1 γy 1, x γy ) = (µ T µ T )((x 1, x )γ(y 1, y )) (µ T µ T )(y 1, y ) = mn{µ T (y 1 ), µ T (y )}. µ I (x 1 γy 1 )+ν I (x γy ) = (µ I µ I )((x 1, x )γ(y 1, y )) (µ I µ I )(y 1, y ) = µi (y 1 )+µ I (y ). max{µ F (x 1 γy 1 ), µ F (x γy )} = (µ F µ F )(x 1 γy 1, x γy ) = (µ F µ F )((x 1, x )γ(y 1, y )) (µ F µ F )(y 1, y ) = max{µ F (y 1 ), µ F (y )}. Takng x 1 = x, x = 0, y 1 = y and y = 0, we obtan µ T (xγy) µ T (y) µ I (xγy) µ I (y) µ F (xγy) µ F (y). Hence µ s a neutrosophc left deal of S. Defnton Let µ and ν be two neutrosophc sets of a Γ-semrng S. composton of µ and ν by Defne µ T oν T (x) = sup {mn{µ T (a ), ν T (b )}} x= a γ b = 0, f x cannot be expressed as above

9 Journal of New Results n Scence 6 (014) µ I oν I (x) = sup x= a γ b µ I (a )+ν I (b ) = 0, f x cannot be expressed as above µ F oν F (x) = nf {max{µ F (a ), ν F (b )}} x= a γ b where x, a, b S and γ Γ, for = 1,..., n. = 0, f x cannot be expressed as above Theorem If µ and ν be two neutrosophc left deals of S then µoν s also a neutrosophc left deal of S. Proof. Suppose µ, ν be two neutrosophc deals of S and x, y S, γ Γ. If x + y cannot be expressed as a γ b, for a, b S and γ Γ, then there s nothng to prove. So, assume that x + y have such an expresson. Then (µ T oν T )(x + y) = sup {mn{µ T (a ), ν T (b )}} x+y= a γ b x= sup c δ d, y = e η f {mn {µ T (c ), ν T (d ), µ T (e ), ν T (f )}} = mn{ sup {mn {µ T (c ), νt (d )}}, sup {mn{µ T (e ), ν T (f )}}} x= c δ d y= e η f = mn{(µ T oν T )(x), (µ T oν T )(y)}. (µ I oν I )(x + y) = sup x+y= a γ b x= 1 [ x= sup c δ d, y = sup c δ d = (µi oν I )(x)+(µ I oν I )(y). µ I (a )+ν I (b ) e η f µ I (c )+ν I (d ) µ I (c )+ν I (d )+µ I (e )+ν I (f ), sup y= e η f µ I (e )+ν I (f ) ]

10 Journal of New Results n Scence 6 (014) (µ F oν F )(x + y) = nf {max{µ F (a ), ν F (b )}} x+y= a γ b x= nf c δ d, y = e η f {max {µ F (c ), ν F (d ), µ F (e ), ν F (f )}} = max{ nf {max{µ F (c ), ν F (d )}}, nf {max{µ F (e ), ν F (f )}}} x= c δ d y= e η f = max{(µ F oν F )(x), (µ F oν F )(y)}. (µ T oν T )(xγy) = sup {mn{µ T (a ), ν T (b )}} xγy= a α b xγy= y= sup {mn{µ T (xγe ), ν T (f )}} xγe η f sup {mn{µ T (e ), ν T (f )}} = (µ T oν T )(y). e η f (µ I oν I )(xγy) = sup xγy= a α b xγy= y= sup xγe η f sup e η f µ I (a )+ν I (b ) µ I (xγe )+ν I (f ) µ I (e )+ν I (f ) = (µ I oν I )(y). (µ F oν F )(xγy) = nf {max{µ F (a ), ν F (b )}} xγy= a α b xγy= y= nf {max{µ F (xγe ), ν F (f )}} xγe η f nf {max{µ F (e ), ν F (f )}} = (µ F oν F )(y). e η f Hence µoν s a neutrosophc left deal of S.

11 Journal of New Results n Scence 6 (014) Concluson In ths paper, we have studed neutrosophc deals of Γ-semrngs n the sense of Smarandache[14] wth some operatons on them and obtan some of ts characterzatons. Our next am s to use these results to study some other propertes such prme neutrosophc deal, semprme neutrosophc deal, radcals etc.. References [1] K. Atanassov, Intutonstc fuzzy sets, Fuzzy Sets and Systems, 0 (1986) [] S. Broum, F. Smarandache and P. K. Maj, Intutonstc neutrosphc soft set over Rngs, Mathematcs and Statstcs, Vol., No. 3 (014) [3] S. Broum, I. Del and F. Smarandache, Relatons on nterval valued neutrosophc soft sets, Journal of New Results n Scence, 5 (014) [4] N. J. Cutland, Nonstandard Analyss and ts Applcatons, Cambrdge Unversty Press, Cambrdge, [5] J. S. Golan, Semrngs and ther applcatons, Kluwer Academc Publshers, Dodrecht, [6] G. J. Klr and B. Yuan, Fuzzy sets and fuzzy logc- Theory and Applcatons, Prentce Hal P T R Upper Saddle Rver, New Jersey, [7] U. Hebsch, J. Wenert, Semrngs algebrac theory and applcaton n computer scence, World Scentfc,1998. [8] P. K. Maj, Neutrosophc soft set, Annals of Fuzzy Mathematcs and Informatcs, Vol. 5, No. 1 (013) [9] D. Mandal, Neutrosophc deals of semrngs, submtted. [10] M.M.K. Rao, Γ-semrngs - 1, Southeast Asan Bull. of Math., 19 (1995), [11] A. Robnson, Nonstandard Analyss, Prnceton Unversty Press, Prnceton, [1] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), [13] T. J. Ross, J. M. Booker, V. J. Parknson, Fuzzy logc and probablty : Brdgng the gap, ASA-SIAM Seres on Statstcs and Appled Probablty, Alexandra, 00. [14] F. Smarandache, Neutrosophc set- a generalzaton of the ntutonstc fuzzy sets, Int. J. Pure Appl. Math., 4 (005) [15] L. A. Zadeh, Fuzzy sets, Informaton and Control, 8 (1965)