Soft Neutrosophic Bi-LA-semigroup and Soft Neutrosophic N-LA-seigroup

Transcription

1 Neutrosophc Sets and Systems, Vol. 5, Soft Neutrosophc B-LA-semgroup and Soft Mumtaz Al, Florentn Smarandache, Muhammad Shabr 3,3 Department of Mathematcs, Quad--Azam Unversty, Islamabad, 44000,Pakstan. -mal: Unversty of New Mexco, 705 Gurley Ave., Gallup, New Mexco 8730, USA -mal: Abstract. Soft set theory s a general mathematcal tool for dealng wth uncertan, fuzzy, not clearly defned objects. In ths paper we ntroduced soft neutrosophc b- LA-semgroup,soft neutosophc sub b-la-semgroup, soft neutrosophc N -LA-semgroup wth the dscusson of some of ther characterstcs. We also ntroduced a new type of soft neutrophc b-lasemgroup, the so called soft strong neutrosophc b-lasemgoup whch s of pure neutrosophc character. Ths s also extend to soft neutrosophc strong N-LA-semgroup. We also gven some of ther propertes of ths newly born soft structure related to the strong part of neutrosophc theory. Keywords: Neutrosophc b-la-semgroup,neutrosophc N -LA-semgroup, Soft set, Soft neutrosophc b-lasemgroup. Soft Neutrosophc N -LA-semgroup. Introducton Florentne Smarandache for the frst tme ntroduced the concept of neutrosophy n 995, whch s bascally a new branch of phlosophy whch actually studes the orgn, nature, and scope of neutraltes. The neutrosophc logc came nto beng by neutrosophy. In neutrosophc logc each proposton s approxmated to have the percentage of truth n a subset T, the percentage of ndetermnacy n a subset I, and the percentage of falsty n a subset F. Neutrosophc logc s an extenson of fuzzy logc. In fact the neutrosophc set s the generalzaton of classcal set, fuzzy conventonal set, ntutonstc fuzzy set, and nterval valued fuzzy set. Neutrosophc logc s used to overcome the problems of mprecseness, ndetermnate, and nconsstences of date etc. The theory of neutrosophy s so applcable to every feld of algebra. W.B. Vasantha Kandasamy and Florentn Smarandache ntroduced neutrosophc felds, neutrosophc rngs, neutrosophc vector spaces, neutrosophc groups, neutrosophc bgroups and neutrosophc N -groups, neutrosophc semgroups, neutrosophc bsemgroups, and neutrosophc N -semgroups, neutrosophc loops, nuetrosophc bloops, and neutrosophc N -loops, and so on. Mumtaz al et. al. ntroduced nuetrosophc LA -semgroups. Soft neutrosophc LA-semgroup has been ntroduced by Florentn Smarandache et.al. Molodtsov ntroduced the theory of soft set. Ths mathematcal tool s free from parameterzaton nadequacy, syndrome of fuzzy set theory, rough set theory, probablty theory and so on. Ths theory has been appled successfully n many felds such as smoothness of functons, game theory, operaton research, emann ntegraton, Perron ntegraton, and probablty. ecently soft set theory attaned much attenton of the researchers snce ts appearance and the work based on several operatons of soft set ntroduced n,9,0. Some propertes and algebra may be found 5. By n. Feng et al. ntroduced soft semrngs n means of level soft sets an adjustable approach to fuzzy soft set can be seen n 6. Some other concepts together wth fuzzy set and rough set were shown n 7,8. In ths paper we ntroduced soft nuetrosophc b-lasemgroup and soft neutrosophc N -LA-semgroup and the related strong or pure part of neutrosophy wth the notons of soft set theory. In the proceedng secton, we defne soft neutrosophc b-la-semgroup, soft neutrosophc strong b-la-semgroup, and some of ther propertes are dscussed. In the last secton soft neutrosophc N -LAsemgroup and ther correspondng strong theory have been constructed wth some of ther propertes. Neutrosophc N-LA-segroup

2 46 Neutrosophc Sets and Systems, Vol. 5, 04 Fundamental Concepts Defnton. Let ( BN( S),, ) be a nonempty set wth two bnary operatons and. ( BN( S),, ) s sad to BN( S) P P be a neutrosophc b-la-semgroup f where atleast one of ( P, ) or ( P, ) s a neutrosophc LA-semgroup and other s just an LA- semgroup. P and P are proper subsets of If both ( P, ) and ( P, ) n the above defnton are neutrosophc LA-semgroups then we call ( BN( S),, ) a strong neutrosophc b-la-semgroup. Defnton. Let ( BN( S) P P;:, ) be a neutrosophc b-la-semgroup. A proper subset ( T,, ) s sad to be a neutrosophc sub b-la-semgroup of BN( S ) f. T T T where T P T and T PT and (, ) ( T, ) s a neutrosoph-. At least one of T or c LA-semgroup. Defnton 3. Let ( BN( S) P P,, ) be a neutrosophc b-la-semgroup. A proper subset ( T,, ) s sad to be a neutrosophc strong sub b-la-semgroup of BN( S ) f. T T T where T P T and T PT and. ( T, ) and ( T, ) are neutrosophc strong LAsemgroups. Defnton 4. Let ( BN( S) P P,, ) be any neutrosophc b-la-semgroup. Let J be a proper subset of BN( S ) such that J J P and J J P are deals of P and P respectvely. Then J s called the neutrosophc bdeal of Defnton 5. Let ( BN( S),, ) be a strong neutrosophc b-la-semgroup where BN( S) P P wth ( P, ) and ( P, ) be any two neutrosophc LA-semgroups. Let J be a proper subset of BN( S ) where I II wth I I P and I I P are neutrosophc deals of the neutrosophc LA-semgroups P and P respectvely. Then I s called or defned as the strong neutrosophc bdeal of Defnton 6. Let { SN ( ),,..., } be a non-empty set wth N -bnary operatons defned on t. We call SN ( ) a neutrosophc N -LA-semgroup ( N a postve nteger) f the followng condtons are satsfed. ) S( N) S... SN where each S s a proper subset of SN ( ).e. S Sj or Sj S f j. ) ( S, ) s ether a neutrosophc LA-semgroup or an LA-semgroup for,,3,..., N. If all the N -LA-semgroups ( S, ) are neutrosophc LA-semgroups (.e. for,,3,..., N ) then we call SN ( ) to be a neutrosophc strong N -LA-semgroup. Defnton 7. Let S( N) { S S... S N,,,..., N } be a neutrosophc N -LA-semgroup. A proper subset P {P P...P,,,..., } of SN ( ) s sad N to be a neutrosophc sub N -LA-semgroup f P P S,,,..., N are sub LA-semgroups of S n whch atleast some of the sub LA-semgroups are neutrosophc sub LA-semgroups. Defnton 8. Let S( N) { S S... S N,,,..., N } be a neutrosophc strong N -LA-semgroup. A proper subset T {T T... T,,,..., } of SN ( ) s N sad to be a neutrosophc strong sub N -LA-semgroup f each ( T, ) s a neutrosophc sub LA-semgroup of ( S, ) for,,..., N where T S T. Defnton 9. Let S( N) { S S... S N,,,..., N } be a neutrosophc N -LA-semgroup. A proper subset P {P P... P N,,,..., N } of SN ( ) s sad to be a neutrosophc N -deal, f the followng condtons are true,. P s a neutrosophc sub N -LA-semgroup of N N

3 Neutrosophc Sets and Systems, Vol. 5, ach P S P,,,..., N s an deal of S. Defnton 0. Let S( N) { S S... S N,,,..., N } be a neutrosophc strong N -LA-semgroup. A proper subset J {J J...J,,,..., } where N Jt J St for t,,..., N s sad to be a neutrosophc strong N -deal of SN ( ) f the followng condtons are satsfed. ) ach t s a neutrosophc sub LA-semgroup of St, t,,..., N.e. It s a neutrosophc strong N- sub LA-semgroup of ) ach t s a two sded deal of S t for t,,..., N. Smlarly one can defne neutrosophc strong N -left deal or neutrosophc strong rght deal of A neutrosophc strong N -deal s one whch s both a neutrosophc strong N -left deal and N -rght deal of Soft Sets Throughout ths subsecton U refers to an ntal unverse, s a set of parameters, PU ( ) s the power set of U, and A, B. Molodtsov defned the soft set n the followng manner: Defnton. A par ( FA, ) s called a soft set over U where F s a mappng gven by F : A P( U ). In other words, a soft set over U s a parameterzed famly of subsets of the unverse U. For a A, F(a) may be consdered as the set of a -elements of the soft set ( FA, ), or as the set of a -approxmate elements of the soft set. xample. Suppose that U s the set of shops. s the set of parameters and each parameter s a word or sentence. Let hgh rent,normal rent,. n good condton,n bad condton Let us consder a soft set ( FA, ) whch descrbes the attractveness of shops that Mr.Z s takng on rent. Suppose that there are fve houses n the unverse N U { s, s, s3, s4, s 5} under consderaton, and that A { a, a, a } be the set of parameters where 3 a stands for the parameter 'hgh rent, a stands for the parameter 'normal rent, a 3 stands for the parameter 'n good condton. Suppose that F( a) { s, s 4}, F( a ) { s, s }, 5 F( a ) { s }. 3 3 The soft set ( FA, ) s an approxmated famly { F( a ),,,3} of subsets of the set U whch gves us a collecton of approxmate descrpton of an object. Then ( FA, ) s a soft set as a collecton of approxmatons over U, where F( a ) hgh rent { s, s }, F( a ) normal rent { s, s }, 5 F( a ) n good condton { s }. 3 3 Defnton. For two soft sets ( FA, ) and ( HB, ) over U, ( FA, ) s called a soft subset of ( HB, ) f. A B and. F( a) H( a ), for all x A. Ths relatonshp s denoted by ( A) ( H, B ). Smlarly ( FA, ) s called a soft superset of ( HB, ) f ( HB, ) s a soft subset of ( FA, ) whch s denoted by ( A) ( H, B ). Defnton 3. Two soft sets ( FA, ) and ( HB, ) over U are called soft equal f ( FA, ) s a soft subset of ( HB, ) and ( HB, ) s a soft subset of ( FA, ). Defnton 4. Let ( FA, ) and ( KB, ) be two soft sets over a common unverse U such that A B. Then ther restrcted ntersecton s denoted by ( A) ( K, B) ( H, C ) where ( HC, ) s defned as H( c) F( c) c) for all c C A B.

4 48 Neutrosophc Sets and Systems, Vol. 5, 04 Defnton 5. The extended ntersecton of two soft sets ( FA, ) and ( KB, ) over a common unverse U s the soft set ( HC, ), where C A B, and for all c C, Hc () s defned as F( c) f c A B, H( c) G( c) f c B A, F( c) G( c) f c A B. We wrte ( A) ( K, B) ( H, C ). Defnton 6. The restrcted unon of two soft sets ( FA, ) and ( KB, ) over a common unverse U s the soft set ( HC, ), where C A B, and for all c C, () H c F c G c for all c C. We wrte t as ( A) ( K, B) ( H, C). Hc s defned as ( ) ( ) ( ) Defnton 7. The extended unon of two soft sets ( FA, ) and ( KB, ) over a common unverse U s the soft set ( HC, ), where C A B, and for all c C, Hc () s defned as F( c) f c A B, H( c) G( c) f c B A, F( c) G( c) f c A B. We wrte ( A) ( K, B) ( H, C ). 3 Soft Neutrosophc B-LA-semgroup Defnton 8. Let BN( S ) be a neutrosophc b-la- semgroup and ( FA, ) be a soft set over Then ( FA, ) s called soft neutrosophc b-la-semgroup f and only f Fa ( ) s a neutrosophc sub b-la-semgroup of BN( S ) for all a A. xample. Let BN( S) { SI S I } be a neutrosophc b-la-semgroup where S I,,3,4, I, I,3 I,4I s a neutrosophc LA-semgroup wth the followng table. * 3 4 I I 3I 4I 4 3 I 4I I 3I 3 4 3I I 4I I I I 3I I I 3I I 4I I I 4I I 3I I 4I I 3I I 3I I 4I I 3I I 4I I 3I 4I I 3I I 4I I 3I I 4I I 3I I 4I I 3I I 4I S I,,3, I, I,3I be another neutrosophc b-la-semgroup wth the followng table. * 3 I I 3I I 3I 3I I 3I 3I I 3I 3I I 3I 3I 3I 3I 3I 3I I 3I 3I 3I 3I 3I 3I 3I I 3I 3I I 3I 3I A { a, a, a } be a set of parameters. Then clearly Let 3 ( FA, ) s a soft neutrosophc b-la-semgroup over BN( S ), where F( a ) {, I} {,3,3 I}, F( a ) {, I} {,3, I,3 I}, F( a ) {4,4 I} { I,3 I}. 3 K be two soft neu- Proposton. Let ( FA, ) and (, D) trosophc b-la-semgroups over Then

5 Neutrosophc Sets and Systems, Vol. 5, Ther extended ntersecton ( A) ( K,D) s soft neutrosophc b-la-semgroup over. Ther restrcted ntersecton ( A) ( K,D) s soft neutrosophc b-la-semgroup over 3. Ther AND operaton ( A) ( K,D) neutrosophc b-la-semgroup over K be two soft neutro- emark. Let ( FA, ) and (, D) sophc b-la-semgroups over Then s soft. Ther extended unon ( A) ( K,D) soft neutrosophc b-la-semgroup over. Ther restrcted unon ( A) ( K,D) soft neutrosophc b-la-semgroup over 3. Ther O operaton ( A) ( K,D) soft neutrosophc b-la-semgroup over One can easly proved (),(), and (3) by the help of examples. Defnton 9. Let ( FA, ) and (K,D) be two soft neutrosophc b-la-semgroups over Then (K,D) s called soft neutrosophc sub b-la-semgroup of ( FA, ), f. D A.. K( a ) s a neutrosophc sub b-la-semgroup of Fa ( ) for all a A. xample 3. Let ( FA, ) be a soft neutrosophc b-la- semgroup over BN( S ) n xample (). Then clearly ( KD, ) s a soft neutrosophc sub b-la-semgroup of ( FA, ) over BN( S ), where K( a ) {, I} {3,3 I}, K( a ) {, I} {, I}. Theorem. Let A be a soft neutrosophc b-lasemgroup over ( ) BN S and j j H, B : j J be a non-empty famly of soft neutrosophc sub b-lasemgroups of A. Then ) H Bj jj s a soft neutrosophc sub b-lasemgroup of A. ) H Bj jj s a soft neutrosophc sub b-lasemgroup of A. 3) H Bj jj s a soft neutrosophc sub b-lasemgroup of A f j k k J. B B for all FA be a soft set over a neutrosoph- FA s called soft Fa s Defnton 0. Let (, ) c b-la-semgroup Then (, ) neutrosophc bdeal over BN( S ) f and only f ( ) a neutrosophc bdeal of BN( S ), for all a A. xample 4. Let BN( S) { SI S I } be a neutrosophc b-la-semgroup, where S I,,3, I, I,3I be another neutrosophc b-la-semgroup wth the followng table. * 3 I I 3I I 3I 3I I 3I 3I I 3I 3I I 3I 3I 3I 3I 3I 3I I 3I 3I 3I 3I 3I 3I 3I I 3I 3I I 3I 3I

6 50 Neutrosophc Sets and Systems, Vol. 5, 04,,3,,,3 be another neutrosophc LA-semgroup wth the followng table. And S I I I I. 3 I I 3I 3 3 3I 3I I I I I 3 I I I I 3I 3I I 3I 3I I I I I I I I I 3I I I I I I I Let A { a, a} be a set of parameters. Then, a soft neutrosophc bdeal over BN( S ), where F a,i,3,3 I {, I}, F a,3, I,3 I {,3, I,3I}. F A s Proposton. very soft neutrosophc bdeal over a neutrosophc b-la-semgroup s trvally a soft neutrosophc b-la-semgroup but the conver true n general. One can easly see the converse by the help of example. Proposton 3. Let A and KD, be two soft neutrosophc bdeals over Then ) Ther restrcted unon F A, ( K, D) a soft neutrosophc bdeal over ) Ther restrcted ntersecton A K, D soft neutrosophc bdeal over 3) Ther extended unon A K, D s a a soft neutrosophc bdeal over 4) Ther extended ntersecton A K, D soft neutrosophc bdeal over s a Proposton 4. Let A and ( KD, ) be two soft neutrosophc bdeals over Then. Ther O operaton A K, D a soft neutrosophc bdeal over. Ther AND operaton A K, D s a soft neutrosophc bdeal over Defnton. Let A and (K,D) be two soft neutrosophc b- LA-semgroups over Then ( KD, ) s called soft neutrosophc bdeal of A, f ) B A, and ) Ka ( ) s a neutrosophc bdeal of ( ) a A. Fa, for all xample 5. Let A be a soft neutrosophc b-lasemgroup over BN( S ) n xample (*). Then ( KD, ) A over BN( S ), s a soft neutrosophc bdeal of where K a I,3 I {, I}, K a,3, I,3 I { I,3I}. Theorem. A soft neutrosophc bdeal of a soft neutrosophc b-la-semgroup over a neutrosophc b- LA_semgroup s trvally a soft neutosophc sub b-lasemgroup but the converse true n general. G B are soft neutro- Proposton 5. If F, A and, sophc bdeals of soft neutrosophc b-la-semgroups A and GB, over neutrosophc b-lasemgroups F ', A ' G ', B ' NS and N T respectvely. Then s a soft neutrosophc bdeal of soft neutrosophc b-la-semgroup A G, B N S N T. over Theorem 3. Let A be a soft neutrosophc b-lasemgroup over ( ) BN S and j j H, B : j J be a non-empty famly of soft neutrosophc bdeals of A. Then

7 Neutrosophc Sets and Systems, Vol. 5, 04 5 s a soft neutrosophcb deal of ) H Bj jj A. ) H Bj s a soft neutrosophc bdeal of j J A. 3) H Bj jj s a soft neutrosophc bdeal of A. 4) H Bj s a soft neutrosophc bdeal of j J A. 4 Soft Neutrosophc Storng B-LA-semgroup Defnton. Let BN( S ) be a neutrosophc b-lasemgroup and ( FA, ) be a soft set over Then ( FA, ) s called soft neutrosophc strong b-lasemgroup f and only f Fa ( ) s a neutrosophc strong sub b-la-semgroup for all a A. xample 6. Let BN( S ) be a neutrosophc b-lasemgroup n xample (). Let A { a, a} parameters. Then (, ) LA-semgroup over BN( S ),where be a set of FA s a soft neutrosophc strong b- F( a ) { I, I,3 I,4 I} { I,3 I}, F( a ) { I, I,3 I,4 I} { I,3 I}. K be two soft neu- Proposton 6. Let ( FA, ) and (, D) trosophc strong b-la-semgroups over Then. Ther extended ntersecton ( A) ( K,D) s soft neutrosophc strong b-la-semgroup over. Ther restrcted ntersecton ( A) ( K,D) s soft neutrosophc strong b-la-semgroup over 3. Ther AND operaton ( A) ( K,D) s soft neutrosophc strong b-la-semgroup over K be two soft neutro- emark. Let ( FA, ) and (, D) sophc strong b-la-semgroups over Then. Ther extended unon ( A) ( K,D) soft neutrosophc strong b-la-semgroup over. Ther restrcted unon ( A) ( K,D) soft neutrosophc strong b-la-semgroup over 3. Ther O operaton ( A) ( K,D) soft neutrosophc strong b-la-semgroup over One can easly proved (),(), and (3) by the help of examples. Defnton 3. Let ( FA, ) and (K,D) be two soft neutrosophc strong b-la-semgroups over Then (K,D) s called soft neutrosophc strong sub b-lasemgroup of ( FA, ), f. B A.. K( a ) s a neutrosophc strong sub b-lasemgroup of Fa ( ) for all a A. Theorem 4. Let A be a soft neutrosophc strong b- LA-semgroup over ( ) BN S and j j H, B : j J be a non-empty famly of soft neutrosophc strong sub b- LA-semgroups of A. Then s a soft neutrosophc strong. H Bj jj sub b-la-semgroup of A. s a soft neutrosophc strong sub. H Bj jj b-la-semgroup of A. s a soft neutrosophc strong 3. H Bj jj

8 5 Neutrosophc Sets and Systems, Vol. 5, 04 sub b-la-semgroup of A f B B for all k j k J. Defnton 4. Let ( FA, ) be a soft set over a neutrosophc b-la-semgroup Then ( FA, ) s called soft neutrosophc strong bdeal over BN( S ) f and only f Fa ( ) s a neutrosophc strong bdeal of BN( S ), for all a A. xample 7. Let BN( S ) be a neutrosophc b-la- A { a, a } be a set of semgroup n xample (*). Let parameters. Then clearly (A) s a soft neutrosophc strong bdeal over BN( S ), where F a I,3 I {I, I,3 I}, F a I,3 I {I,3I}. Theorem 5. very soft neutrosophc strong bdeal over BN( S ) s a soft neutrosophc bdeal but the converse s not true. We can easly see the converse by the help of example. Proposton 7. very soft neutrosophc strong bdeal over a neutrosophc b-la-semgroup s trvally a soft neutrosophc strong b-la-semgroup but the converse s not true n general. Proposton 8. very soft neutrosophc strong bdeal over a neutrosophc b-la-semgroup s trvally a soft neutrosophc b-la-semgroup but the converse true n general. One can easly see the converse by the help of example. Proposton 9. Let A and KD, be two soft neutrosophc strong bdeals over Then. Ther restrcted unon F A, ( K, D) a soft neutrosophc strong bdeal over. Ther restrcted ntersecton A K, D s a soft neutrosophc strong bdeal over 3. Ther extended unon A K, D a soft neutrosophc strong bdeal over 4. Ther extended ntersecton A K, D s a soft neutrosophc strong bdeal over 5. Ther O operaton A K, D a soft neutrosophc bdeal over 6. Ther AND operaton A K, D s a soft neutrosophc bdeal over Defnton 5. Let A and (K,D) be two soft neutrosophc strong b- LA-semgroups over Then ( KD, ) s called soft neutrosophc strong bdeal of A, f. D A, and Ka s a neutrosophc strong bdeal of A.. ( ) Fa ( ), for all a Theorem 6. A soft neutrosophc strong bdeal of a soft neutrosophc strong b-la-semgroup over a neutrosophc b-la_semgroup s trvally a soft neutosophc strong sub b-la-semgroup but the converse true n general. Proposton 0. If F, A and G, B are soft neutrosophc strong bdeals of soft neutrosophc b-lasemgroups A and GB, over neutrosophc b- LA-semgroups F ', A ' G ', B ' NS and N T respectvely. Then s a soft neutrosophc strong bdeal of soft neutrosophc b-la-semgroup A G, B over N S N T. Theorem 7. Let, LA-semgroup over ( ) F A be a soft neutrosophc strong b- BN S and j j H, B : j J be a non-empty famly of soft neutrosophc strong bdeals of A. Then s a soft neutrosophc strong b. H Bj jj

9 Neutrosophc Sets and Systems, Vol. 5, deal of A.. H Bj s a soft neutrosophc strong j J bdeal of A. s a soft neutrosophc strong 3. H Bj jj bdeal of A. 4. H Bj s a soft neutrosophc strong j J bdeal of A. 5 Soft Neutrosophc N-LA-semgroup Defnton 6. Let { SN ( ),,,..., N} be a neutrosophc N-LA-semgroup and ( FA, ) be a soft set over Then ( FA, ) s called soft neutrosophc N-LA-semgroup f and only f Fa ( ) s a neutrosophc sub N-LA- semgroup of SN ( ) for all a A. xample 8. Let S(N) {S SS 3,,, 3} be a neutrosophc 3-LA-semgroup where S,,3,4, I, I,3 I,4I s a neutrosophc LA- semgroup wth the followng table. * 3 4 I I 3I 4I 4 3 I 4I I 3I 3 4 3I I 4I I I I 3I I I 3I I 4I I I 4I I 3I I 4I I 3I I 3I I 4I I 3I I 4I I 3I 4I I 3I I 4I I 3I I 4I I 3I I 4I I 3I I 4I S,,3, I, I,3I be another neutrosophc b-la- semgroup wth the followng table. * 3 I I 3I I 3I 3I I 3I 3I I 3I 3I I 3I 3I 3I 3I 3I 3I I 3I 3I 3I 3I 3I 3I 3I I 3I 3I I 3I 3I And S I I I 3,,3,,,3 s another neutrosophc LA- semgroup wth the followng table.. 3 I I 3I 3 3 3I 3I I I I I 3 I I I I 3I 3I I 3I 3I I I I I I I I I 3I I I I I I I A { a, a, a } be a set of parameters. Then clearly Let 3 ( FA, ) s a soft neutrosophc 3-LA-semgroup over SN ( ), where F( a ) {, I} {,3,3 I} {,I}, F( a ) {, I} {,3, I,3 I} {,3,I,3I}, F( a ) {4,4 I} { I,3 I} {I,3I}. 3 K be two soft neu- SN. Then Proposton. Let ( FA, ) and (, D) trosophc N-LA-semgroups over ( )

10 54 Neutrosophc Sets and Systems, Vol. 5, 04. Ther extended ntersecton ( A) ( K,D) s soft neutrosophc N-LA-semgroup over. Ther restrcted ntersecton ( A) ( K,D) s soft neutrosophc N-LA-semgroup over 3. Ther AND operaton ( A) ( K,D) neutrosophc N-LA-semgroup over K be two soft neutro- SN. Then emark 3. Let ( FA, ) and (, D) sophc N-LA-semgroups over ( ) s soft. Ther extended unon ( A) ( K,D) soft neutrosophc N-LA-semgroup over. Ther restrcted unon ( A) ( K,D) soft neutrosophc N-LA-semgroup over 3. Ther O operaton ( A) ( K,D) soft neutrosophc N-LA-semgroup over One can easly proved (),(), and (3) by the help of examples. Defnton 7. Let ( FA, ) and (K,D) be two soft neutrosophc N-LA-semgroups over Then (K,D) s called soft neutrosophc sub N-LA-semgroup of ( FA, ), f. D A.. K( a ) s a neutrosophc sub N-LA-semgroup of Fa ( ) for all a A. Theorem 8. Let A be a soft neutrosophc N-LAsemgroup over ( ) SN and j j H, B : j J be a non-empty famly of soft neutrosophc sub N-LAsemgroups of A. Then s a soft neutrosophc sub N-. H Bj jj LA-semgroup of A. s a soft neutrosophc sub N-. H Bj jj LA-semgroup of A. s a soft neutrosophc sub N- 3. H Bj jj LA-semgroup of A f j k all k J. B B for FA be a soft set over a neutrosoph- FA s called soft Fa s Defnton 8. Let (, ) c N-LA-semgroup Then (, ) neutrosophc N-deal over SN ( ) f and only f ( ) a neutrosophc N-deal of SN ( ) for all a A. xample 9. Consder xample (***).Let A { a, a} be a set of parameters. Then ( FA, ) s a soft neutrosophc 3- deal over SN ( ), where F( a ) {, I} {3,3 I} {, I}, F( a ) {, I} { I,3 I} {,3,3I}. Proposton. very soft neutrosophc N-deal over a neutrosophc N-LA-semgroup s trvally a soft neutrosophc N-LA-semgroup but the converse true n general. One can easly see the converse by the help of example. Proposton 3. Let A and KD, be two soft neutrosophc N-deals over Then. Ther restrcted unon F A, ( K, D) a soft neutrosophc N-deal over. Ther restrcted ntersecton A K, D s a soft neutrosophc N-deal over 3. Ther extended unon A K, D a soft neutrosophc N-deal over

11 Neutrosophc Sets and Systems, Vol. 5, Ther extended ntersecton A K, D s a soft neutrosophc N-deal over Proposton 5. Let A and ( KD, ) be two soft neutrosophc N-deals over Then. Ther O operaton A K, D s a not soft neutrosophc N-deal over. Ther AND operaton A K, D s a soft neutrosophc N-deal over Defnton 9. Let A and (K,D) be two soft neutrosophc N- LA-semgroups over Then ( KD, ) s called soft neutrosophc N-deal of A, f. B A, and. Ka ( ) s a neutrosophc N-deal of ( ) a A. Fa for all Theorem 8. A soft neutrosophc N-deal of a soft neutrosophc N-LA-semgroup over a neutrosophc N-LAsemgroup s trvally a soft neutosophc sub N-LAsemgroup but the converse true n general. Proposton 6. If F, A and G, B are soft neutrosophc N-deals of soft neutrosophc N-LA-semgroups A and GB, over neutrosophc N-LAsemgroups F ', A ' G ', B ' NS and N T respectvely. Then s a soft neutrosophc N-deal of soft neutrosophc N-LA-semgroup A G, B N S N T. over Theorem 9. Let A be a soft neutrosophc N-LAsemgroup over ( ) SN and j j H, B : j J be a non-empty famly of soft neutrosophc N-deals of A. Then s a soft neutrosophc N-deal. H Bj jj of A.. H Bj s a soft neutrosophc N-deal of j J A. 3. H Bj jj s a soft neutrosophc N-deal of A. 4. H Bj s a soft neutrosophc N-deal of j J A. 6 Soft Neutrosophc Strong N-LA-semgroup Defnton 30. Let { SN ( ),,,..., N} be a neutrosophc N-LA-semgroup and ( FA, ) be a soft set over SN ( ). Then ( FA, ) s called soft neutrosophc strong N-LA-semgroup f and only f Fa ( ) s a neutrosophc strong sub N-LA-semgroup of SN ( ) for all a A. xample 0. Let S(N) {S SS 3,,, 3} be a neutrosophc 3-LA-semgroup n xample 8. Let A { a, a, a } be a set of parameters. Then clearly 3 ( FA, ) s a soft neutrosophc strong 3-LA-semgroup over SN ( ), where F( a ) { I} { I,3 I} {I}, F( a ) { I} { I,3 I} {I,3I}, SN s a neutrosophc strong N-LA- FA s also a soft neutrosophc strong Theorem 0. If ( ) semgroup, then (, ) N-LA-semgroup over K be two soft neu- SN. Then Proposton 7. Let ( FA, ) and (, D) trosophc strong N-LA-semgroups over ( ). Ther extended ntersecton ( A) ( K,D) s soft neutrosophc strong N-LA-semgroup over

12 56 Neutrosophc Sets and Systems, Vol. 5, 04. Ther restrcted ntersecton ( A) ( K,D) s soft neutrosophc strong N-LA-semgroup over 3. Ther AND operaton ( A) ( K,D) neutrosophc strong N-LA-semgroup over K be two soft neutro- SN. Then emark 4. Let ( FA, ) and (, D) sophc strong N-LA-semgroups over ( ) s soft. Ther extended unon ( A) ( K,D) soft neutrosophc strong N-LA-semgroup over. Ther restrcted unon ( A) ( K,D) soft neutrosophc strong N-LA-semgroup over 3. Ther O operaton ( A) ( K,D) soft neutrosophc strong N-LA-semgroup over One can easly proved (),(), and (3) by the help of examples. Defnton 3. Let ( FA, ) and (K,D) be two soft neutrosophc strong N-LA-semgroups over Then (K,D) s called soft neutrosophc strong sub N-LAsemgroup of ( FA, ), f 3. D A. 4. K( a ) s a neutrosophc strong sub N-LAsemgroup of Fa ( ) for all a A. Theorem. Let A be a soft neutrosophc strong N- LA-semgroup over ( ) SN and j j H, B : j J be a non-empty famly of soft neutrosophc strong sub N-LAsemgroups of A. Then s a soft neutrosophc strong. H Bj jj sub N-LA-semgroup of A. s a soft neutrosophc strong sub. H Bj jj N-LA-semgroup of A. s a soft neutrosophc strong sub 3. H Bj jj N-LA-semgroup of A f j k for all k J. B B FA be a soft set over a neutrosoph- FA s called soft SN f and only f SN for all Defnton 3. Let (, ) c N-LA-semgroup Then (, ) neutrosophc strong N-deal over ( ) Fa ( ) s a neutrosophc strong N-deal of ( ) a A. Proposton 8. very soft neutrosophc strong N-deal over a neutrosophc N-LA-semgroup s trvally a soft neutrosophc strong N-LA-semgroup but the converse s not true n general. One can easly see the converse by the help of example. Proposton 9. Let A and KD, be two soft neutrosophc strong N-deals over Then. Ther restrcted unon F A, ( K, D) a soft neutrosophc strong N-deal over. Ther restrcted ntersecton A K, D s a soft neutrosophc N-deal over 3. Ther extended unon A K, D s also a not soft neutrosophc strong N-deal over 4. Ther extended ntersecton A K, D s a soft neutrosophc strong N-deal over 5. Ther O operaton A K, D s a not soft neutrosophc strong N-deal over 6. Ther AND operaton A K, D s a

13 Neutrosophc Sets and Systems, Vol. 5, soft neutrosophc strong N-deal over Defnton 33. Let A and (K,D) be two soft neutrosophc strong N- LA-semgroups over Then ( KD, ) s called soft neutrosophc strong N-deal of A, f. B A, and. Ka ( ) s a neutrosophc strong N-deal of Fa ( ) for all a A. Theorem. A soft neutrosophc strong N-deal of a soft neutrosophc strong N-LA-semgroup over a neutrosophc N-LA_semgroup s trvally a soft neutosophc strong sub N-LA-semgroup but the converse true n general. Theorem 3. A soft neutrosophc strong N-deal of a soft neutrosophc strong N-LA-semgroup over a neutrosophc N-LA_semgroup s trvally a soft neutosophc strong N- deal but the converse true n general. Proposton 0. If F, A and G, B are soft neutrosophc strong N-deals of soft neutrosophc strong N- LA-semgroups A and GB, over neutrosophc N-LA-semgroups F ', A ' G ', B ' NS and N T respectvely. Then s a soft neutrosophc strong N-deal of soft neutrosophc strong N-LA-semgroup A G, B N S N T. over Theorem 4. Let A be a soft neutrosophc strong N-LA-semgroup over ( ) SN and j j H, B : j J be a non-empty famly of soft neutrosophc strong N-deals of A. Then s a soft neutrosophc strong N-. H Bj jj deal of A.. H Bj s a soft neutrosophc strong N- j J deal of A. s a soft neutrosophc strong N- 3. H Bj jj deal of A. 4. H Bj Concluson s a soft neutrosophc strong N- j J deal of A. Ths paper we extend soft neutrosophc bsemgroup, soft neutrosophc N -semgroup to soft neutrosophc b-la-semgroup, and soft neutrosophc N -LAsemgroup. Ther related propertes and results are explaned wth many llustratve examples. The notons related wth strong part of neutrosophy also establshed. eferences ) H. Aktas, N. Cagman, Soft sets and soft groups, Inf. Sc. 77 (007) ) K. Atanassov, Intutonstc fuzzy sets, Fuzzy Sets Syst. 64()(986) ) M. Shabr, M. Al, M. Naz, F. Smarandache, Soft neutrosophc group, Neutrosophc Sets and Systems. (03) 5-. 4) M. Al, F. Smarandache,M. Shabr, M. Naz, Soft neutrosophc Bgroup, and Soft Neutrosophc N-group, Neutrosophc Sets and Systems. (04) ) M. I. Al, F. Feng, X. Lu, W. K. Mn, M. Shabr, On some new operatonsn soft set theory. Comp. Math. Appl., 57(009), ) S. Broum, F. Smarandache, Intutonstc Neutrosophc Soft Set, J. Inf. & Comput. Sc. 8(03) ) D. Chen,.C.C. Tsang, D.S. Yeung, X. Wang, The parameterzaton reducton of soft sets and ts applcatons,comput. Math. Appl. 49(005) ) F. Feng, M. I. Al, M. Shabr, Soft relatons appled to semgroups, Flomat 7(7)(03) ) M.B. Gorzalzany, A method of nference n approxmate reasonng based on nterval-valued fuzzy sets, Fuzzy Sets Syst. (987) -7.

14 58 Neutrosophc Sets and Systems, Vol. 5, 04 0) W. B. V. Kandasamy, F. Smarandache, Basc Neutrosophc Algebrac Structures and ther Applcatons to Fuzzy and Neutrosophc Models, Hexs (004). ) W. B. V. Kandasamy, F. Smarandache, N-Algebrac Structures and S-N-Algebrac Structures, Hexs Phoenx (006). ) W. B. V. Kandasamy, F. Smarandache, Some Neutrosophc Algebrac Structures and Neutrosophc N- Algebrac Structures, Hexs (006). 3) P.K. Maj,. Bswas and A.. oy, Soft set theory, Comput. Math. Appl. 45(003) ) P. K. Maj, Neutrosophc Soft Sets, Ann. Fuzzy Math. Inf. 5()(03) ) D. Molodtsov, Soft set theory frst results, Comput. Math. Appl. 37(999) ) Z. Pawlak, ough sets, Int. J. Inf. Comp. Sc. (98) ) F. Smarandache, A Unfyng Feld n Logcs. Neutrosophy: Neutrosophc Probablty, Set and Logc. ehoboth: Amercan esearch Press (999). 8) F. Smarandache, M. Al, M. Shabr, Soft Neutrosophc Algebrac Structures and Ther Generalzaton. ducaton Publshng, Oho 43, USA (04). 9) L.A. Zadeh, Fuzzy sets, Inf. Cont. 8(965) eceved: July 5, 04. Accepted: August 0, 04.