Neutrosophc Sets Systems, Vol. 4, 04 9 Neutrosophc B-LA-Semgroup Neutrosophc N-LA- Semgroup Mumtaz Al *, Florentn Smarache, Muhammad Shabr 3 Munazza Naz 4,3 Department of Mathematcs, Quad--Azam Unversty, Islamabad, 44000,Pakstan. E-mal: mumtazal770@yahoo.com, mshabrbhatt@yahoo.co.uk Unversty of New Mexco, 705 Gurley Ave., Gallup, New Mexco 8730, USA E-mal: fsmarache@gmal.com 4 Department of Mathematcal Scences, Fatma Jnnah Women Unversty, The Mall, Rawalpnd, 46000, Pakstan. E-mal: munazzanaz@yahoo.com Abstract. In ths paper we defne neutrosophc b-lasemgroup neutrosophc N-LA-semgroup. Infact ths paper s an extenson of our prevous paper neutrosophc left almost semgroup shortly neutrosophc LAsemgroup. We also extend the neutrosophc deal to neutrosophc bdeal neutrosophc N-deal. We also fnd some new type of neutrosophc deal whch s related to the strong or pure part of neutrosophy. We have gven suffcent amount of examples to llustrate the theory of neutrosophc b-la-semgroup, neutrosophc N-LAsemgroup dsplay many propertes of them ths paper. Keywords: Neutrosophc LA-semgroup, neutrosophc deal, neutrosophc b-la-semgroup, neutrosophc bdeal, neutrosophc N-LA-semgroup, neutrosophc N-deal. Introducton Neutrosophy s a new branch of phlosophy whch studes the orgn features of neutraltes n the nature. Florentn Smarache n 980 frstly ntroduced the concept of neutrosophc logc where each proposton n neutrosophc logc s approxmated to have the percentage of truth n a subset T, the percentage of ndetermnacy n a subset I, the percentage of falsty n a subset F so that ths neutrosophc logc s called an extenson of fuzzy logc. In fact neutrosophc set s the generalzaton of classcal sets, con- n- ventonal fuzzy set, ntutonstc fuzzy set terval valued fuzzy set 3. Ths mathematcal tool s used to hle problems lke mprecse, ndetermnacy nconsstent data etc. By utlzng neutrosophc theory, Vasantha Kasamy Florentn Smarache dg out neutrosophc algebrac structures n. Some of them Kazm M. Naseeruddn 3 n 97. Ths structure s bascally a mdway structure between a groupod a commutatve semgroup. Ths structure s also termed as Able-Grassmann s groupod abbrevated as AG -groupod 6. Ths s a non assocatve non commutatve algebrac structure whch closely resemble to commutatve semgroup. The generalzaton of semgroup theory s an LA-semgroup ths structure has wde applcatons n collaboraton wth semgroup. We have tred to develop the deal theory of LAsemgroups n a logcal manner. Frstly, prelmnares basc concepts are gven for neutrosophc LA-semgroup. Then we presented the newly defned notons results n neutrosophc b-la-semgroups neutrosophc N- LA-semgroups. Varous types of neutrosophc bdeals neutrosophc N-deal are defned elaborated wth the help of examples. are neutrosophc felds, neutrosophc vector spaces, neutrosophc groups, neutrosophc bgroups, neutrosophc N- Prelmnares groups, neutrosophc semgroups, neutrosophc bsemgroups, neutrosophc N-semgroup, neutrosophc loops, S I a bi : a, b S. The neutrosophc LA- Defnton. Let S, be an LA-semgroup let neutrosophc bloops, neutrosophc N-loop, neutrosophc semgroup s generated by S I under denoted as groupods, neutrosophc bgroupods so on. N S S I,, where I s called the A left almost semgroup abbrevated as LA-semgroup s neutrosophc element wth property I I. For an an algebrac structure whch was ntroduced by M.A. nteger n, n I ni are neutrosophc elements Mumtaz Al, Florentn Smarache, Muhammad Shabr Munazza Naz, Neutrosophc B-LA-Semgroup Neutrosophc N-LA-Semgroup
0 Neutrosophc Sets Systems, Vol. 4, 04 I 0. I 0., the nverse of I s not defned hence does not exst. Smlarly we can defne neutrosophc RA-semgroup on the same lnes. Defnton. Let NH be a proper subset of NS N H operaton of NS. N S be a neutrosophc LA-semgroup. Then s called a neutrosophc sub LA-semgroup f N H tself s a neutrosophc LA-semgroup under the Defnton 3. A neutrosophc sub LA-semgroup NH s called strong neutrosophc sub LA-semgroup or pure neutrosophc sub LA-semgroup f all the elements of N H are neutrosophc elements. Defnton 4. Let NK be a subset of NS. Then called Left (rght) neutrosophc deal of N S N K N K { N, K N S N K }. If N S be a neutrosophc LA-semgroup N K s N S f N K s both left rght neutrosophc deal, then N K s called a two sded neutrosophc deal or smply a neutrosophc deal. Defnton 5. A neutrosophc deal NK s called strong neutrosophc deal or pure neutrosophc deal f all of ts elements are neutrosophc elements. 3 Neutrosophc B-LA-Semgroup Defnton 6. Let ( BN( S),, ) be a non-empty set wth two bnary operatons. ( 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 other s just an LA- semgroup. P P are proper subsets of BN( S ). Smlarly we can defne neutrosophc b-ra-semgroup on the same lnes. Theorem. All neutrosophc b-la-semgroups contans the correspondng b-la-semgroups. Example. 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 3 4 3 4I I 3I I 4 3 4 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 3 3 3 3I 3I 3I 3 3 3 3I 3I 3I 3 3 3 I 3I 3I I 3I 3I 3I 3I 3I 3I I 3I 3I 3I 3I 3I 3I 3I I 3I 3I I 3I 3I Defnton 7. 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 T PT (, ) ( T, ) s a neutrosoph-. At least one of T or c LA-semgroup. Example : BN( S ) be a neutrosophc b-lasemgroup n Example. Then P {, I} {3,3 I} Q {, I} {, I} are neutrosophc sub b-la-semgroups of BN( S ). Mumtaz Al, Florentn Smarache, Muhammad Shabr Munazza Naz, Neutrosophc B-LA-Semgroup Neutrosophc N-LA-Semgroup
Neutrosophc Sets Systems, Vol. 4, 04 Theorem. Let BN S be a neutrosophc b-lasemgroup NH be a proper subset of BN S. Then NH s a neutrosophc sub b-la-semgroup of BN S f N H. N H N H. Defnton 8. 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 J J P are deals of P P respectvely. Then J s called the neutrosophc bdeal of BN( S ). Example 3. 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 3 3 3 3I 3I 3I 3 3 3 3I 3I 3I 3 3 3 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 I,,3,,,3 be 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 Then Q I I P,I,3,3 I {, I},,3,,3 {,3, I,3I} are neutrosophc bdeals of BN( S ). Proposton. Every neutrosophc bdeal of a neutrosophc b-la-semgroup s trvally a Neutrosophc sub b-la-semgroup but the conver s not true n general. One can easly see the converse by the help of example. 3 Neutrosophc Strong B-LA-Semgroup Defnton 9: If both ( P, ) ( P, ) n the Defnton 6. are neutrosophc strong LAsemgroups then we call ( BN( S),, ) s a neutrosophc strong b-la-semgroup. Defnton 0. 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 T PT. ( T, ) ( T, ) are neutrosophc strong LA-semgroups. Example 4. Let BN( S ) be a neutrosophc b- LA-semgroup n Example 3. Then P I,3 I { I}, Q I,3 I{I,3I} are neutrosophc strong sub b- LA-semgroup of BN( S ). Theorem 4: Every neutrosophc strong sub b- LA-semgroup s a neutrosophc sub b-lasemgroup. Defnton. Let ( BN( S),, ) be a strong neutrosophc b-la-semgroup where BN( S) P P wth ( P, ) ( P, ) be any two neutrosophc LAsemgroups. Let J be a proper subset of BN( S ) where I I I wth I I P I I P are neutrosophc deals of the neutrosophc LA-semgroups P P respectvely. Then I s called or defned as the Mumtaz Al, Florentn Smarache, Muhammad Shabr Munazza Naz, Neutrosophc B-LA-Semgroup Neutrosophc N-LA-Semgroup
Neutrosophc Sets Systems, Vol. 4, 04 neutrosophc strong bdeal of BN( S ). Theorem 5: Every neutrosophc strong bdeal s trvally a neutrosophc sub b-la-semgroup. Theorem 6: Every neutrosophc strong bdeal s a neutrosophc strong sub b-la-semgroup. Theorem 7: Every neutrosophc strong bdeal s a neutrosophc bdeal. Example 5. Let BN( S ) be a neutrosophc b-la semgroup n Example (*).Then P I,3 I { I}, Q I,3 I{I,3I} are neutrosophc strong bdeal of BN( S ). 4 Neutrosophc N-LA-Semgroup Defnton. 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. Example 6. 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 LAsemgroup wth the followng table. * 3 4 I I 3I 4I 4 3 I 4I I 3I 3 4 3I I 4I I 3 4 3 4I I 3I I 4 3 4 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-lasemgroup wth the followng table. * 3 I I 3I 3 3 3 3I 3I 3I 3 3 3 3I 3I 3I 3 3 3 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 LAsemgroup 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 Theorem 8 All neutrosophc N-LA-semgroups contans the correspondng N-LA-semgroups. Defnton 3. 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. Example 7: Let S(N) {S SS 3,,, 3} be a neutrosophc 3-LA-semgroup n above Example 6. Then clearly P {, I} {,3,3 I} {, I}, Q {, I} {,3, I,3 I} {,3, I,3I}, N Mumtaz Al, Florentn Smarache, Muhammad Shabr Munazza Naz, Neutrosophc B-LA-Semgroup Neutrosophc N-LA-Semgroup
Neutrosophc Sets Systems, Vol. 4, 04 3 R {4,4 I} { I,3 I} {I,3I} are neutrosophc sub 3-LA-semgroups of S(N). Theorem 9. Let N( S ) be a neutrosophc N-LAsemgroup NH be a proper subset of N( S ). Then NH s a neutrosophc sub N-LA-semgroup of N( S ) f N H. N H N H. Defnton 4. 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 SN ( ).. Each P S P,,,..., N s an deal of S. Example 8. Consder Example 6. Then I {, I} {3,3 I} {, I}, I {, I} { I,3 I} {,3,3I} are neutrosophc 3- deals of SN ( ). Theorem 0: Every neutrosophc N-deal s trvally a neutrosophc sub N-LA-semgroup but the converse s not true n general. One can easly see the converse by the help of example. 5 Neutrosophc Strong N-LA-Semgroup Defnton 5: If all the N -LA-semgroups ( S, ) n Defnton ( ) are neutrosophc strong LA-semgroups (.e. for,,3,..., N ) then we call SN ( ) to be a neutrosophc strong N -LA-semgroup. Defnton 6. 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 strong sub LA-semgroup of ( S, ) for,,..., N where T S T. Theorem : Every neutrosophc strong sub N-LAsemgroup s a neutrosophc sub N-LA-semgroup. N Defnton 7. 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. ) Each t s a neutrosophc sub LA-semgroup of St, t,,..., N.e. It s a neutrosophc strong N- sub LA-semgroup of SN ( ). ) Each 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 SN ( ). A neutrosophc strong N -deal s one whch s both a neutrosophc strong N -left deal N -rght deal of SN ( ). Theorem : Every neutrosophc strong Ndeal s trvally a neutrosophc sub N-LA-semgroup. Theorem 3: Every neutrosophc strong N-deal s a neutrosophc strong sub N-LA-semgroup. Theorem 4: Every neutrosophc strong N-deal s a N- deal. Concluson In ths paper we extend neutrosophc LA-semgroup to neutrosophc b-la-semgroup neutrosophc N-LAsemgroup. The neutrosophc deal theory of neutrosophc LA-semgroup s extend to neutrosophc bdeal neutrosophc N-deal. Some new type of neutrosophc deals are dscovered whch s strongly neutrosophc or purely neutrosophc. Related examples are gven to llustrate neutrosophc b-la-semgroup, neutrosophc N-LAsemgroup many theorems propertes are dscussed. References [] M. Al, M.Shabr, M. Naz F. Smarache, Neutrosophc Left Almost Semgroup, Neutrosophc Sets Systems, 3 (04), 8-8. [] Florentn Smarache, A Unfyng Feld n Logcs. Neutrosophy, Neutrosophc Probablty, Set Logc. Rehoboth: Amercan Research Press, (999). [3] W. B. Vasantha Kasamy & Florentn Smarache, Some Neutrosophc Algebrac Structures Neutrosophc N-Algebrac Struc- N Mumtaz Al, Florentn Smarache, Muhammad Shabr Munazza Naz, Neutrosophc B-LA-Semgroup Neutrosophc N-LA-Semgroup
4 Neutrosophc Sets Systems, Vol. 4, 04 tures, 9 p., Hexs, 006. [4] W. B. Vasantha Kasamy & Florentn Smarache, N-Algebrac Structures S-N- Algebrac Structures, 09 pp., Hexs, Phoenx, 006. [5] W. B. Vasantha Kasamy & Florentn Smarache, Basc Neutrosophc Algebrac Structures ther Applcatons to Fuzzy Neutrosophc Models, Hexs, 49 pp., 004. [6] P. Holgate: Groupods satsfyng a smple nvertve law, The Math. Student 6 (99). [7] M. Kazm M. Naseeruddn: On almost semgroups, Alg. Bull. Math. (97). [8] Q. Mushtaq M. S. Kamran: On left almost groups, Proc. Pak. Acad. Sc. 33 (996), 53-55. [9] M. Shabr, S. Naz, Pure spectrum of an aggroupod wth left dentty zero, World Appled Scences Journal 7 (0) 759-768. [0] Protc, P.V N. Stevanovc, AG-test some general propertes of Abel-grassmann s groupods,pu. M. A, 4,6 (995), 37 383. [] Madad Khan N. Ahmad, Characterzatons of left almost semgroups by ther deals, Journal of Advanced Research n Pure mathematcs, (00), 6 73. [] Q. Mushtaq M. Khan, Ideals n left almost semgroups, Proceedngs of 4 th Internatonal Pure mathematcs Conference, 003, 65 77. Receved: June 8, 04. Accepted: June 30, 04. Mumtaz Al, Florentn Smarache, Muhammad Shabr Munazza Naz, Neutrosophc B-LA-Semgroup Neutrosophc N-LA-Semgroup