Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 2(1), 67-70, January (2013) Res.J.Recent Sci. International Science Congress Association 67 Review Paper On Introduction of New Classes of AG-groupoidsShah M., Ahmad I. and Ali A.Department of Mathematics, Quaid-e-Azam University, Islamabad, PAKISTAN Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, PAKISTANAvailable online at: www.isca.in Received 12th August 2012, revised 8th September 2012, accepted 18th September 2012Abstract We discover eight new subclasses of AG-groupoids namely; Type-1, Type-2, Left Type-3, Right Type-3, Type-3, Backward Type-4, Forward Type-4, and Type-4. We provide examples of each type to prove their existence. We also give their enumer-ation up to order 6 and prove some of their basic properties and relations with each other and with other known classes. Keywords: AG-groupoid, LA-group, AG-group, types of AG-groupoid, enumeration. Introduction A groupoid is called AG-groupoid if it satisfies the left invertive law: cbab = . An AG-groupoid is an AG-groupoid satisfying the identityacbc = . An AG-groupoid with left identity is called AG-monoid. Every AG-monoid is AGgroupoid. An AG-groupoid always satisfies the medial law2; Lemma 1.1 (i): )()(bdaccdab = while an AG-monoid satises paramedial law2; Lemma 1.2 (ii): )()(cadbcdab = . Note that the name right modular groupoid is also used for AG-groupoid. An AG-groupoid with left identity e is an AG-groupoid. An AG-groupoid which satisfiesacab = , for all Î , is called AGgroupoid. An AG-groupoid is called Bol-groupoid if it satises the identitybcab × = × . An element a of an AG-groupoid is called idempotent if . An AG-groupoid is called idempotent or AG-2-band or simply AG-band if its every element is idempotent. An AG-groupoid is called AG-3-band if its every element satisfies aaaa = = . An AG-groupoid is called AG-group if contains left identity and inverses with respect to this identity. For detailed studies of this concept we refer the reader to reference5, 6. AG-groupoids (also called LA-semigroups), generalize commutative semigroups, have applications in flock theory and some geometrical applications. For additional sources on AG-groupoids, we suggest reference8, 9 and for the semigroup concept we refer the reader to follow the book of Howie10. Recently we have discovered eight new interesting subclasses of AG-groupoid11. We introduce here more eight new subclasses of AG-groupoids which initially we call types. We give their counting up to order 6 and prove some relations between them and to other subclasses of AG-groupoids. We prove that every AG-3-band is T -AG-groupoid and T -AG-groupoid is T AG-groupoid and every T -AG-groupoid is Bol-AG-groupoid. For T -AG-groupoid we prove that square of every element is idempotent and if it has left identity also then it becomes a unitary AG-group. As in semigroup theory the concept of zero-semigroup and zero-group exists, we find a similar concept for zero-AG-groupoid and zero-AG-group. Table-1 Presents the counting of new subclasses of AG-groupoids. Note that only the number of non-associative AG-groupoids is shown. Table-1 Classification and enumeration results for new subclasses of AG- groupoids of orders 3–6 Order 3 4 5 6 Total number of AG-groupoids 20 331 31913 40104513 1 T -AG-groupoids 2 14 101 783 2 T -AG-groupoids 1 3 8 16 -AG-groupoids 2 17 135 1272 -AG-groupoids 3 36 374 5150 3 T -AG-groupoids 2 16 111 870 -AG-groupoids 1 13 90 784 -AG-groupoids 0 1 6 11 4 T -AG-groupoids 0 1 7 7 Type-1, Type-2, Type -3 and Type-4 AG-Groupoids Denition 1: An AG-groupoid is called a Type-1 AG-groupoid denoted by T -AG-groupoid if , allfor dcbacdab Î =  = The following is now an obvious fact. Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 67-70, January (2013) Res. J. Recent Sci. International Science Congress Association 68 Proposition 1: Let be an AG-groupoid. Then the following are equivalent: i.bdaccdab Î " =  = , ; ii. dbcacdab Î " =  = , . Denition 2: An AG-groupoid is called a Type-2 AG-groupoid denoted by T -AG-groupoid ifbdaccdab Î " =  = , . Denition 3: An AG-groupoid is called a Left Type-3 AG-groupoid denoted by -AG-groupoid if Î , allfor , cabaacab =  = Denition 4: An AG-groupoid is called a Right Type-3 AG-groupoid denoted by -AG-groupoid if Î , allfor , acabcaba =  = Denition 5: An AG-groupoid is called a Type-3 AG-groupoid denoted by T -AG-groupoid if it is both -AG-groupoid and -groupoid. Denition 6: An AG-groupoid is called a Forward Type-4 AG-groupoid denoted by -AG-groupoid if Î , allfor , cbadcdab =  = Denition 7: An AG-groupoid is called a Backward Type-4 AG-groupoid denoted by -AG-groupoid if Î " , , bcdacdab =  = Denition 8: An AG-groupoid S is called a Type-4 AG-groupoid denoted by T -AG-groupoid if it is both -AG-groupoid and -AG-groupoid. Proposition 2: Let be an AG-groupoid. Then S is a commutative semigroup if any of the following holds: i. , bcadcdab Î " =  = ii. , cbdacdab Î " =  = Proof: Since Î " the equation abab = trivially holds. Now an application of either of (i) and (ii) proves commutativity in . Since any commutative AG-groupoid is associative, thus becomes commutative semigroup. There are some other cases but either they become semigroups or are identical to the cases that we have already considered. The following are examples or counter examples of some of the above considered types of AG-groupoid. Example 1: (i) A T -AG-groupoid of order 3. (ii) A T -AG-groupoid of order 4 which is not T -AG-groupoid. (iii) A T AG-groupoid of order 4 which is not T -AG-groupoid. (iv) A T -AG-groupoid of order 4 which is neither T -AG-groupoid nor T -AG-groupoid. Let us first put the previous known facts involving these types into the new format. Thus the result12; Theorem 2.7 now becomes: Theorem 1: Every AG-monoid is T -AG-groupoid. Two generalizations of Theorem 1 have been considered by M. Shahthat can be read in the new scenario as: Theorem 2: Let S be an AG-groupoid. Then S is a T -AG-groupoid if S has a cancellative element. More generally, Theorem 3: Let S be an AG-groupoid. Then S is a T -AG-groupoid if S has a left invertive left cancellative element. Regarding T -AG-groupoid the following fact is known. Theorem 4: Every AG-band is T -AG-groupoid. The following theorem generalizes the previous theorem to AG-3-band. Theorem 5: Every AG-3-band S is T -AG-groupoid × 1 2 3 × 1 2 3 4 × 1 2 3 4 × 1 2 3 4 1 1 2 3 1 1 2 3 4 1 1 2 3 4 1 1 1 3 3 2 3 1 2 2 2 1 4 3 2 1 1 3 4 2 1 1 3 3 3 2 3 1 3 4 2 2 1 3 4 4 1 3 3 3 3 1 1 4 3 4 1 2 4 3 3 4 1 4 3 3 1 2 (i) (ii) (iii) (iv) Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 67-70, January (2013) Res. J. Recent Sci. International Science Congress Association 69 Proof.Let Î . In order to prove that is-AG-groupoid let acab = Then cbaacbaccbabacabbacaaccaaccaabcacbcaNow to prove that is AG-groupoid, let caba = Then accabaabEquivalently is T -AG-groupoid. An AG-group is said to be unitary if square of every element is equal to left identity. Theorem 6:Let S be a T -AG-groupoid. Then i. Square of every element of S is idempotent; ii. If S is an AG-monoid then S is a unitary AG-group. Proof: Let S be T -AG-groupoid. Then i. Obviously the identity aaaa = holds trivially for every a in an AG-groupoid. Since is a T AG-groupoid, it becomesaaaa = . Hence is locally associative. ii. Let has left identity e then for all a in S we trivially have e ae e ae × = × , which by the property of T -AG-groupoid implies that ee ae ae = × , which by medial law impliesee, which then by cancellativity of e implies that . Hence the result. Theorem 7:Every T -AG-groupoid is Bol-AG-groupoid. Proof: Let S be a T -AG-groupoid and Î . Then abdcab × = × (by left invertive law) dcabab × = ×  (by denition of T -AG-groupoid) dcab × = ×  (by left invertive law) bcab × = ×  (by left invertive law) ),bcab × = ×  (by denition of T -AG-groupoid) Hence the result. Remark 1. The converse of the above theorem is not true as the BolAG-groupoid given in example 2 is not T -AG-groupoid. Example 2. A Bol-AG-groupoid. × 1 2 3 1 1 1 1 2 1 1 1 3 1 2 2 From table 1 this is obvious that right Type-3-AG-groupoid is not necessarily left Type-3-AG-groupoid but one may guess the impression that the converse may be true. The following example shows that the converse is also false. Example 3. A -AG-groupoid of order 4 which is not AG-groupoid. × 1 2 3 4 1 1 1 1 1 2 1 1 1 1 3 1 1 1 2 4 1 1 3 2 Theorem 8.The following facts always hold, i. A T -AG-groupoid is T -AG-groupoid; ii. A T AG-groupoid is T AG-, iii. groupoid; iv. A T -AG-groupoid is, a.-AG-groupoid, b. -AG-groupoid; c. T -AG-groupoid. Proof: i. Let Î and let cdab = which by definition of T -AG-groupoid implies thatbdac = . But then obviouslyacbd = Applying the same definition again we have dcba = . Hence is T -AG-groupoid. ii. Let Î and let cdab = which by denition of AG-groupoid implies thatcbad = . Now applying denition of -AG-groupoid we havedcba = . Hence is T -AG-groupoid. iii. (a) Apply denition of T -AG-groupoid with , c a = b. is similar to (a) and (c) follows from (a) and (b). As a corollary we immediately have the following: Corollary 1: The following facts always hold, i. A T -AG-groupoid is, (a) -AG-groupoid;(b)-AG-groupoid;(c) T -AG-groupoid. (ii) A T -AG-groupoid is, (a) -AG-groupoid;(b)-AG-groupoid; (c) T -AG-groupoid. Zero-AG-Groupoid and Zero-AG-GroupAs in the case of semigroups see for instance the book of Howie10, there exists a zero-semigroup and zero-group, we prove the existence of Zero-AG-groupoid and zero-AG-group. Let us first define them. Denition 9. An AG-groupoid is called a zero-AG-groupoid if there exists an element z in such that without z is an AG-group and for all x in we have that . z zx xz = = Denition 10. An AG-groupoid is called a zero-AG-group if Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 2(1), 67-70, January (2013) Res. J. Recent Sci. International Science Congress Association 70 there exists an element z in such that S without z is a semigroup and for all x in we have that . z zx xz = = Now we provide some examples to show the existence of these concepts. Example 4.(i) A zero-AG-groupoid of order 4. (ii) A zero-AG-group of order 3. (i) × 1 2 3 4 1 1 1 1 1 2 1 2 2 2 3 1 2 2 2 4 1 3 3 3 (ii) × 1 2 3 4 1 1 1 1 1 2 1 2 3 4 3 1 4 21 31 4 1 3 42 2 Theorem 9: Let be an AG-group. Then aGGa Î " = = Proof: Clearly, GGGa Í Í Conversely, let Î and let e be the left identity of then, Gagaaaeg Therefore, Ga Í Hence Ga Í Next clearly GGaG Í Í Conversely, let Î then, aGgeaagegeeeTherefore, Ga Í Hence aG = Corollary 2: Let be an AG-group having left identity e . Then13GeeG = = Corollary 3: Let be an AG-group. Then for all Î , there exist Î such that yaax = = . Proposition 3:If an AG-groupoid S with 0 is a zero-AG-groupoid AG-group then },0{\ Î " aSSa = = Proof:}0{ È = is a zero-AG-groupoid-AG-group where}0{\ = . Let }.0{\}0{\ = Î  Î As is an AG-group, so by Theorem 9 GaaG = = Now }0{}0{aGaS = È = È = and }0{}0{GaSa = È = È = Hence aSSa = = ConclusionThis article launches and investigates eight new classes of AG-groupoids. Enumeration of each class has also been done up to order 6. Relations of these newly discovered classes with each other and with previously known classes have been investigated to some extent. The readers are motivated to study these new classes in more detail. References 1.Kazim M.A. and Naseerudin M., On almost semigroups, Portugaliae Mathematica,36(1), (1977) 2.Cho J.R., Pusan Jezek J. and Kepka T., Praha, Paramedial Groupoids, Czechoslovak Mathematical Journal, 49(124), (1996) Praha 3.Stevanovic N. and Protic P.V., Abel-grassmann’s bands, Quasigroups and Related Systems, 11(1) 95–101 2004. 4.Stevanovic N. and Protic P.V., Composition of Abel-Grassmann’s 3-bands, Novi Sad J. 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