Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 1(11), 47-49, November (2012) Res.J.Recent Sci. International Science Congress Association 47 Short Communication Discovery of New Classes of AG-groupoidsShah M., Ahmad I. and Ali A.Department of Mathematics, Quad-i-Azam University, Islamabad, PAKISTAN Department of Mathematics, University of Malakand, PAKISTAN Available online at: www.isca.in Received 29th July 2012, revised 8th August 2012, accepted 18th September 2012Abstract We discover eight new subclasses of AG-groupoids namely; anti-commutative AG-groupoid, transitively commutative AG-groupoid, self-dual AG-groupoid, unipotent AG-groupoid, left alternative AG-groupoid, right alternative AG-groupoid, alternative AG-groupoid and exible AG-groupoid. We prove their existence by providing examples to these classes. We also prove some basic results of these classes and present a table of their enumeration up to order 6. Keywords: AG-groupoid, LA-group, AG-group, types of AG-groupoid, enumeration. Introduction AG-groupoids have been enumerated up to order 6. Using GAP the Cayley tables have been obtained in the above-mentioned enumeration. We investigate eight new interesting subclasses of AG-groupoids. These classes are anti-commutative AG-groupoid, transitively commutative AG-groupoid, self-dual AG-groupoid, unipotent AG-groupoid, left alternative AG-groupoid, right alternative AG-groupoid, alternative AG-groupoid and exible AG-groupoid. We prove here the existence of the above classes by providing their cayley tables. We investigate some relations between them and to some other known classes of AG-groupoid. Table 1 provides counting of the newly found classes of AG-groupoids for the non-associative AG-groupoids. Section 2 is about anti-commutative AG-groupoids and transitively commutative AG-groupoids. Here we prove that every anti-commutative AG-groupoid and every cancellative AG**- groupoid is transitively commutative. Also the equivalence of AG-band and locally associativity is proved for anti-commutative AG-groupoid. In section 3 we introduce the concept of alternativity and exibility from loop theory into AG-groupoids. Here we prove two basic facts that every AG-3-band is exible and that in a right alternative AG-groupoid, square of every element commute with every element. Section 4 is about the existence of self-dual AG-groupoid and unipotent AG-groupoids, where we prove that a self-dual AG-groupoid with left identity becomes commutative monoid and also that in a left alternative self-dual AG-groupoid, square of every element commutes with every element. Preliminaries: A groupoid is called AG-groupoid if it satisfies the left invertive law: (ab) c = (cb)a. An AG**-groupoid is an AG-groupoid satisfying the identity ()(). abcbac = An AG-groupoid with left identity is called AG-monoid. Every AG-monoid is AG**- groupoid. An AG-groupoid always satisfies the medial law4;Lemma 1.1 (i) ()()()() abcdacbd = , while an AG-monoid satisfies paramedial law4;Lemma1.1 (ii): )()(cadbcdab = . An AG-groupoids with left identity e is an AG**-groupoid. An AG-groupoid which satisfy ),acbc = for all Î , is called AG-groupoid. An AG-groupoid is called Bol-groupoid if it satisfies the identitybcab × × = × . An element of an AG-groupoid is called idempotent if = a. 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 satisesaaaa = = . Some applications of AG-groupoid in theory of ocks are given inand some of its applications in geometry have been investigated by Shah. AG-groupoid (also called LA-semigroup), is the generalization of commutative semigroups. For additional sources on AG-groupoids, we suggest arcticles9,10, while for the semigroup concept we refer the reader to a book by Howie11. We present counting of the new subclasses of AG-groupoids in table 1. Note that only the number of non-associative AG-groupoids is shown. Anti-commutativity and Transitively Commutativity of AG-groupoidsActually the notion of anti-commutativity and transitively commutativity had been defined for AG-bands, which is a very small class of AG-groupoids. We make these denitions global for the whole AG-groupoids and prove their existence in example 1 and example 2. Denition 1: An AG-groupoid is called anti-commutative if for all baab = Î , implies that = Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(11), 47-49, November (2012) Res. J. Recent Sci. International Science Congress Association 48 Table-1 Classication and enumeration results for new subclasses of AG-groupoids of orders 3–6 Order 3 4 5 6 Total 20 331 31913 40104513 Anti-commutative AG-groupoids 1 2 4 0 Transitively commutative AG-groupoids 3 61 2937 1239717 Self-dual AG-groupoids 0 8 133 4396 Left alternative AG-groupoids 0 5 171 12029 Right alternative AG-groupoids 2 33 997 139225 Alternative AG-groupoids 0 2 59 4447 Flexible AG-groupoids 1 19 447 32770 Unipotent AG-groupoids 5 74 3946 1739186 Example-1: An anti-commutative AG-groupoid of order 4. Example 2: A transitively commutative AG-groupoid (a non-anticommutative AG-groupoid) of order 4. × 1 2 3 4 1 1 1 1 1 2 1 1 1 1 3 1 1 1 1 4 2 2 2 1 We now give an interesting relation of AG**-groupoid with transitively commutative AG-groupoid. Theorem 1. Every cancellative AG**-groupoid S is transitively commutative. Proof. Let Î such that cbbcbaab = = , Then consider cabaabcbbcac = = = = = , which by left cancellativity implies that ca ac = . Corollary 1: Every AG-group is transitively commutative. The following theorem shows that the class of transitively commutative AG-groupoids always contain the class of anti-commutative AG-groupoids. Theorem 2. Every anti-commutative AG-groupoid S is transitively commutative. Proof. Let S be an anti-commutative AG-groupoid and let Î , such that cbbcbaab = = Then by definition of anti-commutativity, this implies that = = , But this implies that c a = and which further implies that . ca ac = Hence S is transitively commutative. Conjecture 1: Every anti-commutative AG-groupoid S is cancellative but the converse is not true. We now give the following result to show that each AG-band is locally associative. Theorem 3: Let S be an anti-commutative AG-groupoid. Then the following are equivalent. i. S is AG-band; ii. S is locally associative. Proof: (i) (ii) is always true, ii. (i). By denition of locally associativity and anti-commutativity, for every Î , we haveaa Alternative and Flexible AG-groupoids In an attempt to bring AG-groupoids a bit closer to quasigroups and loops, the concept of nucleus of AG-groupoids was introduced by Shah and by doing so, six new classes of AG-groupoids have been dened. Here we introduce the concept of exibility and alternativity from loops. This will give us four more classes of AG-groupoids. Denition 3: An AG-groupoid is called exible if it satises the identity, . yx x x xy × = × Denition 4: An AG-groupoid is called left alternative if it satisfies the identity, . xy x y xx × = × Denition 5: An AG-groupoid is called right alternative if it satisfies the identity, . . . yy x y xy = Denition 6: An AG-groupoid is called alternative if it is both left alternative and right alternative. Example 3: i. A left alternative AG-groupoid of order 4 and ii. A right alternative AG-groupoid of order 3, iii. An alternative AG-groupoid of order 4. × 1 2 3 4 1 1 1 1 1 2 1 1 1 1 3 1 1 1 1 4 1 1 3 1 (i) × 1 2 3 1 1 1 1 2 1 1 1 3 1 2 1 (ii) × 1 2 3 4 1 1 1 1 1 2 1 1 1 1 3 1 1 1 2 4 3 1 1 1 (iii) The following results illustrate some basic properties of these new classes. × 1 2 3 4 1 1 3 4 2 2 4 2 1 3 3 2 4 3 1 4 3 1 2 4 Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(11), 47-49, November (2012) Res. J. Recent Sci. International Science Congress Association 49 Proposition 1: Every AG-3-band is exible. Proof: Let be an AG-3-band and Î . Then xyxyyxxxyx Hence is exible. Proposition 2: In a right alternative AG-groupoid, square of every element commute with every element. Proof: Let S be an AG-groupoid and Î . Then yxyxThe following now easily follows. Corollary 2: (i) Every right alternative AG-groupoid is locally associative. (ii) A right alternative AG-monoid is commutative monoid. Though a non-associative left alternative can be AG-monoid (see the following Example) but then it cannot contain inverses because a left alternative AG-group is abelian group. Example 4. A left alternative AG-monoid of order 4. × 1 2 3 4 1 1 1 1 1 2 1 1 1 3 3 1 1 1 2 4 1 2 3 4 Theorem 4.A right alternative AG-groupoid having left identity is commutative semigroup. Proof: Let be a right alternative AG-groupoid, and let Î then by denition of right alternative AG-groupoid, we have, ababab by left invertive law. Now let = we have aeae Thus has right identity and hence is commutative semigroup. Self-dual AG-groupoids and Unipotent AG-groupoids Here we introduce the notion of Self-duality from the theory of semi-group into AG-groupoids. (Left) AG-groupoid and right AG-groupoid can easily be seen dual to each other. Thus the transpose of the multiplication table of an AG-groupoid becomes right AG-groupoid. There are AG-groupoids whose transpose is also an AG-groupoid. In this section we discuss such AG-groupoids. Denition 7. An AG-groupoid is called self-dual if it is also right AG-groupoid. Note: Though not studied as a class but the name “almost semigroup” has been used for what we call self-dual AG-groupoid. It is easy to prove that: Proposition 3. A self-dual AG-groupoid with left identity becomes commutative monoid. Denition 8. An AG-groupoid is called unipotent if for every Î we have Example 5: A self-dual AG-groupoid that is also unipotent. × 1 2 3 4 1 1 1 1 1 2 1 1 1 1 3 1 1 1 2 4 1 3 1 1 Theorem 5: In a left alternative self-dual AG-groupoid, square of every element commutes with every element. Proof: Let be an AG-groupoid and Î . Then yxxyConclusionNew eight classes of AG-groupoids have been discovered and investigated. Enumeration of each class up to order 6 has also been provided in a table. Some of the interesting relations of these new discovered classes with each other and with other previously known classes have been investigated. The researchers of the field are motivated to investigate these new classes more in detail. References 1.Distler A., Shah M. and Sorge V., Enumeration of AG-groupoids, Lecture Notes in Computer Science, Volume 6824/2011, 1-14 (2011) 2.GAP: Groups Algorithm and Programming, Version 4.4.12, 2008, (2012) 3.Kazim M.A. and Naseerudin M., On almost semigroups, Portugaliae Mathematica.,36(1) (1977) 4.Cho J.R., Pusan, Jezek J. and T. Kepka, Praha, Paramedial Groupoids, Czechoslovak Mathematical Journal, 49(124) (1996) Praha5.Stevanovic N. and Protic P.V., Abel-grassmann’s bands, Quasigroups and Related Systems,11(1), 95–101 (2004) 6.Stevanovic N. and Protic P.V., Composition of Abel-Grassmann’s 3 -bands, Novi Sad J. Math., 34(2), 175-182 (2004) 7.Naseeruddin M., Some studies on almost semigroups and flocks, PhD Thesis, The Aligarh Muslim University, India (1970) 8.Shah M., A theoretical and computational investigations of AG-groups, PhD thesis, Quaid-i-Azam University Islamabad, (2012) 9.Mushtaq Q. and Yusuf S.M., On Locally Associative LA-semigroup, J. Nat. Sci. Math., XIX(1), 57–62 (1979) 10.Shah M., Shah T. and Ali A., On the cancellativity of AG-groupoids, International Mathematical Forum,6(44), 2187–2194 (2011) 11.Howie J.M., Fundamentals of Semigroup Theory, Clarendon Press, Oxford, (2003)