Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 42, 2065 - 2070 On Quasi-Cancellativity of AG-Groupoids M. Shah Department of Mathematics Quad-i-Azam University, Islamabad, Pakistan shahmaths problem@hotmail.com I. Ahmad Department of Mathematics University of Malakand, Pakistan iahmaad@hotmail.com A. Ali Department of Mathematics Quad-i-Azam University, Islamabad, Pakistan shahmaths problem@hotmail.com Abstract Quasi-cancellativity is the generalization of cancellativity. We introduce the notion of quasi-cancellativity of semigroup into AG-groupoids. We prove that every AG-band is quasi-cancellative. We also prove the famous Burmistrovich’s Theorem for AG-groupoids that states “An AG∗∗ -groupoid S is a quasi-cancellative if and only if S is a semilattice of cancellative AG∗∗ -groupoids”. Keywords: AG-groupoid, Quasi-cancellativity, Burmistrovich’s Theorem 1 Introduction By definition, an AG-groupoid (also called an LA-semigroup) S is a set with a binary operation satisfying the left invertive law: (xy)z = (zy)x for all x, y, z ∈ G. AG-groupoids were initiated by M. Naseeruddin and Kazim in [12]. AG-groupoid is a generalization of commutative semigroups and have applications in flock theory, see for example [3] and some of its applications in geometry have been investigated in [8]. An AG-groupoid S is called AG∗∗ groupoid if a(bc) = b(ac), ∀a, b, c ∈ S. An AG-groupoid S is called AG-Band 2066 M. Shah, I. Ahmad and A. Ali if aa = a, ∀a ∈ S [6]. An AG-groupoid S is called AG-3-band if its every element satisfies a(aa) = (aa)a = a [11]. An AG-groupoid with left identity is called AG-monoid. Every AG-monoid is AG∗∗ -groupoid. An AG-groupoid S always satisfies the medial law: (ab)(cd) = (ac)(bd) [9, Lemma 1.1 (i)] while an AG-monoid satisfies paramedial law: (ab)(cd) = (db)(ca) [9, Lemma 1.2 (ii)]. Note that in [9] the name right modular groupoid is used for AG-groupoid. For further studies in AG-groupoids, we suggest [1, 2, 7] and [8] while for the semigroup terminology we refer to [10]. It is easy to see that every cancellative AG-groupoid is quasi-cancellative but the converse is not true see Example 1. We introduce the notion of quasicancellativity of semigroup [5] into AG-groupoids. We prove that every AGband is quasi-cancellative. We also prove Burmistrovich’s Theorem for AGgroupoids that states “An AG∗∗ -groupoid S is a quasi-cancellative if and only if S is a semilattice of cancellative AG∗∗ -groupoids”. Table 1 illustrates the enumeration of newly discovered subclass of AGgroupoids. Order 3 4 5 6 Total 20 331 31913 40104513 Quasi-cancellative AG-groupoids 1 6 18 66 Table 1: Enumeration of Quasi-cancellative AG-groupoids of orders 3–6. 2 Quasi-cancellativity of AG-groupoids Here in this section we introduce the notion of quasi-cancellativity of semigroups into AG-groupoids. Quasi-cancellativity is the generalization of cancellativity. We begin with the following definition: Definition 1. An AG-groupoid S is quasi-cancellative if for any x, y ∈ S, (i) x = xy and y 2 = yx imply that x = y, (ii) x = yx and y 2 = xy imply that x = y. Using the under construction package AGGROUPOIDS [8], the two parts of the above definition are equivalent for AG-groupoids up to order 6. Thus: Conjecture 1. Conditions (i) and (ii) of Definition 1 are equivalent for AGgroupoids. Example 1. A quasi-cancellative AG∗∗ -groupoid of order 5. 2067 On quasi-cancellativity of AG-groupoids · 1 2 3 4 5 1 1 1 1 1 1 2 1 2 2 5 4 3 1 2 3 5 4 4 1 4 4 2 5 5 1 5 5 4 2 Proposition 1. Every AG-band is quasi-cancellative. Proof. Let S, be an AG-band such that for any x, y ∈ S we prove the two conditions of the above definition as follows: (i) x2 = xy and y 2 = yx which by definition of AG-band become x = xy and y = yx. Now x = xy = xy · y = y 2 x = yx = y 2 = y. (ii) x2 = yx and y 2 = xy which by definition of AG-band become x = yx and y = xy. Now x = yx = y 2 x = xy · y = y 2 = y. Hence S is quasicancellative. It should be noted that we have enumerated quasi-cancellative AG-groupoids up to order 6, our data shows that every AG-3-band is quasi-cancellative. However we are unable to prove or disprove the fact. Thus we have the following Conjecture 2. Every AG-3-band is quasi-cancellative. Lemma 1. In a quasi-cancellative paramedial AG-groupoid S, for any x, y, a, b ∈ S the following statements hold; (i) xa = xb iff ax = bx; (ii) x2 a = x2 b implies that xa = xb; (iii) (xy)a = (xy)b implies that (yx)a = (yx)b. Proof. Let S be a quasi-cancellative paramedial AG-groupoid. (i) Let xa = xb, then (xa)(xa) = (xb)(xa) and (xa)(xb) = (xb)(xb) so that (ax)2 = (ax)(bx) and (bx)2 = (bx)(ax) which by Definition 1 implies that ax = bx. The opposite implication follows by symmetry. (ii) If x2 a = x2 b, then (x2 a)a = (x2 b)a and (x2 a)b = (x2 b)b which implies that a2 x2 = (ab)x2 or (ax)2 = (ax)(bx), and (1) (ba)x2 = b2 x2 or (bx)2 = (bx)(ax) (2) Thus from (1) and (2) by Definition 1(i) we have ax = bx, and hence by (i), xa = xb. 2068 M. Shah, I. Ahmad and A. Ali (iii) Let (xy)a = (xy)b, then a2 (xy) (xy)2 a2 (yx)2 a2 ⇒ [a(yx)]2 = = = = (ab)(xy) (xy)2 (ab) (yx)2 (ab) [b(yx)][a(yx)] (3) Similarly, let (xy)a = (xy)b then, (ba)(xy) ⇒ (xy)2 (ba) ⇒ (yx)2 b2 ⇒ [b(yx)]2 ⇒ [b(yx)]2 = = = = = b2 (xy) (xy)2 b2 (yx)2 (ba) (ab)(yx)2 [a(yx)][b(yx)] (4) From (3) and (4) by Definition 1(ii) we have a(yx) = (yx)b. 3 Burmistrovich’s Theorem for AG-groupoids Here we prove that the famous Burmistrovich’s Theorem [5] of semigroups also holds in AG-groupoids. AG∗∗ -groupoids play an important role in the theory of AG-groupoids. It has been investigated that AG∗∗ -groupoid is paramedial [8]. Theorem 1. An AG∗∗ -groupoid S is quasi-cancellative if and only if S is a semilattice of cancellative AG∗∗ -groupoids. Proof. Necessity: On S define the relation σ by xσy if for any a, b ∈ S, xa = xb if and only if ya = yb. It is clear that σ is an equivalence relation. Let xσy and z ∈ S. If (xz)a = =⇒ =⇒ =⇒ =⇒ =⇒ (xz)b (az)x = (bz)x by left invertive law x(az) = x(bz) by Lemma 1(i) y(az) = y(bz) by hypothesis a(yz) = b(yz) by definition of AG∗∗ -groupoid (yz)a = (yz)b by Lemma 1(i) By symmetry (yz)a = (yz)b implies that (xz)a = (xz)b thus it follows that xzσyz. Now, (zx)a = (zx)b On quasi-cancellativity of AG-groupoids 2069 ⇒ (xz)a = (xz)b by Lemma 1(iii) ⇒ (yz)a = (yz)b as above ⇒ (zy)a = (zy)b by Lemma 1(iii) By symmetry (zy)a = (zy)b implies that (zx)a = (zx)b. Thus it follows that zxσzy and hence σ is congruence. Further, Lemma 1(i) & (ii) imply that S/σ is an AG-band and while Lemma 1(iii) implies that S/σ is commutative. Therefore σ is a semilattice congruence. Suppose that zx = zy and xσz and yσz. Since xσz, zx = zy implies that x2 = xy and since yσz , it implies that yx = y 2. But then Definition 1(i) yields x = y. If xz = yz with xσz and yσz then by Lemma 1(i) zx = zy, and this reduces to the case just considered. Hence each σ-class is cancellative. Sufficiency: Suppose S is a semilattice of cancellative AG∗∗ -groupoids. Let x and y be elements such that x2 = xy and y 2 = yx. Let β be the component of S that contains xy. Since S is commutative being semilattice we have yx ∈ β as well. Thus x2 , y 2 ∈ β. Since β is an AG∗∗ -groupoid so by the closure property in β, we have x, y ∈ β. But β is cancellative, and therefore the equality xx = xy implies x = y. A similar argument applies if x2 = yx and y 2 = xy. Let us verify this by an example. Consider the Example 1. Since S = {1, 2, 3, 4, 5} is quasi-cancellative AG∗∗ -groupoid. So we can write S = {A = {1} , B = {3} , C = {2, 4, 5}}. Here A, B and C are cancellative AG∗∗ groupoids such that they commute with each other and A2 = A, B 2 = B, C 2 = C. We remark that there are 1, 4, 12 non-associative quasi-cancellative AG∗∗ groupoids of order 3, 4, 5 respectively and we have verified them all manually for this theorem. References [1] Q. Mushtaq and S. M. Yusuf, On LA-semigroups, Alig. Bull. Math., 8(1), 65-70. [2] Q. Mushtaq and S. M. Yusuf, On Locally Associative LA-semigroup, J. Nat. Sci. Math. Vol. XIX, No.1, April 1979, pp. 57–62. [3] M. Naseeruddin, Some studies on almost semigroups and flocks, Ph.D Thesis, The Aligarh Muslim University, India, 1970. [4] M. Shah, A. Ali, Some structural properties of AG-group, International Mathematical Forum 6 (2011), no. 34, 1661–1667. [5] I. E. Burmistrovich, The commutative bands of cancellative semigroups, Sib. Mat. Zh., 6:284–299, 1965, Russian. 2070 M. Shah, I. Ahmad and A. Ali [6] N. Stevanovic, P.V. Protic, Abel-grassmann’s bands, Quasigroups and Related Systems, 11(1):95–101, 2004. [7] M. Shah, T. Shah, A. Ali, On the cancellativity of AG-groupoids, International Mathematical Forum 6 (2011), no. 44, 2187–2194. [8] M. Shah, A theoretical and computational investigations of AG-groups, PhD thesis, Quaid-i-Azam University Islamabad, 2012. [9] J. R. Cho, Pusan, J. Jezek and T. Kepka, Praha,Paramedial Groupoids, Czechoslovak Mathematical Journal, 49(124)(1996), Praha. [10] J. M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 2003. [11] N. Stevanovic, P. V. Protic, Composition Of Abel-Grassmann’s 3-bands, Novi Sad J. Math. Vol. 34, No. 2, 2004, 175 − 182. [12] M. A. Kazim and M. Naseerudin, On almost semigroups, Portugaliae Mathematica. Vol.36-Fase. 1 − 1977. Received: June, 2012