On some classes of Abel-Grassmann's groupoids 1

Journal of Advanced Research in Pure Mathematics Online ISSN: 1943-2380

Vol. 3, Issue. 4, 2011, pp. 109-119 doi: 10.5373/jarpm.670.121410

On some classes of Abel-Grassmann’s groupoids Madad Khan∗ , Faisal, Venus Amjad Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan.

Abstract. In this paper, we have introduced the notion of (2, 2)-regular and strongly regular AG-groupoids and investigated these structures. We have shown that regular AG-groupoid is the most generalized class of an AG-groupoid. It has proved that weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular and (2, 2)-regular AG-groupoids with left identity coincide. Also we have proved that every intra-regular AG -groupoid with left identity (AG ∗∗ -groupoid) is regular but the converse is not true in general. Further we have shown that nonassociative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2, 2)-regular and strongly regular AG ∗ -groupoids do not exist. Keywords: An AG-groupoid; Left invertive law; Medial law and paramedial law. Mathematics Subject Classification 2010: 20M10, 20N99.

1

Introduction

The concept of a left almost semigroup (LA-semigroup) [2] was first introduced by M. A. Kazim and M. Naseeruddin in 1972. In [1], the same structure is called a left invertive groupoid. P. V. Protić and N. Stevanović called it an Abel-Grassmann’s groupoid (AGgroupoid) [9]. An AG-groupoid [9], is a groupoid S holding the left invertive law (ab)c = (cb)a, for all a, b, c ∈ S.

(1.1)

This left invertive law has been obtained by introducing braces on the left of ternary commutative law abc = cba.

∗

Correspondence to: Madad Khan, Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan. Email: [email protected], [email protected] and [email protected] † Received: 14 December 2010, revised: 11 February 2011, accepted: 02 June 2011. http://www.i-asr.com/Journals/jarpm/

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c ⃝2011 Institute of Advanced Scientific Research

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On some classes of Abel-Grassmann’s groupoids

In an AG-groupoid, the medial law holds [2] (ab)(cd) = (ac)(bd), for all a, b, c, d ∈ S.

(1.2)

In an AG-groupoid S with left identity, the paramedial law holds [6] (ab)(cd) = (dc)(ba), for all a, b, c, d ∈ S.

(1.3)

Further if an AG-groupoid contains a left identity, the following law holds [6] a(bc) = b (ac) , for all a, b, c ∈ S.

(1.4)

If an AG-groupoid (without left identity) satisfies (4), then [5] it is called an AG ∗∗ groupoid. AG-groupoids have been widely explored in [3], [6] and [9]. An AG-groupoid is a non-associative algebraic structure mid way between a groupoid and a commutative semigroup with wide applications in theory of flocks [8]. This structure is closely related with a commutative semigroup, because if an AGgroupoid contains a right identity, then it becomes a commutative semigroup [6]. Define the binary operation ”•” on a commutative inverse semigroup S as a • b = ba−1 , for all a, b ∈ S, then (S, •) becomes an AG-groupoid [7]. The connection between AG-groupoids and vector spaces over finite fields has been given in [3] as: Let W be a sub-space of a vector space V over a field F of cardinal pn for pn −1

some prime p ̸= 2. Define the binary operation ⊙ on W as follows: u ⊙ v = α 2 u + v, where α is a generator of F \{0} and u, v ∈ W . Then (W, ⊙) is an AG-groupoid with left identity 0. An AG-groupoid (S, ·) with left identity becomes a semigroup S under new binary operation ”◦” defined as x ◦ y = (xa)y, for all x, y ∈ S. It is easy to show that “◦” is associative (x ◦ y) ◦ z = (((xa)y)a)z = (za)((xa)y) = (xa)((za)y) = (xa)((ya)z) = x ◦ (y ◦ z). Hence (S, ◦) is a semigroup [10]. The Connections discussed above make this nonassociative structure interesting and useful. Many interesting structural properties of AG-groupoids have been investigated in [4]. An AG-groupoid is proved to be a union of groups if and only if every element has a partial inverse with which it commutes. Also it has been shown in [4], that every AG union of groups is a union of disjoint, maximal abelian groups.

Madad Khan, Faisal and Venus Amjad

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In this paper, we have investigated several important properties of an AG-groupoid, and studied some well known classes of semigroup in this non-associative algebraic structure which is the generalization of semigroup [6], and shown that all of these classes coincide in an AG-groupoid with left identity but except a regular class. It is important to note that all the mention classes in this paper do not coincide in a semigroup or in a monoid. An element a of an AG-groupoid S is called a regular element of S if there exists x ∈ S such that a = (ax)a and S is called regular if all elements of S are regular. An element a of an AG-groupoid S is called a weakly regular element of S if there exist x, y ∈ S such that a = (ax)(ay) and S is called weakly regular if all elements of S are weakly regular. An element a of an AG-groupoid S is called an intra-regular element of S if there exist x, y ∈ S such that a = (xa2 )y and S is called intra-regular if all elements of S are intra-regular. An element a of an AG-groupoid S is called a right regular element of S if there exists x ∈ S such that a = a2 x = (aa)x and S is called right regular if all elements of S are right regular. An element a of an AG-groupoid S is called a left regular element of S if there exists x ∈ S such that a = xa2 = x(aa) and S is called left regular if all elements of S are left regular. An element a of an AG-groupoid S is called a left quasi regular element of S if there exist x, y ∈ S such that a = (xa)(ya) and S is called left quasi regular if all elements of S are left quasi regular. An element a of an AG-groupoid S is called a completely regular element of S if a is regular, left regular and right regular. S is called completely regular if it is regular, left and right regular. An element a of an AG-groupoid S is called a (2,2)-regular element of S if there exists x ∈ S such that a = (a2 x)a2 and S is called (2, 2)-regular AG-groupoid if all elements of S are (2, 2)-regular. An element a of an AG-groupoid S is called a strongly regular element of S if there exists x ∈ S such that a = (ax)a and ax = xa. S is called strongly regular AG-groupoid if all elements of S are strongly regular. An AG-groupoid S is called an AG ∗ -groupoid if the following holds [5] (ab)c = b(ac), for all a, b, c ∈ S.

(1.5)

In an AG ∗ -groupoid S, the following law holds [9] (x1 x2 )(x3 x4 ) = (xp(1) xp(2) )(xp(3) xp(4) ),

(1.6)

where {p(1), p(2), p(3), p(4)} means any permutation on the set {1, 2, 3, 4}. It is an easy consequence that if S = S 2 , then S becomes a commutative semigroup. An AG-groupoid may or may not contains a left identity. The left identity of an AG-groupoid allow us to introduce the inverses of elements in an AG-groupoid. If an AG-groupoid contains a left identity, then it is unique [6].

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Example 1.1. Let us consider an AG-groupoid S = {a, b, c} in the following multiplication table.

. a b c

a c c c

b c c c

c c a a

Clearly S is non-commutative and non-associative, because bc ̸= cb and (cc)a ̸= c(ca). Note that S has no left identity. Example 1.2. Let us consider an AG-groupoid S = {a, b, c, d, e, f } with left identity e in the following Cayley’s table.

. a b c d e f

a a a a a a a

b a b b b b b

c a b f f c f

d a b f f d f

e a b d c e f

f a b f f f f

Clearly S is non-commutative and non-associative, because ed ̸= de and (de)e ̸= d(ee). Lemma 1.1. If S is a regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2, 2)-regular or strongly regular AG-groupoid, then S = S 2 . Proof. Let S be a regular AG-groupoid, then S 2 ⊆ S is obvious. Let a ∈ S, then since S is regular so there exists x ∈ S such that a = (ax)a. Now a = (ax)a ∈ SS = S 2 . Similarly if S is weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2, 2)-regular or strongly regular, then we can show that S = S 2. The converse is not true in general, because in Example 1.2, S = S 2 holds but S is not regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2, 2)-regular and strongly regular, because d ∈ S is not regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2, 2)-regular and strongly regular.


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Theorem 1.2. If S is an AG-groupoid with left identity (AG ∗∗ -groupoid), then S is intra-regular if and only if for all a ∈ S, a = (xa)(az) holds for some x, z ∈ S. Proof. Let S be an intra-regular AG-groupoid with left identity (AG ∗∗ -groupoid), then for any a ∈ S there exist x, y ∈ S such that a = (xa2 )y. Now by using Lemma 1.1, y = uv for some u, v ∈ S. Thus by using (1.4), (1.1) and (1.3), we have a = (xa2 )y = (x(aa))y = (a(xa))y = (y(xa))a = (y(xa))((xa2 )y) = ((uv)(xa))((xa2 )y) = ((ax)(vu))((xa2 )y) = ((ax)t)((xa2 )y) = (((xa2 )y)t)(ax) = ((ty)(xa2 ))(ax) = ((a2 x)(yt))(ax) = ((a2 x)s)(ax) = ((sx)(aa))(ax) = ((aa)(xs))(ax) = ((aa)w)(ax) = ((wa)a)(ax) = (za)(ax) = (xa)(az), where vu = t, yt = s, xs = w and wa = z for some t, s, w, z ∈ S. Conversely, let for all a ∈ S, a = (xa)(az) holds for some x, z ∈ S. Now by using (1.4), (1.1), (1.2) and (1.3), we have a = (xa)(az) = a((xa)z) = ((xa)(az))((xa)z) = (a((xa)z))((xa)z) = (((xa)z)((xa)z))a = (((xa)(xa))(zz))a = (((ax)(ax))(zz))a = ((a((ax)x))(zz))a = (((zz)((ax)x))a)a = ((z 2 ((ax)x))a)a = (((ax)(z 2 x))a)a = ((((z 2 x)x)a)a)a = (((x2 z 2 )a)a)a = (a2 (x2 z 2 ))a = (a(x2 z 2 ))(aa) = (at)(aa), where x2 z 2 = t for some t ∈ S. Now by using (1.3) and (1.1), we have a = (at)(aa) = (((at)(aa))t)(aa) = (((aa)(ta))t)(aa) = ((a2 (ta))t)(aa) = ((t(ta))a2 )(aa) = (ua2 )v, where t(ta) = u and aa = v for some u, v ∈ S. Thus S is intra-regular. Theorem 1.3. If S is an AG-groupoid with left identity (AG ∗∗ -groupoid), then the following are equivalent. (i) S is weakly regular. (ii) S is intra-regular.

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Proof. (i) =⇒ (ii) Let S be a weakly regular AG-groupoid with left identity (AG ∗∗ groupoid), then for any a ∈ S there exist x, y ∈ S such that a = (ax)(ay) and by Lemma 1.1, x = uv for some u, v ∈ S. Let vu = t ∈ S. Now by using (1.3), (1.1), (1.4) and (1.2), we have a = (ax)(ay) = (ya)(xa) = (ya)((uv)a) = (ya)((av)u) = (av)((ya)u) = (a(ya))(vu) = (a(ya))t = (y(aa))t = (ya2 )t. Thus S is intra-regular. (ii) =⇒ (i) It follows from Lemma 1.1, (1.2), (1.4), (1.1) and (1.3). Theorem 1.4. If S is an AG-groupoid (AG ∗∗ -groupoid), then the following are equivalent. (i) S is weakly regular. (ii) S is right regular. Proof. (i) =⇒ (ii) Let S be a weakly regular AG-groupoid (AG ∗∗ -groupoid), then for any a ∈ S there exist x, y ∈ S such that a = (ax)(ay) and let xy = t for some t ∈ S. Now by using (1.2), we have a = (ax)(ay) = (aa)(xy) = a2 t. Thus S is right regular. (ii) =⇒ (i) It follows from Lemma 1.1 and (2). Theorem 1.5. If S is an AG-groupoid with left identity (AG ∗∗ -groupoid), then the following are equivalent. (i) S is weakly regular. (ii) S is left regular. Proof. (i) =⇒ (ii) Let S be a weakly regular AG-groupoid with left identity (AG ∗∗ groupoid), then for any a ∈ S there exist x, y ∈ S such that a = (ax)(ay). Now by using (1.2) and (1.3), we have a = (ax)(ay) = (aa)(xy) = (yx)(aa) = (yx)a2 = ta2 , where yx = t for some t ∈ S. Thus S is left regular. (ii) =⇒ (i) It follows from Lemma 1.1, (1.3) and (1.2). Theorem 1.6. If S is an AG-groupoid with left identity (AG ∗∗ -groupoid), then the following are equivalent.


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(i) S is weakly regular. (ii) S is left quasi regular. Proof. The proof of this Lemma is straight forward, so is omitted. Theorem 1.7. If S is an AG-groupoid with left identity, then the following are equivalent. (i) S is (2, 2)-regular. (ii) S is completely regular. Proof. (i) =⇒ (ii) Let S be a (2, 2)-regular AG-groupoid with left identity, then for a ∈ S there exists x ∈ S such that a = (a2 x)a2 . Now a = (a2 x)a2 = ya2 , where a2 x = y ∈ S, and by using (1.3), we have a = (a2 x)(aa) = (aa)(xa2 ) = a2 z, where xa2 = z ∈ S. And by using (1.3), (1.1) and (1.4), we have a = (a2 x)(aa) = (aa)(xa2 ) = (aa)((ex)(aa)) = (aa)((aa)(xe)) = (aa)(a2 t) = ((a2 t)a)a = (((aa)t)a)a = (((ta)a)a)a = ((aa)(ta))a = ((at)(aa))a = (a((at)a))a = (ay)a, where xe = t ∈ S and (at)a = y ∈ S. Thus S is left regular, right regular and regular, so S is completely regular. (ii) =⇒ (i) Assume that S is a completely regular AG-groupoid with left identity, then for any a ∈ S there exist x, y, z ∈ S such that a = a(xa), a = a2 y and a = za2 . Now by using (1.1), (1.4) and (1.3), we have a = (ax)a = ((a2 y)x)(za2 ) = ((xy)a2 )(za2 ) = ((za2 )a2 )(xy) = ((a2 a2 )z)(xy) = ((xy)z)(a2 a2 ) = a2 (((xy)z)a2 ) = (ea2 )(((xy)z)a2 ) = (a2 ((xy)z))(a2 e) = (a2 ((xy)z))((aa)e) = (a2 ((xy)z))((ea)a) = (a2 ((xy)z))(aa) = (a2 t)a2 , where (xy)z = t ∈ S. This shows that S is (2, 2)-regular.

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Lemma 1.8. Every weakly regular AG-groupoid with left identity (AG ∗∗ -groupoid) is regular. Proof. Assume that S is a weakly regular AG-groupoid with left identity (AG ∗∗ -groupoid), then for any a ∈ S there exist x, y ∈ S such that a = (ax)(ay). Let xy = t ∈ S and t((yx)a) = u ∈ S. Now by using (1.1), (1.2), (1.3) and (1.4), we have a = (ax)(ay) = ((ay)x)a = ((xy)a)a = (ta)a = (t((ax)(ay)))a = (t((aa)(xy)))a = (t((yx)(aa)))a = (t(a((yx)a)))a = (a(t((yx)a)))a = (au)a. Thus S is regular. The converse of Lemma 1.8 is not true in general, as can be seen from the following example. Example 1.3. [9] Let us consider an AG-groupoid S = {1, 2, 3, 4} with left identity 3 in the following Cayley’s table.

. 1 2 3 4

1 2 2 1 1

2 2 2 2 2

3 4 2 3 1

4 4 2 4 2

Clearly S is regular, because 1 = (1.3).1, 2 = (2.1).2, 3 = (3.3).3 and 4 = (4.1).4, but S is not weakly regular, because 1 ∈ S is not a weakly regular element of S. Theorem 1.9. If S is an AG-groupoid with left identity (AG ∗∗ -groupoid), then the following are equivalent. (i) S is weakly regular. (ii) S is completely regular. Proof. (i) =⇒ (ii) It follows from Theorems 1.4, 1.5 and Lemma 1.8. (ii) =⇒ (i) It follows from Theorem 1.5. Lemma 1.10. Every strongly regular AG-groupoid with left identity (AG ∗∗ -groupoid) is completely regular.


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Proof. Assume that S is a strongly regular AG-groupoid with left identity (AG ∗∗ -groupoid), then for any a ∈ S there exists x ∈ S such that a = (ax)a and ax = xa. Now by using (1.1), we have a = (ax)a = (xa)a = (aa)x = a2 x. This shows that S is right regular and by Theorems 1.4 and 1.9, it is clear to see that S is completely regular. Note that a completely regular AG-groupoid need not to be a strongly regular AGgroupoid, as can be seen from the following example. Example 1.4. Let S = {a, b, c, d, e, f, g} be an AG-groupoid with the following multiplication table.

. a b c d e f g

a b e a d g c f

b d g c f b e a

c f b e a d g c

d a d g c f b e

e c f b e a d g

f e a d g c f b

g g c f b e a d

Clearly S is completely regular. Indeed, S is regular, as a = (a.e).a, b = (b.a).b, c = (c.d).c, d = (d.g).d, e = (e.c).e, f = (f.f ).f , g = (g.b).g, also S is right regular, as a = (a.a).f, b = (b.b).f, c = (c.c).f, d = (d.d).f, e = (e.e).f, f = (f.f ).f, g = (g.g).f, and S is left regular, as a = g.(a.a), b = d.(b.b), c = a.(c.c), d = e.(d.d), e = b.(e.e), f = f.(f.f ), g = c.(g.g), but S is not strongly regular, because ax ̸= xa for all a ∈ S. Theorem 1.11. In an AG-groupoid S with left identity (AG ∗∗ -groupoid), the following are equivalent. (i) S is weakly regular. (ii) S is intra-regular. (iii) S is right regular. (iv) S is left regular. (v) S is left quasi regular. (vi) S is completely regular. (vii) For all a ∈ S, there exist x, y ∈ S such that a = (xa)(ay).

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Proof. (i) =⇒ (ii) It follows from Theorem 1.3. (ii) =⇒ (iii) It follows from Theorems 1.3 and 1.4. (iii) =⇒ (iv) It follows from Theorems 1.4 and 1.5. (iv) =⇒ (v) It follows from Theorems 1.5 and 1.6. (v) =⇒ (vi) It follows from Theorems 1.6 and 1.9. (vi) =⇒ (i) It follows from Theorem 1.9. (ii) ⇐⇒ (vii) It follows from Theorem 1.2. Remark 1.12. Every intra-regular, right regular, left regular, left quasi regular and completely regular AG-groupoids with left identity (AG ∗∗ -groupoids) are regular. The converse of above is not true in general. Indeed, from Example 1.3, regular AG-groupoid with left identity is not necessarily intra-regular. Theorem 1.13. In an AG-groupoid S with left identity, the following are equivalent. (i) S is weakly regular. (ii) S is intra-regular. (iii) S is right regular. (iv) S is left regular. (v) S is left quasi regular. (vi) S is completely regular. (vii) For all a ∈ S, there exist x, y ∈ S such that a = (xa)(ay). (viii) S is (2, 2)-regular. Proof. (vi) ⇐⇒ (viii) It follows from Theorem 1.7. The rest of proof is same as proof in Theorem 1.11. Remark 1.14. (2, 2)-regular and strongly regular AG-groupoids with left identity are regular. The converse of above is not true in general, as can be seen from Example 1.4. Theorem 1.15. Regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2, 2)-regular and strongly regular AG ∗ -groupoids becomes semigroups. Proof. It follows from (1.6) and Lemma 1.1.


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