IJMMS 27:6 (2001) 387–389 PII. S0161171201010997 http://ijmms.hindawi.com
© Hindawi Publishing Corp.
FINITE AG-GROUPOID WITH LEFT IDENTITY AND LEFT ZERO
QAISER MUSHTAQ and M. S. KAMRAN (Received 3 October 2000)
Abstract.A groupoidGwhose elements satisfy the left invertive law:(ab)c=(cb)ais known as Abel-Grassman’s groupoid (AG-groupoid). It is a nonassociative algebraic struc- ture midway between a groupoid and a commutative semigroup. In this note, we show that ifGis a finite AG-groupoid with a left zero then, under certain conditions,Gwithout the left zero element is a commutative group.
2000 Mathematics Subject Classification. 20N99.
1. Preliminaries. An Abel-Grassman’s groupoid [6], abbreviated as AG-groupoid, is a groupoidG whose elements satisfy the left invertive law:(ab)c=(cb)a. It is also called a left almost semigroup [2,3,4,5]. In [1], the same structure is called left invertive groupoid. In this note we call it AG-groupoid.
It is a nonassociative algebraic structure midway between a groupoid and a com- mutative semigroup. The structure is medial [5], that is,(ab)(cd)=(ac)(bd)for all a,b,c,d∈G. It has been shown in [5] that if an AG-groupoid contains a left identity then it is unique. It has been proved also that an AG-groupoid with right identity is a commutative monoid, that is, a semigroup with identity element. An elementa0of an AG-groupoidGis called a left (right) zero ifa0a=a0(aa0=a0)for alla∈G.
Leta,b,c, anddbelong to an AG-groupoid with left identity andab=cd. Then it has been shown in [5] thatba=dc.
An element a−1 of an AG-groupoid with left identitye is called a left inverse if a−1a=e. It has been shown in [5] that ifa−1is a left inverse ofathen it is unique and is also the right inverse ofa.
If for alla,b,cin an AG-groupoidG,ab=acimplies thatb=c, thenGis known as left cancellative. Similarly, ifba=ca, implies thatb=c, thenGis called right can- cellative. It is known [5] that every left cancellative AG-groupoid is right cancellative but the converse is not true. However, every right cancellative AG-groupoid with left identity is left cancellative.
In this note, we show that ifGis a finite AG-groupoid with left identity and a left zeroa0, under certain conditionsG\{a0}is a commutative group without a left zero.
2. Results. We need the following theorem from [4] for our main result.
Theorem 2.1 [4]. A cancellative AG-groupoid G is a commutative semigroup if a(bc)=(cb)afor alla,b,c∈G.
388 Q. MUSHTAQ AND M. S. KAMRAN We now state and prove our main result.
Theorem2.2. Let(G,◦)be a finite AG-groupoid with at least two elements. Suppose that it contains a left identity and a left zeroa0. ThenG0=G\{a0}is a commutative group under the binary operation(◦)provided there is another binary operation(∗) such that
(i) (G,∗)is an AG-groupoid with left identity and left inverses, (ii) a0∗a=a, for alla∈G,
(iii) (a∗b)◦c=(a◦c)∗(b◦c), for alla, b, c∈G,
(iv) a◦b=a0implies that eithera=a0orb=a0for alla, b∈G, (v) a◦(b◦c)=(c◦b)◦a, for alla,b,c∈G.
Proof. Suppose thatG= {a0, a1, . . . , am}, wheremis a positive integer, is an AG- groupoid with left identity under the binary operation(◦). Letebe the identity element ofG. It is certainly different froma0 because of (ii) and becausea0is the left zero under(◦). The left invertive law together with (iv) implies that(a◦a0)◦e=(e◦a0)◦a= a0◦a=a0, wheree≠a0. That is,
a0◦a=a◦a0=a0. (2.1)
Now consider the subsetG0ofG which is obtained from it by deletinga0, so that G0 = {ai: i=1,2, . . . , m}. In view of the facts that a0 is a zero under the binary operation(◦)and it is the left identity under(∗)and that(G,◦)is a finite AG-groupoid with left identity.(G0,◦)is also a finite AG-groupoid with left identity having the same eas the left identity in which all elements are distinct.
We now examine whether an elementaofG0has an inverse inG0under(◦)or not.
We construct a setHk= {ak◦a1, ak◦a2, . . . , ak◦am}, whereak≠a0. Ifak=a0, then becausea0is a left zero inGunder(◦)and the left identity under(∗), the ultimate form of the setHkwill be{a0}. Therefore it validates our supposition thatak≠a0.
We assert thatHkcontainsmelements. Suppose otherwise and let
ak◦ar=ak◦as, (2.2)
for some r , s=1,2, . . . , m andr ≠s. Since Hk is an AG-groupoid with left identity under(◦), therefore (2.2) implies that
ar◦ak=as◦ak, (2.3)
for somer , s=1,2, . . . , mandr≠s. Consider now the element(as∗a−1r )◦ak, which is certainly an element ofG, wherea−r1is the left inverse ofar inGwith respect to (∗). Now,
as∗a−1r
◦ak= as◦ak
∗
a−1r ◦ak
=
ar◦ak
∗
a−1r ◦ak
=
ar∗a−r1
◦ak=a0◦ak=a0. (2.4) Because of (iii), equation (2.3) and the facts thata−1r is the inverse ofar under(∗).
Thus(as∗a−1r )◦ak=a0. Sinceak≠a0, therefore because of (iv),as∗a−1r =a0. Next (as∗a−r1)◦ar=a0∗arimplies that(as∗a−r1)◦ar=arbecausea0is the left identity inGunder(∗). Hence,ar=(as∗a−1r )∗ar =(ar∗a−1r )∗as=a0∗as=as, that is, ar=as. Since|Hk| =m, therefore the resultar=ascontradicts our assumption; thus
FINITE AG-GROUPOID WITH LEFT IDENTITY AND LEFT ZERO 389 proving thatHkcontains distinct elements. SinceHkis contained inG0and|G0| =m we haveHk=G0.
Also, sinceG0is an AG-groupoid under (◦)with the left identitye, so is Hk and henceHkcontains the left identitye. So,ewill be of the formai◦aj, that is,e=ai◦aj
implying thatai is the left inverse ofaj under the binary operation (◦). But in an AG-groupoid with left identity, if it contains left inverses, every left inverse is a right inverse. Thusajis the right inverse ofajunder(◦).
Sincek=1,2, . . . , mhas been chosen arbitrarily, we have shown thatG0is an AG- groupoid with left identity and inverses under the binary operation(◦).
If ai, aj, ak∈G0such that ai◦ak=aj◦ak, then (ai◦ak)◦a−k1=(aj◦ak)◦a−k1 implies that (a−1k ◦ak)◦ai=(a−1k ◦ak)◦aj and so ai =aj. ThusG0 is right can- cellative under(◦). ButG0being right cancellative under(◦), is left cancellative also, thereforeG0is cancellative. SinceG0is cancellative whose elements satisfy condition (v), therefore by applyingTheorem 2.1, we conclude thatG0is a commutative group under(◦).
Corollary2.3. If(G,◦)is a finite AG-groupoid with left identity and a left zeroa0, then(G\{a0},◦)is a cancellative AG-groupoid with left identity and inverses provided there is another binary operation(∗)such that
(i) (G,∗)is an AG-groupoid with left identity and left inverses, (ii) a0∗a=a, for alla∈G,
(iii) (a∗b)◦c=(a◦c)∗(b◦c), for alla, b, c∈G,
(iv) a◦b=a0implies that eithera=a0orb=a0for alla,b∈G.
Proof. The proof is analogous to the proof ofTheorem 2.2.
Acknowledgement. The authors are grateful to the referee for his invaluable suggestions.
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Qaiser Mushtaq: Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
E-mail address:[email protected]
M. S. Kamran: Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
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