A Note on Abel-Grassmann ' s Groupoids

It is also called a left almost semigroup (Kazim and Naseeruddin, 1972; Mushtaq and Iqbal, 1990). In Holgate (1992) it is called a left invertive groupoid. In this study we shall call it an AG-groupoid. It has been shown in Mushtaq and Yusuf (1978) that if an AGgroupoid contains a left identity then the left identity 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. It is a useful nonassociative algebraic structure, midway between a groupoid and a commutative semigroup. An AG-groupoid S is medial (Kazim and Naseeruddin, 1972) that is:


INTRODUCTION
An Abel-Grassmann's groupoid (Protić and Stevanović, 2004), abbreviated as an AG-groupoid, is a groupoid S whose elements satisfy the invertive law: . , , all for , ) ( It is also called a left almost semigroup (Kazim and Naseeruddin, 1972;Mushtaq and Iqbal, 1990).In Holgate (1992) it is called a left invertive groupoid.In this study we shall call it an AG-groupoid.It has been shown in Mushtaq and Yusuf (1978) that if an AGgroupoid contains a left identity then the left identity 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.It is a useful nonassociative algebraic structure, midway between a groupoid and a commutative semigroup.
An AG-groupoid S is medial (Kazim and Naseeruddin, 1972) An AG-groupoid is called an AG-band if all its elements are idempotents.
In Stevanović and Protić (2004) Connection discussed above make this non-associative structure interesting and useful.

PRELIMINARIES
Here we construct AG-groupoids by defining new operations on vector spaces over finite fields.AGgroupoids constructed from finite fields are very interesting.It is well known that a multiplicative group of a finite field is a cyclic group generated by a single element.By using these generators we have drawn the Cayley diagrams for such AG-groupoids which have been constructed from finite fields.The diagrams are either bi-partite (that is, their vertices can be colored by using two minimum colors) or tri-partite (that is, they can be colored using three minimum colors).
Here we begin with the examples of AG-groupoids having n o left identity.
Following is an example of an AG-groupoid with the left identity.
, the binary operation be defined on S as follows:

Then
) , ( ⋅ S is an AG-groupoid with left identity 7. A graph G is a finite non-empty set of objects called vertices (the singular is vertex) together with a (possibly empty) set of unordered pairs of distinct vertices of G called edges.The vertex set is denoted by V (G), while the edge set is denoted by E (G).
A graph G is connected if every two of its vertices are connected.A graph G that is not connected is disconnected.A graph is planar if it can be embedded in the plane.
A directed graph or digraph D is a finite non-empty set of objects called vertices together with a (possibly empty) set of ordered pairs of distinct vertices of D called arcs or directed edges.
(called partite sets) such that every element of , such graphs are called bi-partite graphs.
Theorem 1: Let W be a sub-space of a vector space V over a field F of cardinal 2r such that r>1.Define the binary operation o on W as follows: Proof: Clearly W is closed.Next we will show that W satisfies left invertive law: From ( 3) and ( 4 ., some for , so , and , Remark 1: An AG-groupoid ) , ( o W is referred to as an AG-groupoid defined by the vector space ) , , ( + ⋅ V .
Remark 2: If we take , taking α as a generator of F and cardinal of F is 2r, then ) , ( o F is said to be an AG-groupoid defined by Galois field.
An element a of an AG-groupoid S is called an idempotent if and only if 2 a a = .
An AG-groupoid is called AG-band if all its elements are idempotents.

CAYLEY DIAGRAMS
A Cayley graph (also known as a Cayley colour graph and named after A. Cayley), is a graph that encodes the structure of a group.
Specifically, let | be a presentation of the finitely generated group G with generators X and relations R. We define the Cayley graph ) , That is, the vertices of the Cayley graph are precisely the elements of G and two elements of G are connected by an edge if and only if some generator in X transfers the one to the other.He has proposed the use of colors to distinguish the edges associated with different generators.
Remark 3: If we put the value of 2 = r , in remark 2, we get Galois field of order 4.
Further we need to construct a field of 4 elements, for this take an irreducible polynomial 1 The table of this field is given by: , and F t ∈ = α , we get the following table of an AG-groupoid: We can draw the Cayley diagram for it as under, which is a tri-partite, planar disconnected graph (Fig. 1).
Theorem 2: Let W be a sub-space of a vector space V over a field F of cardinal n p for some prime 2 ≠ p .
Define the binary operation ⊗ on W as follows: , which is a bi-partite, planar disconnected graph (Fig. 2).

Fig. 1 :
Fig. 1: Tri-pertite and planar graph is an AG-groupoid with left identity 0.Proof: Clearly W is closed.Next we will show that W satisfies left invertive law: