Abstract
One of the best approaches to study one type of algebraic structure is to connect it with other type of algebraic structure which is better explored. In this paper we have accomplished this aim by connecting \({\mathcal{AG }}\)-groupoids with some useful associative and commutative algebraic structures. We have also introduced a fully regular class of an \( {\mathcal{AG }}\)-groupoid and shown that an \({\mathcal{AG }}\)-groupoid \(\mathcal S \) with left identity is fully regular if and only if \(\mathcal{L=L }^{i+1}\), for any left ideal \(\mathcal L \) of \(\mathcal S \), where \(i=1,\ldots ,n\).
Highlights
The concept of an Abel-Grassmann’s groupoid (AG-groupoid) was first given by Kazim and Naseeruddin in 1972 [3] and they have called it a left almost semigroup (LA-semigroup)
Remark 2 An AG-groupoid S with left identity (AG **-groupoid) is regular if S is fully regular but the converse is not valid in general which can be followed from Example 6
Lemma 2 A non-empty subset A of a fully regular AG-groupoid S with left identity is a left ideal of S if and only if it is a right ideal of S
Summary
The concept of an Abel-Grassmann’s groupoid (AG-groupoid) was first given by Kazim and Naseeruddin in 1972 [3] and they have called it a left almost semigroup (LA-semigroup). In [2], the same structure is called a left invertive groupoid. Further if an AG-groupoid S contains a left identity, the following law holds [7]. An AG-groupoid is nonassociative and non-commutative in general, there is a close relation with semigroup as well as with commutative structures. It has been investigated in [7] that if an AG-groupoid contains a right identity, it becomes a commutative monoid. We see that AG-groupoids have very closed links with semigroups and vector spaces which make an AG -groupoid to be among the most interesting non-associative algebraic structure
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