On Structural Properties Of AG-groupoids And Their Generalizations

The main motivation behind our work is to study different structural properties of a non-associative structure called an Abel-Grassmann’s groupoid (briefly an AG-groupoid) as it hasn’t attracted much attention compared to associative structures. It is often important to use the behavior and character of one algebraic structure to study another for the sake of having more and better informative results. This thesis consists of several problems in the theory of AG-groupoids with the common feature that they are all best tackled using semi-group theory.The thesis comprises seven chapters. The first chapter contains a brief history and com-parison of semigroups and AG-groupoids. Further, we have established several conditions for an AG-groupoid to become a semigroup (commutative semigroup, commutative monoid). We have also generated AG-groupoids from some abelian groups.In chapter 2, the concept of soft inverses in an AG**-groupoid is introduced. The system known as partially inverse AG**-groupoid has been defined. Some basic properties of par-tially inverse AG**-groupoids are investigated. In particular, AG**-groupoids whose idem-potents form a semilattice or a rectangular band are studied. Some special congruences on partially inverse AG**-groupoids are considered, such as unitary congruences and idempo-tent separating congruences. We have studied separative commutative image and maximal anti-separative commutative image of a locally associative AG**-groupoid.In chapter 3, we have given a new definition for a completely inverse AG**-groupoid and characterized it by using strong inverses. We have looked at the problems of finding some useful results for a completely inverse AG**-groupoid. We have established a condi-tion to connect a completely inverse AG**-groupoid with sandwich sets. We have defined a natural partial order relation on the set of idempotents E of an AG**-groupoid and find a greatest lower bound of E. We have also introduced the concept of a completely AMnverse AG**-groupoid and studied some of the fundamental properties. Furthermore, we have in-vestigated the maximum idempotent separating congruence in a completely N-inverse AG**-groupoid.In chapter 4, we have shown that the concepts of strongly regular and intra-regular classes coincide in a unitary AG-groupoid. Further, we have shown that every strongly regular element of an AG**-groupoid has a strong inverse. We have provided a condition for a strongly regular AG-groupoid to become an AG**-groupoid. We have characterized a strongly regular AG-groupoid in terms of left (right) ideals. We have also investigated some useful properties of a completely regular AG-groupoid.In chapter 5, we have introduced the concept of (m, n)-ideals in an ordered AG-groupoid. We have characterized (0,2)-ideals and (1,2)-ideals of an ordered AG-groupoid in term of left ideals. The results obtained extend the results on an AG-groupoid without order.In chapter 6, we have given the concept of pure left identity and studied an (m, n)-regular class of an AG-hypergroupoid. We have also characterized an AG-hypergroupoid. Further, we have given the connection of ordered and hyper theories of an AG-groupoid.In chapter 7, we have shown that a Γ-AG-groupoid with left identity becomes an AG-groupoid with left identity. Finally, we have also given a method to construct Γ-AG-groupoid.