Theory of Abel Grassmann's Groupoids

arXiv (Cornell University), 2014

A groupoid that satisfies the left invertive law: ab•c = cb • a is called an AG-groupoid. We extend the concept of left abelian distributive groupoid (LAD) and right abelian distributive groupoid (RAD) to introduce new subclasses of AG-groupoid, left abelian distributive AG-groupoid and right abelian distributive AG-groupoid. We give their enumeration up to order 6 and find some basic relations of these new classes with other known subclasses of AG-groupoids and other relevant algebraic structures. We establish a method to test an arbitrary AG-groupoid for these classes.

Afrika Matematika, 2012

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 AG-groupoids with some useful associative and commutative algebraic structures. We have also introduced a fully regular class of an AG-groupoid and shown that an AG-groupoid S with left identity is fully regular if and only if L = L i+1 , for any left ideal L of S, where i = 1,. .. , n.

Symmetry

The quasi-cancellativity of Abel Grassmann‘s groupoids (AG-groupoids) are discussed and two conjectures are partially solved. First, the following conjecture is proved to be true: every AG-3-band is quasi-cancellative. Moreover, a new notion of AG-(4,1)-band is proposed, and it is also proved that every AG-(4,1)-band is quasi-cancellative. Second, the notions of left (right) quasi-cancellative AG-groupoids and power-cancellative AG-groupoids are proposed, and the following results are obtained: for an AG*-groupoid or AG**-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative; for a power-cancellative and locally power-associative AG-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative. Finally, a general result is proved, that for any AG-groupoid, if it is left quasi-cancellative then it is right quasi-cancellative.

arXiv (Cornell University), 2014

An AG-groupoid is an algebraic structure that satisfies the left invertive law: (ab)c = (cb)a. We prove that the class of left transitive AG-groupoids (AG-groupoids satisfying the identity, ab • ac = bc) coincides with the class of T 2-AG-groupoids. We also develop a simple procedure to test whether an arbitrary groupoid is left transitive AG-groupoid or not. Further we prove that, (i). Every left transitive AG-groupoid is transitively commutative AG-groupoid (ii) For left transitive AG-groupoid the properties of flexibility, right alternativity, AG * , right nuclear square, middle nuclear square and commutative semigroup are equivalent. 1. introduction An AG-groupoid is an algebraic structure that satisfies the left invertive law, (ab)c = (cb)a[1]. This structure is a generalization of commutative semigroups. Every AG-groupoid satisfies the medial law, (ab) (cd) = (ac) (bd). An AG-groupoid S with the left identity element e is called an AG-monoid. Every AG-monoid is paramedial, (ab) (cd) = (db) (ca). Recently many researchers have been motivated to the field and a considerable work has been done on various aspects. Various classes have been discovered and enumerated up to order 6 [5, 6, 10, 11]. AG-groupoids have applications in the theory of flocks [1] and some in geometry[4]. A groupoid G satisfying the identity ac • bc = ab ∀a, b, c ∈ G is called right transitive groupoid [2], and G is called left transitive groupoid if it satisfying the identity ab • ac = bc[13]. In Section 2 we prove that nonassociative right transitive AG-groupoids do not exist. In Section 3 we prove the coincidence of the left transitive AG-groupoids with the class of T 2-AG-groupoids. We prove that every left transitive AG-groupoid is transitively commutative AGgroupoid. We also investigate that on what conditions some left transitive AG-groupoids become commutative semigroups. The following table contains various AG-groupoids with their defining identities that will be used in the rest of this article.

Bulletin of the Karaganda University Mathematics series, 2020

In this article, we have presented some important charcterizations of the ordered non-associative semigroups in relation to their ideals. We have initially characterized the ordered AG-groupoid through the properties of the their ideals, then we characterized the two important classes of these AG-groupoids, namely the regular and intragregular non-associative AG-groupoids. Our aim is also to encourage the research and the maturity of the associative algebraic structures by studying a class of non-associative and non-commutative algebraic structures called the ordered AG-groupoid.

Advances in Mathematics, 2010

Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universalétale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C * -algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality.

2009

In the present paper we have studied the concept of fuzzification in AG-groupoids. The equivalent statement for an AG-groupoid to be a commutative semigroup is proved. Fuzzy points have been defined in an AG-groupoid and has been shown the representation of smallest fuzzy left ideal generated by a fuzzy point. The set of all fuzzy left ideals, which are idempotents,

2011

An AG-groupoid is a non-associative groupoid in general in which the identity (ab)c = (cb)a holds. In this paper we study some structural properties of AG-groupoids with respect to the cancellativity. We prove that cancellative and non-cancellative elements of an AG-groupoid S partition S and the two classes are AG-subgroupoids of S if S has left identity e. Cancellativity and invertibility coincide in a finite AG-groupoid S with left identity e. For a finite AG-groupoid S with left identity e having at least one non-cancellative element, the set of non-cancellative elements form a maximal ideal. We also prove that for an AG-groupoid S, the conditions (i) S is left cancellative (ii) S is right cancellative (iii) S is cancellative, are equivalent.

Zenodo (CERN European Organization for Nuclear Research), 2023

An AG-groupoid is the midway between commutative semigroup and groupoid. The core structure of Flock theory is an AG-groupoid, which focuses on motion replication and distance optimization and has numerous applications in physics and biology. Unfortunately, in many cases, modelling real-world problems in domains like computer science, operations research, artificial intelligence, control engineering, and robotics can be risky. Different theories, such as fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and others, have been created to deal with similar situations. In this paper, We define the notions of neutrosophic κ-ideal structures in an AG-groupoid and investigate their properties. We also obtain equivalent assertion of neutrosophic κ-ideals and product of neutrosophic κ-structures in AG-groupoid.

International Journal of Mathematics and Mathematical Sciences, 2001

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 structure 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.