Enumeration of Bi-commutative AG-groupoids

A groupoid satisfying the left invertive law: $ab\cdot c=cb\cdot a$ is called an AG-groupoid and is a generalization of commutative semigroups. We consider the concept of bi-commutativity in AG-groupoids and thus introduce left commutative AG-groupoids, right commutative AG-groupoids and bi-commutative AG-groupoids.Comment: Though the old version had good results but had some serious mistakes in Theorem 1 and some misprints. So it is revise

Some General Properties of LAD and RAD AG-groupoids

A groupoid that satisfies the left invertive law: $ab\cdot c=cb\cdot a$ 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.Comment: 10 page

Left Transitive AG-groupoids

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 T2-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AG*, right nuclear square, middle nuclear square and commutative semigroup are equivalent.Comment: This paper has been withdrawn by the author(s) due to crucial errors in the table on page

AG-groups and other classes of right Bol quasigroups

By a result of Sharma, right Bol quasigroups are obtainable from right Bol loops via an involutive automorphism. We prove that the class of AG-groups, introduced by Kamran, is obtained via the same construction from abelian groups. We further introduce a new class of Bol* quasigroups, which turns out to correspond, as above, to the class of groups. Sharma's correspondence allows an efficient implementation and we present some enumeration results for the above three classes

Linear representation of Abel-Grassmann groups

We describe an linear representation for Abel-Grassmann groups. As a consequence, we obtain or improve many previous results. In particular, enumeration of Abel-Grassmann groups up to isomorphism is obtained for orders <512

The Multiplication Group of an AG-group

We investigate the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: the multiplication group of an AG-group of order n is non-abelian group of order 2n and its left section is an abelian group of order n. The inner mapping group of an AG-group of any order is a cyclic group of order 2.Comment: 9 page

The zeta function of a finite category

We define the zeta function of a finite category. And we propose a conjecture which states the relationship between the Euler characteristic of finite categories and the zeta function of finite categories. This conjecture is verified when categories are finite groupoids, finite acyclic categories, categories with 2-objects and finite categories satisfying certain condition

Computing invariants via slicing groupoids: Gel'fand MacPherson, Gale and positive characteristic stable maps

We offer a groupoid-theoretic approach to computing invariants. We illustrate this approach by describing the Gel'fand-MacPherson correspondence and the Gale transform as well as giving Zariski-local descriptions of the moduli space of ordered points in P^1. We give an explicit description of the moduli space M_0(P^1,2) over Spec Z. In characteristic 2, there is a singularity at the totally ramified cover which is isomorphic to the affine cone over the Veronese embedding P^1 --> P^4

Amalgamation and Symmetry: From Local to Global Consistency in The Finite

Amalgamation patterns are specified by a finite collection of finite template structures together with a collection of partial isomorphisms between pairs of these. The template structures specify the local isomorphism types that occur in the desired amalgams; the partial isomorphisms specify local amalgamation requirements between pairs of templates. A realisation is a globally consistent solution to the locally consistent specification of this amalgamation problem. This is a single structure equipped with an atlas of distinguished substructures associated with the template structures in such a manner that their overlaps realise precisely the identifications induced by the local amalgamation requirements. We present a generic construction of finite realisations of amalgamation patterns. Our construction is based on natural reduced products with suitable groupoids. The resulting realisations are generic in the sense that they can be made to preserve all symmetries inherent in the specification, and can be made to be universal w.r.t. to local homomorphisms up to any specified size. As key applications of the main construction we discuss finite hypergraph coverings of specified levels of acyclicity and a new route to the lifting of local symmetries to global automorphisms in finite structures in the style of Herwig-Lascar extension properties for partial automorphisms.Comment: A mistake in the proposed construction from [arXiv:1211.5656], cited in Theorem 3.21, was discovered by Julian Bitterlich. This version relies on the new approach to this construction as presented in the new version of [arXiv:1806.08664

The Ihara Zeta function for infinite graphs

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and percolation graphs. Making use of Connes' non-commutative integration theory we construct a Zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara Zeta function via the normalized version on finite graphs in the sense of Benjamini-Schramm follows as a special case. Our framework not only unifies corresponding earlier results occurring in the literature. It likewise provides extensions to rich new classes of objects such as percolation graphs