Using Mathematica to Explore Abstract Algebra

Discrete Mathematics, 2013

The quasivariety of groupoids (N, * ) satisfying the implication a * b = c * d ⇒ a * d = c * b = a * b generalises rectangular semigroups and central groupoids. We call them rectangular groupoids and find three combinatorial structures based upon arrays, matrices and graphs that are closely related.

Enumeration and classification of mathematical entities is an important part of mathematical research in particular in finite algebra. For algebraic structures that are more general than groups this task is often only feasible by use of computers due to the sheer number of structures that have to be considered. In this paper we present the enumeration and partial classification of AG-groupoids — groupoids in which the identity (ab)c = (cb)a holds — of up to order 6. The results are obtained with the help of the computer algebra system GAP and the constraint solver Minion by making use of both algebraic techniques as well as search pruning via symmetry breaking.

Information Processing Letters, 1972

Proceedings of the London Mathematical Society, 1972

Applied general topology, 2021

The aim of this paper is to obtain a group-2-groupoid as a 2-groupoid object in the category of groups and also as a special kind of an internal category in the category of group-groupoids. Corresponding group2-groupoids, we obtain some categorical structures related to crossed modules and group-groupoids and prove categorical equivalences between them. These results enable us to obtain 2-dimensional notions of group-groupoids. 2010 MSC: 20L05; 18D05; 18D35; 20J15.

2010

In this paper we study properties of left (right) division (cancellative) groupoids with associative-like identities, namely, with cyclic associative identity (x • (y • z) = (z • x) • y) and Tarki (x • (z • y) = (x • y) • z) identities.

Semigroup Forum, 2013

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state and prove precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, as well as for their homomorphisms, by means of Leech's cohomology groups of monoids.

Study of algebraic structures built using [0, n) happens to be one of an interesting and innovative research. Here in this book authors define non associative algebraic structures using the interval [0, n).

Symmetry

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum me...

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.