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Rectangular groupoids and related structures.

¹Institute for Algebra, Johannes Kepler Universität Linz, Altenbergerstrasse 69, A-4040 Linz, Austria.

Discrete Mathematics

|July 10, 2013

Summary

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Researchers introduce rectangular groupoids, a novel algebraic structure generalizing rectangular semigroups. These groupoids exhibit strong connections to isotopy and are constructed using arrays, matrices, and graphs.

Area of Science:

Abstract Algebra
Universal Algebra
Combinatorial Structures

Background:

Groupoids are algebraic structures with a single binary operation.
Rectangular semigroups and central groupoids are specific types of groupoids.
Understanding generalizations of known algebraic structures is crucial for advancing the field.

Purpose of the Study:

To introduce and define a new class of algebraic structures called rectangular groupoids.
To explore the combinatorial structures related to rectangular groupoids.
To investigate the relationship between rectangular groupoids, isotopy, and group constructions.

Main Methods:

Defining a quasivariety of groupoids based on a specific implication.
Exploring combinatorial representations using arrays, matrices, and graphs.

Keywords:

Dualities General algebra Groupoids Isotopy Matrix identities Path property Quasivarieties Subvarieties Transversals

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Analyzing connections to isotopy and group theory.

Main Results:

The defined quasivariety, termed rectangular groupoids, generalizes existing structures.
Three related combinatorial structures (arrays, matrices, graphs) are identified.
The quasivariety generates the variety of all groupoids, satisfying no nontrivial equations.
Strong connections with isotopy are established, with rectangular groupoids being closed under isotopy.

Conclusions:

Rectangular groupoids offer a new perspective on algebraic structures and their combinatorial counterparts.
The existence of a regular automorphism is equivalent to the groupoid being derived from a group via a Cayley graph construction.
This research contributes to the understanding of algebraic structures and their diverse applications.