The spectrum of the variety of anti-rectangular Abel Grassmann bands

TL;DR

This paper characterizes the finite and countable anti-rectangular Abel Grassmann bands, showing their orders are powers of four and establishing their unique isomorphism properties.

Contribution

It proves the set of all finite anti-rectangular Abel Grassmann bands' orders are powers of four and establishes their unique isomorphism characteristics.

Findings

Orders of finite algebras are powers of four

Unique countable anti-rectangular Abel Grassmann band exists

Countable band is isomorphic to a proper subgroupoid

Abstract

We prove that the set of all orders of finite algebras in the groupoid variety of anti-rectangular Abel Grassmann bands consists of all powers of four. We also prove that any groupoid anti-isomorphic to a finite or countable anti-rectangular Abel Grassmann band G is isomorphic to G. It is proved that within isomorphism there is only one countable anti-rectangular Abel Grassmann band and that it is isomophic to a proper subgroupoid of itself.