Groupoid identities common to four abelian group operations

TL;DR

This paper proves that a specific variety of medial groupoids generated by four abelian group operations has a finite basis of identities, including the medial law and five others, confirming a conjecture and providing bases for related subvarieties.

Contribution

It establishes a finite basis for a natural variety of medial groupoids generated by four abelian group operations, advancing understanding of their algebraic identities.

Findings

The variety is finitely based with a specific set of identities.

Finite bases are provided for subvarieties generated by subsets of the four groupoids.

The proof supports a conjecture about finite basis property, despite its difficulty.

Abstract

We exhibit a finite basis M for a certain variety $\mathbf{V}$ of medial groupoids. The set M consists of the medial law (xy)(zt)=(xz)(yt) and five other identities involving four variables. The variety $\mathbf{V}$ is generated by the four groupoids $\pm x\pm y$ on the integers. Since $\mathbf{V}$ is a very natural variety, proving it to be finitely based should be of interest. In an earlier paper, we made a conjecture which implies that $\mathbf{V}$ is finitely based. In this paper, we show that $\mathbf{V}$ is finitely based by proving that M is a basis. Based on our proof, we think that our conjecture will be difficult to prove. We used four medial groupoids to define $\mathbf{V}$. We also present a finite basis for the variety generated by any proper subset of these four groupoids. In an earlier paper with R. Padmanabhan, we gave the corresponding finite bases when the constant zero is allowed.