Groupoid identities common to four abelian group operations

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.