Groupoid identities common to four abelian group operations

We exhibit a finite basis M for a certain variety V of medial groupoids. The setM consists of the medial law (xy)(zt) = (xz)(yt) and five other identities involving four variables. The variety V is generated by the four groupoids ±x± y on the integers. Since 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 V is finitely based. In this paper, we show that 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. As we explain in the paper, the variety V corresponds to the Klein 4-group. We use this group to show that V has a basis consisting of interchange laws. (We define “interchange law” in the introduction.) We give more examples of finite groups where such a basis exists for the corresponding groupoid variety. We also give examples of finite groups where such a basis is impossible. The second case is a further challenge to anyone who tries to prove