Linear groupoids and the associated wreath products

A groupoid identity is said to be linear of length 2k if the same k variables appear on both sides of the identity exactly once. We classify and count all varieties of groupoids defined by a single linear identity. For k=3, there are 14 nontrivial varieties and they are in the most general position with respect to inclusion. Hentzel et al. [Hentzel, I.R., Jacobs, D.P., Muddana, S.V., 1993. Experimenting with the identity (xy)z=y(zx). J. Symbolic Comput. 16, 289–293] showed that the linear identity (xy)z=y(zx) implies commutativity and associativity in all products of at least five factors. We complete their project by showing that no other linear identity of any length behaves this way, and by showing how the identity (xy)z=y(zx) affects products of fewer than five factors; we include distinguishing examples produced by the finite model builder Mace4. The corresponding combinatorial results for labelled binary trees are given. We associate a certain wreath product with any linear identity. Questions about linear groupoids can therefore be transferred to groups and attacked by group-theoretical computational tools, e.g., GAP. Systematic notation and diagrams for linear identities are devised. A short equational basis for Boolean algebras involving the identity (xy)z=y(zx) is presented, together with a proof produced by the automated theorem prover OTTER.