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. [3] showed that the linear identity (xy)z = y(zx) implies commutativity and associativity in all products of at least 5 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 5 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.