Linear groupoids and the associated wreath products
| dc.creator | Phillips, J. D. | |
| dc.creator | Vojtěchovský, Petr | |
| dc.date | 2007-01-24 | |
| dc.date.accessioned | 2026-07-07T07:42:53Z | |
| dc.date.available | 2026-07-07T07:42:53Z | |
| dc.description | 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. 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. | |
| dc.description | 19 pages | |
| dc.identifier | https://arxiv.org/abs/math/0701713 | |
| dc.identifier | http://arxiv.org/abs/math/0701713 | |
| dc.identifier | Journal of Symbolic Computation 40 (2005), no. 3, 1106-1125 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/122613 | |
| dc.subject | Group Theory | |
| dc.subject | 20N05, 18B40, 20B40 | |
| dc.title | Linear groupoids and the associated wreath products | |
| dc.type | text |