An application of lattice basis reduction to polynomial identities for algebraic structures

The authors’ recent classification of trilinear operations includes, among other cases, a fourth family of operations with parameter q∈Q∪{∞}, and weakly commutative and weakly anticommutative operations. These operations satisfy polynomial identities in degree 3 and further identities in degree 5. For each operation, using the row canonical form of the expansion matrix E to find the identities in degree 5 gives extremely complicated results. We use lattice basis reduction to simplify these identities: we compute the Hermite normal form H of Et, obtain a basis of the nullspace lattice from the last rows of a matrix U for which UEt = H, and then use the LLL algorithm to reduce the basis.