On free partially associative triple systems

A triple system is partially associative (by definition) if it satisfies the identity (abc)de + a(bcd)e + ab(cde) ≡ 0. This paper presents a computational study of the free partially associative triple system on one generator with coefficients in the ring Z of integers. In particular, the Z-module structure of the homogeneous submodules of (odd) degrees ≤ 11 is determined, together with explicit generators for the free and torsion components in degrees ≤ 9. Elements of additive order 2 exist in degrees ≥ 7, and elements of additive order 6 exist in degrees ≥ 9. The most difficult case (degree 11) requires finding the row-reduced form over Z of a matrix of size 364 × 273. These computations were done with Maple V.4 on a Sun workstation.