On Free Partially Associative Triple Systems

A triple system is partially associative (by deenition) if it satisses 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 coeecients 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 diicult case (degree 11) requires nding the row-reduced form over Z of a matrix of size 364273. These computations were done with Maple V.4 on a Sun workstation. PRELIMINARIES A triple system is a Z-module A together with a Z-module homomorphism t: A Z A Z A ?! A: We will usually write abc instead of t(abc). There are two distinct generalizations of the familiar (binary) associative identity (ab)c ? a(bc) 0 to the ternary case: total associativity, deened by the identities (abc)de ? a(bcd)e 0; a(bcd)e ? ab(cde) 0; and partial associativity, deened by the identity I(a; b; c; d; e) := (abc)de + a(bcd)e + ab(cde) 0: We call I(a; b; c; d; e) the ternary partial associator. (Note that monomials in a triple system must have odd degree). For more information on total and partial associativity in k-ary rings, see the work of Gnedbaye G1{3]. It is easy to motivate consideration of totally associative triple systems: the total associativity identities imply (by a straightforward extension of the inductive argument from the binary case) that any product a 1 a 2m+1 (m 2) depends only on the order of the factors, not the placement of parentheses. There is also an