Identities for generalized Lie and Jordan products on totally associative triple systems
The authors determine the identities of degree \(\leq 9\) satisfied by the new ternary operations \(abc+bca+cab+acb+cba+bac\) (symmetric sum), \(abc+bca+cab-acb-cba-bac\) (alternating sum), and \(abc+bca+cab\) (cyclic sum) on every triple system satisfying the total associativity identities \((abc)de=a(bcd)e=ab(cde)\). The first two operations are ternary analogs of the Lie and Jordan products, and the identities they satisfy may be regarded as ternary version of the Jacobi and Jordan identities. The authors did computations on computer using arithmetic modulo 103. For this reason, it is theoretically possible that there exist more identities, but these new identities should have all the integral coefficients congruent to zero modulo 103.
identities, triple systems, ternary Lie product
scientific article; zbMATH DE number 1525279