  We take an algorithmic and computational approach to a basic problem in
abstract algebra: determining the correct generalization to dialgebras of a
given variety of nonassociative algebras. We give a simplified statement of the
KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial
identities for algebras to corresponding identities for dialgebras. We apply
the KP algorithm to the defining identities for Jordan triple systems to obtain
a new variety of nonassociative triple systems, called Jordan triple disystems.
We give a generalized statement of the BSO algorithm introduced by Bremner and
Sanchez-Ortega for extending multilinear operations in an associative algebra
to corresponding operations in an associative dialgebra. We apply the BSO
algorithm to the Jordan triple product and use computer algebra to verify that
the polynomial identities satisfied by the resulting operations coincide with
the results of the KP algorithm; this provides a large class of examples of
Jordan triple disystems. We formulate a general conjecture expressed by a
commutative diagram relating the output of the KP and BSO algorithms. We
conclude by generalizing the Jordan triple product in a Jordan algebra to
operations in a Jordan dialgebra; we use computer algebra to verify that
resulting structures provide further examples of Jordan triple disystems. For
this last result, we also provide an independent theoretical proof using Jordan
structure theory.
