  These are the lecture notes from my short course of the same title at the
CIMPA Research School on Associative and Nonassociative Algebras and
Dialgebras: Theory and Algorithms - In Honour of Jean-Louis Loday (1946-2012),
held at CIMAT, Guanajuato, Mexico, February 17 to March 2, 2013. The underlying
motivation is to apply the theory of noncommutative Grobner bases in free
associative algebras to the construction of universal associative envelopes for
nonassociative structures defined by multilinear operations. Trilinear
operations were classified by the author and Peresi in 2007. In her Ph.D.
thesis of 2012, Elgendy studied the universal associative envelopes of
nonassociative triple systems obtained by applying these trilinear operations
to the 2-dimensional simple associative triple system. In these notes I use
computer algebra to extend some aspects of her work to the 4-dimensional and
6-dimensional simple associative triple systems.
