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These are the lecture notes from my short course of the same title at theCIMPA Research School on Associative and Nonassociative Algebras andDialgebras: Theory and Algorithms - In Honour of Jean-Louis Loday (1946-2012),held at CIMAT, Guanajuato, Mexico, February 17 to March 2, 2013. The underlyingmotivation is to apply the theory of noncommutative Grobner bases in freeassociative algebras to the construction of universal associative envelopes fornonassociative structures defined by multilinear operations. Trilinearoperations were classified by the author and Peresi in 2007. In her Ph.D.thesis of 2012, Elgendy studied the universal associative envelopes ofnonassociative triple systems obtained by applying these trilinear operationsto the 2-dimensional simple associative triple system. In these notes I usecomputer algebra to extend some aspects of her work to the 4-dimensional and6-dimensional simple associative triple systems.

Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures