Abstract:First, we provide an introduction to the theory and algorithms for noncommutative Gr\"obner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.
Keywords: free associative algebras; Gr\"obner bases; composition (diamond) lemma; universal associative envelopes; Lie algebras and triple systems; PBW theorem; Jordan algebras and triple systems; trilinear operations; computer algebra
AMS Subject Classification: 16S10 16S30 16W10 16Z05 17A30 17A40 17A42 17B35 17B60 17C50 68W30