Multigraded algebras and non-associative Gröbner bases

The basic notions of a theory of Gröbner bases for ideals in the non-associative, noncommutative algebras K{X} and K{X}$_\infty$ with a unit freely generated by a set X over a field K are discussed. The monomials in this algebras can be identified with PB(X), the set of isomorphism classes of X-labelled finite, planar binary rooted trees, and PRT(X), the set of isomorphism classes of X-labelled finite, planar reduced rooted trees respectively, where X is the set of free algebra generators. The reduced Gröbner basis of the ideal J of relations in the Cayley algebra with respect to a chosen admissible order is computed. The multihomogeneous polynomials of multidegree (2,1,1) in the reduced Gröbner basis of the alternator ideal is computed. Some computations are obtained on Gröbner bases of ideals by use of computer programs Magma and MuPAD

Non-associative Gröbner bases, finitely-presented Lie rings and the Engel condition, II

AbstractWe give an algorithm for constructing a basis and a multiplication table of a finite-dimensional finitely-presented Lie ring. Secondly, we give relations that are equivalent to the n-Engel condition, and only have to be checked for the elements of a basis of a Lie ring. We apply this to construct the freest t-generator Lie rings that satisfy the n-Engel condition, for (t,n)=(2,3),(3,3),(4,3),(2,4)