Admissible orders on quotients of the free associative algebra
An admissible order on a multiplicative basis of a noncommutative algebra A is a term order satisfying additional conditions that allow for the construction of Grobner bases for A -modules. When A is commutative, a finite reduced Grobner basis for an A -module can always be obtained, but when A is not commutative this is not the case; in fact in many cases a Grobner basis theory for A may not even exist. E. Hinson has used position-dependent weights, encoded in so-called admissible arrays, to partially order words in the free associative algebra in a way which produces a length-dominant admissible order on a particular quotient of the free algebra, where the ideal by which the quotient is taken is an ideal generated by pure homogeneous binomial differences and is determined by the array A. This dissertation investigates the properties of two large classes of admissible arrays A. We prove that weight ideals associated to arrays in the first class are finitely generated and we describe the generating sets. We exhibit instances of trivial and nontrivial finitely generated weight ideals associated to arrays in the second class and we partially characterize the corresponding arrays. We also exhibit instances of weight ideals associated to arrays in the second class which do not admit a finite generating set. We identify an algebro-combinatorial property on weight ideals, which we call saturation, that is connected to finite generation. In addition, we look at actions of the multiplicative monoid generated by the set of transvections and diagonal matrices with non-negative entries on the set of equivalence classes of admissible arrays under order-isomorphism and we analyze the stabilizers and orbits of these actions.
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