Weight ideals associated to regular and log-linear arrays

Certain weight-based orders on the free associative algebra R=k〈x1,…,xt〉 can be specified by t×∞ arrays whose entries come from the subring of positive elements in a totally ordered field. If such an array satisfies certain additional conditions, it produces a partial order on R which is an admissible order on the quotient R/I, where the ideal I is a homogeneous binomial ideal called the weight ideal associated to the array. The structure of the weight ideal is determined entirely by the array. This article discusses the structure of the weight ideals associated to two distinct types of arrays which define admissible orders on the associated quotient algebra.