Weight Ideals Associated to Regular and Log-Linear Arrays

TL;DR

This paper explores the structure of weight ideals derived from regular and log-linear arrays in free associative algebras, providing insights into their algebraic properties and the orders they induce.

Contribution

It introduces a framework for associating weight ideals to arrays and analyzes their structure for specific array types, advancing understanding of algebraic orderings.

Findings

Characterization of weight ideals from regular arrays

Analysis of weight ideals from log-linear arrays

Conditions under which these ideals produce admissible orders

Abstract

Certain weight-based orders on the free associative algebra $R = k<x_1, ..., x_t >$ can be specified by $t \times \infty$ arrays whose entries come from the subring of nonnegative elements in a totally ordered field. Such an array $A$ satisfying certain additional conditions produces a partial order on $R$ which is an admissible order on the quotient $R/I_A$, where $I_A$ is a homogeneous binomial ideal called the {\em weight ideal} associated to the array and whose structure is determined entirely by $A$. This article discusses the structure of the weight ideals associated to two distinct sets of arrays whose elements define admissible orders on the associated quotient algebra.