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Composite Linear Quotient Orderings of Ideals and Modified Anticycles

Stephen Landsittel

Abstract

In this paper we give a construction for a linear quotient ordering of a class of products of two ideals which have linear quotients. We apply this construction to give a class of modified anticycle graphs whose square and cube have linear quotients.

Composite Linear Quotient Orderings of Ideals and Modified Anticycles

Abstract

In this paper we give a construction for a linear quotient ordering of a class of products of two ideals which have linear quotients. We apply this construction to give a class of modified anticycle graphs whose square and cube have linear quotients.
Paper Structure (7 sections, 14 theorems, 66 equations)

This paper contains 7 sections, 14 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory
  4. Commutative Algebra
  5. Notation and Examples of Linear Quotient Orderings
  6. Proof of Theorem \ref{['thmA']}: Composite Linear Quotient Orderings
  7. Proof of Theorem \ref{['thmA']}: Demonstration of the Main Lemmas

Key Result

Proposition 1.1

Let $G_0$ be a graph on $[n-1]$, let $F_0$ be a star graph on $[n]$, and let $s\in\mathbb{Z}_{\geq 2}$, and let $H_0$ be the graph on $[n]$ whose edge set is $E(G_0)\cup E(F_0)$. Assume the following two conditions Then $I_{H_0}^s$ has linear quotients by the ordering obtained by concatenating the orderings $\mathcal{O}_s,\ldots,\mathcal{O}_0$.

Theorems & Definitions (46)

  • Proposition 1.1: Construction of composite linear quotient orderings (Lemma \ref{['lem-tech']})
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • proof
  • Remark 3.4
  • Example 3.5
  • ...and 36 more