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Homomorphisms of Partial Fields

Nathaniel Vaduthala

TL;DR

This work advances the algebraic theory of partial fields by (i) precisely characterizing the weak and strong characteristic sets $\chi(\mathbb{P})$ and $\chi_{\text{strong}}(\mathbb{P})$, showing these sets are nonempty with a finite and/or zero-containing structure, (ii) proving that the class of partial fields is not well-quasi-ordered under $\succeq_{\text{Hom}}$, (iii) providing a new ring-theoretic proof of the idempotence of the lift operator and establishing a robust universal property for the Dowling lift, and (iv) clarifying the relationship between fundamental elements and the Dowling lift, along with idempotence results for the Dowling lift. These results deepen the understanding of representability phenomena in matroid theory and enhance the lifting framework used to transfer representability across partial-field contexts, including Dowling geometries. The methods combine model-theoretic techniques (ultraproducts), constructive ring-theoretic constructions, and universal-property arguments to connect algebraic structure with combinatorial representability.

Abstract

A partial field is an algebraic object that allows one to simultaneously abstract several different representability properties of matroids. In this paper we study partial fields as algebraic objects in their own right. We characterize the weak and strong characteristic sets of partial fields and show that the class of partial fields is not well-quasi ordered. We provide a new proof that the lift operator of a partial field is idempotent. We also provide a relation between the fundamental elements of a partial field and its Dowling lift, and show that the Dowling lift operator is idempotent.

Homomorphisms of Partial Fields

TL;DR

This work advances the algebraic theory of partial fields by (i) precisely characterizing the weak and strong characteristic sets and , showing these sets are nonempty with a finite and/or zero-containing structure, (ii) proving that the class of partial fields is not well-quasi-ordered under , (iii) providing a new ring-theoretic proof of the idempotence of the lift operator and establishing a robust universal property for the Dowling lift, and (iv) clarifying the relationship between fundamental elements and the Dowling lift, along with idempotence results for the Dowling lift. These results deepen the understanding of representability phenomena in matroid theory and enhance the lifting framework used to transfer representability across partial-field contexts, including Dowling geometries. The methods combine model-theoretic techniques (ultraproducts), constructive ring-theoretic constructions, and universal-property arguments to connect algebraic structure with combinatorial representability.

Abstract

A partial field is an algebraic object that allows one to simultaneously abstract several different representability properties of matroids. In this paper we study partial fields as algebraic objects in their own right. We characterize the weak and strong characteristic sets of partial fields and show that the class of partial fields is not well-quasi ordered. We provide a new proof that the lift operator of a partial field is idempotent. We also provide a relation between the fundamental elements of a partial field and its Dowling lift, and show that the Dowling lift operator is idempotent.
Paper Structure (9 sections, 12 theorems, 23 equations)

This paper contains 9 sections, 12 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Background Information
  3. Homomorphisms to Fields
  4. Characteristic Sets
  5. Well-Quasi-Ordering of Partial Fields
  6. Lifts of Partial Fields
  7. Lift Ring
  8. The Dowling Lift of a Partial Field
  9. Acknowledgments

Key Result

Proposition 2.6

Let $\phi_1 : \mathbb{P}_1 \to \mathbb{P}_2$ and $\phi_2 : \mathbb{P}_2 \to \mathbb{P}_3$ be partial-field homomorphisms. Then $\phi_2 \circ \phi_1 : \mathbb{P}_1 \to \mathbb{P}_3$ is a partial-field homomorphism.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5: vzThesis
  • Proposition 2.6
  • proof
  • Definition 3.1
  • Definition 3.3
  • Definition 3.4
  • ...and 27 more