Date of Award

January 2025

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Bryce Christopherson

Abstract

This study investigates threshold phenomena in infinite modular lattices, extending classical threshold theory beyond Boolean and distributive lattice settings. Specifically, let\(P\) be a finite graded modular lattice with a finite graded modular lattice with rank function \(r: P \rightarrow \mathbb {N}\). By adapting probabilistic combinational methods, notably the methods used by Janson, Łuczak, and Ruciński (2011), we establish rigorously that every monotone property \(F\subseteq P\) possesses a threshold under the chain-uniform random model.Formally, this threshold is characterized by a critical rank \(M^*\), at which the probability \(\mathbb{P}(\Gamma_M\in F)\) transitions sharply from nearly 0 to nearly 1 as the parameter \(M\) passes through \(M^*\). Significant contributions include deriving explicit thresholds in various algebraic contexts. In particular, for lattices of normal subgroups \(N\triangleleft G\) of finite groups \(G\), thresholds distinguish probabilistically between abelian and anabelian subgroups. Additionally, analogous threshold phenomena are investigated within the lattice of finite fields in terms of cyclic Galois extensions, highlighting their algebraic significance.

Download

Share

COinS