Boolean Rank via Monomial Ideals and Neural Ideals

Authors

• Juliann Marie Geraci, University of Nebraska-LincolnFollow

First Advisor

Alexandra Seceleanu

Degree Name

Doctor of Philosophy (Ph.D.)

Committee Members

Alexander Kunin, Brittany Duncan, Jack Jeffries, Mark Walker

Department

Mathematics

Date of this Version

4-28-2026

Document Type

Dissertation

Citation

A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfillment of requirements for the degree Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Alexandra Seceleanu

Lincoln, Nebraska, May 2026

Comments

Copyright 2026, Evangelina Nikolaidou. Used by permission

Abstract

This dissertation develops new connections between Boolean matrix factorization, combinatorial neural codes, and commutative algebra. The central goal is to understand how structural and algebraic invariants can be used to measure and bound complexity in discrete data. We begin by studying the Boolean matrix factorization (BMF) problem, which seeks to express a binary matrix as a product over the Boolean semiring with minimal inner dimension, known as the Boolean rank. By interpreting binary matrices as bipartite graphs, we relate Boolean rank to biclique covers and introduce algebraic techniques to study this quantity. We associate to a matrix its edge ideal and show that the Castelnuovo–Mumford regularity provides a lower bound for Boolean rank. We further study the isolation number of a matrix and develop an algebraic framework via Stanley–Reisner theory, proving that this invariant can be recovered as the regularity of a quotient by the isolation ideal.

We then turn to the theory of combinatorial neural codes, which encode patterns of neural activity as discrete objects. Using the neural ring construction, we study algebraic and combinatorial properties of codes and the correspondence between morphisms of codes and algebraic maps. This perspective organizes codes into a partially ordered set and allows complexity to be studied through order-theoretic and algebraic invariants. Building on these frameworks, we show that Boolean matrix factorizations arise naturally from morphisms of neural codes. The associated maps form a Galois connection, placing BMF within the broader categorical structure of code morphisms. We further investigate how Boolean rank behaves under reduction and introduce notions such as maximal rank codes and monomial rank. Together, these results demonstrate that Boolean matrix factorization, neural code theory, and homological invariants of monomial ideals are deeply interconnected. By translating between combinatorial and algebraic perspectives, this work provides new tools for analyzing discrete structures and establishes a unified framework for studying complexity across these domains.

Recommended Citation

Geraci, Juliann Marie, "Boolean Rank via Monomial Ideals and Neural Ideals" (2026). Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–. 474.
https://digitalcommons.unl.edu/dissunl/474