Numerical Semigroups

This book offers an exploration of the rich interplay between numerical semigroups and commutative algebra. It fills a notable gap in the literature by bridging numerical semigroup theory with advanced algebraic methods. The book is structured to support both self-study and advanced coursework, and it is divided into two major parts. The first three chapters lay the algebraic groundwork for later applications to numerical semigroups. They introduce readers to foundational topics in commutative algebra, homological methods, CohenMacaulay and canonical modulesall with a focus on the graded structures that arise naturally in semigroup rings. Building on the first three chapters, Chapters 4-6 lead to deep results connecting semigroup properties and invariants to algebraic properties and homological data of semigroup rings.

Throughout, the exposition is enriched with illustrative examples, detailed proofs, and exercises to reinforce understanding. The book is designed for graduate students in mathematics as well as researchers in algebra, number theory, and combinatorics.

Bridges numerical semigroups and commutative algebra in a systematic way Presents concepts in a way that is accessible to non-specialists in commutative algebra Offers the main trends and directions of research

Autorentext

Jürgen Herzog was a professor of mathematics at the University of Duisburg-Essen, Germany. He received his doctorate at Louisiana State University in 1969 and completed his habilitation at the University of Regensburg in 1974. Since 1975, he was a professor at the University of Essen, later Duisburg-Essen, where he retired in 2009. His main research area was in the field of commutative algebra. During the course of his scientific career, he published more than 250 articles and made a significant impact in the development of commutative algebra. Jürgen Herzog sadly passed away on 23 April 2024 before this book could be published. He will be greatly missed in the mathematical community. Somayeh Moradi is associate professor of mathematics at Ilam University, Iran. She received her doctorate from Amirkabir University of Technology - Tehran Polytechnic in 2009. She was appointed assistant professor at Ilam University in 2010 and later in 2017 as associate professor. From march 2023 to March 2025, she was the recipient of the Alexander von Humboldt Research Fellowship at the University of Duisburg-Essen. She is also the recipient of the Maryam Mirzakhani Award 2025. Her main research interests lie in commutative algebra and combinatorics. Masoomeh Rahimbeigi is a researcher at the University of Duisburg-Essen, Germany. She received her doctorate from the University of Kurdistan, Iran in 2019. Her main research area is in commutative algebra.

Inhalt

Basic commutative algebra.- Homological methods (on graded structures).- Cohen-Macaulay modules and the canonical module.- Numerical semigroups and their relations.- Pseudo-Frobenius numbers via minimal graded free resolutions.- Almost symmetric and nearly symmetric numerical semigroups.