Monomial Ideals and Their Decompositions

1. Front Matter

   Pages i-xxiv

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2. Monomial Ideals

   1. Front Matter

      Pages 1-3

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   2. Fundamental Properties of Monomial Ideals

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 5-32

   3. Operations on Monomial Ideals

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 33-79

   4. M-Irreducible Ideals and Decompositions

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 81-109

3. Monomial Ideals and Other Areas

   1. Front Matter

      Pages 111-112

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   2. Connections with Combinatorics

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 115-159

   3. Connections with Other Areas

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 161-216

4. Decomposing Monomial Ideals

   1. Front Matter

      Pages 217-218

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   2. Parametric Decompositions of Monomial Ideals

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 221-260

   3. Computing M-Irreducible Decompositions

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 261-292

5. Commutative Algebra and Macaulay2

   1. Front Matter

      Pages 293-294

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   2. Appendix A: Foundational Concepts

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 297-329

   3. Appendix B: Introduction to Macaulay2

      • W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff

      Pages 331-347

6. Back Matter

   Pages 349-387

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This textbook on combinatorial commutative algebra focuses on properties of monomial ideals in polynomial rings and their connections with other areas of mathematics such as combinatorics, electrical engineering, topology, geometry, and homological algebra. Aimed toward advanced undergraduate students and graduate students who have taken a basic course in abstract algebra that includes polynomial rings and ideals, this book serves as a core text for a course in combinatorial commutative algebra or as preparation for more advanced courses in the area.  The text contains over 600 exercises to provide readers with a hands-on experience working with the material; the exercises include computations of specific examples and proofs of general results. Readers will receive a firsthand introduction to the computer algebra system Macaulay2 with tutorials and exercises for most sections of the text, preparing them for significant computational work in the area. Connections to non-monomial areas of abstract algebra, electrical engineering, combinatorics and other areas of mathematics are provided which give the reader a sense of how these ideas reach into other areas.

 

• Macaulay 2
• combinatorial commutative algebra
• irreducible decompositions
• monomial ideals
• polynomial rings
• simplicial complexes
• modifying monomial ideals
• decompositions of monomial ideals
• vertex covers
• edge ideal construction of Villarreal
• m-irreducible decompositions
• parametric decompositions
• algorithms
• commutative algebra
• Dickson’s Lemma
• Stanley-Reisner ideals
• Phasor Measurement Units
• Cohen-Macaulayness
• Hilbert functions

“The present book is thought as a gentle introduction to monomial ideals … . All the chapters contain exercises and Macaulay 2 material for the computational exploration of the presented notions.” (Christos Tatakis, zbMATH 1476.13002, 2022)

“Primarily directed at advanced undergraduates, the text is also a valuable resource for graduate students and researchers who wish to learn more about the subject, providing an introduction to active research topics in combinatorial commutative algebra and its applications. … the authors' presentation of monomial decompositions and their applications is an exciting, enlightening read and will serve an individual reader or class instructor well.” (Timothy B. P. Clark, Mathematical Reviews, October, 2019)

“Each definition includes examples of reasonably common structures … . This style makes the text accessible to advanced undergraduates. … it will be useful to those who work in symbolic computation and theory.” (Paul Cull, Computing Reviews, May 13, 2019)

• Department of Mathematics, Wake Forest University, Winston-Salem, USA

  W. Frank Moore

• Department of Mathematics, Missouri State University, Springfield, USA

  Mark Rogers

• School of Mathematical and Statistical Sciences, Clemson University, Clemson, USA

  Sean Sather-Wagstaff

W. Frank Moore is an Associate Professor of Mathematics at Wake Forest University. He earned his PhD from the University of Nebraska-Lincoln, and his research is in the homological algebra of commutative and noncommutative rings.

Mark Rogers is a Professor in the Department of Mathematics at Missouri State University. He earned his PhD from Purdue University, and his area of research is commutative algebra.

Sean Sather-Wagstaff is an Associate Professor in Clemson University’s department of Mathematical Sciences. He earned his PhD from the University of Utah, specializing in homological commutative algebra.

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