# Book:J.C. Rosales/Finitely Generated Commutative Monoids

## J.C. Rosales and P.A. García-Sánchez: *Finitely Generated Commutative Monoids*

Published $1999$, **Nova Science Publishers, Inc.**

- ISBN 1-56072-670-9.

### Subject Matter

### Contents

- Preface

- Acknowledgements

- Chapter 1. Basic definitions and results
- Remarks
- Exercises

- Chapter 2. Finitely generated commutative groups
- 1. Bases and rank of a subgroup of $\Z^n$
- 2. Equivalence of matrices with integer entries and invariant factors
- 3. Some practical results concerning the computation of a basis
- Remarks
- Exercises

- Chapter 3. Finitely generated cancellative monoids
- 1. Finitely generated cancellation torsion free monoids
- 2. Finitely generated cancellation reduced monoids
- 3. Finite cancellative monoids
- Remarks
- Exercises

- Chapter 4. Minkowski-Farkas' lemma and its applications to monoids
- 1. Main result and algorithms
- 2. Applications to monoids
- Remarks
- Exercises

- Chapter 5. Finitely generated monoids are finitely presented
- 1. Linear admissible orders
- 2. Rédei's theorem
- 3. The word problem for monoids
- 4. Cyclic monoids
- Remarks
- Exercises

- Chapter 6. The word problem for monoids

- 1. Reduced systems of generators of a congruence
- 2. Canonical systems of generators of a congruence
- 3. The group of units of a monoid
- Remarks
- Exercises

- Chapter 7. Nonnegative integer solutions of systems of linear equations
- 1. Nonnegative integer solutions of a system of linear homogeneous Diophantine equations
- 2. The monoid of nonnegative elements of a subgroup of $\Z^n$
- 3. Nonnegative integer solutions of systems of linear Diophantine equations
- 4. Normal affine semigroups
- Remarks
- Exercises

- Chapter 8. Computing presentations of finitely generated cancellative monoids
- 1. Primitive elements of a congruence
- 2. Computing presentations of finitely generated cancellative monoids
- 3. Deciding whether a monoid is cancellation
- Remarks
- Exercises

- Chapter 9. Minimal presentations of finitely generated cancellative reduced monoids
- 1. Characterisation of minimal presentations of finitely generated cancellative reduced monoids
- 2. The affine case
- Remarks
- Exercises

- Chapter 10. Numerical semigroups
- 1. Minimal presentations of numerical semigroups
- 2. A bound for the cardinality of minimal presentations of numerical semigroups
- 3. Numerical semigroups with maximal embedding dimension
- Remarks
- Exercises

- Chapter 11. Projections of congruences
- 1. Presentations of finitely generated cancellative monoids as projections of affine semigroups
- 2. Lifting some projections
- Remarks
- Exercises

- Chapter 12. Finite torsion free monoids
- 1. Presentations of finite torsion free monoids
- 2. Finite lattices
- 3. Finite Boolean algebras
- 4. Boolean monoids
- Remarks
- Exercises

- Chapter 13. Archimedean Components
- 1. Computing the Archimedean components of a finitely generated monoid
- Remarks
- Exercises

- Chapter 14. Separative monoids
- 1. Separative monoids and their Archimedean components
- 2. Deciding whether the quotient of an ideal of $\N^n$ by a congruence is cancellative
- 3. Elimination
- 4. Deciding whether a finitely generated monoid is separative
- 5. Deciding whether a finitely generated monoid is torsion free
- 6. $\mathcal N$-semigroups
- Remarks
- Exercises

- Appendix A. Graphs

- Bibliography

- Index of notation

- Index of main results and algorithms