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

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

Published $\text {1999}$, Nova Science Publishers, Inc.

ISBN 1-56072-670-9.

### 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
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

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