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

Subject Matter

 * Monoids

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



Source work progress
* : Chapter $1$: Basic Definitions and Results