Finitely generated commutative monoids

202 Pages·1999·10.038 MB·English

by  J. C. Rosales and P. A. García-Sánchez

Nova Science Publishers, Inc.

J. C. ROSALEASN D P. A. GARC~A-SÁNCEHEZ
Nova Science Publishers, Inc.
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Contents
Pref ace
.. .
Acknowledgements
Xlll
Chapter 1. Basic definitions and results
Remarks
Exercises
Chapter 2. Finitely generated commutative groups
1. Bases and rank of a subgroup of Zn
2. Equivalente of matrices with integer entries and invariant factors
3. Some practical results conceming the computation of a basis
Remarks
Exercises
Chapter 3. Finitely generated cancellative monoids
1. Finitely generated cancellative torsion free monoids
2. Finitely generated cancellative reduced monoids
3. Finite cancellative monoids
Rem arks
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
l. Linear admissible orders
2. Rédei7st heorem
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 congmence
2. Canonical systems of generators of a congmence
3. The group of units of a monoid
Remarks
Exercises
Chapter 7. Nonnegative integer solutions of systems of linear equations 73
1. Nonnegative integer solutions of a system of linear homogeneous
Diophantine equations '.1-.5e
2. The monoid of nonnegative elements of a subgroup of Zn 77
3. Nonnegative integer solutions of systems of linear Diophantine equations 80
4. Normal affine semigroups 8 1
Remarks 83
Exercises 84
Chapter 8. Computing presentations of finitely generated cancellative monoids 85
1. Primitive elements of a congruence 85
2. Computing presentations of finitely generated cancellative monoids 87
3. Deciding whether a monoid is cancellative 90
Remarks 91
Exercises 91
Chapter 9. Minimal presentations of finitely generated cancellative reduced
. .
monoids 02
/cl
1. Characterization of minimal presentations of finitely generated cancellative
reduced monoids 93
2. The dfine case
Remarks
Exercis es 103
Chapter 10. Numerical semigroups 105
1. Minimal presentations of numerical semigroups 108
2. A bound for the cardinality of minimal presentations of numerical
semigroups 110
3. Numerical semigroups with maximal embedding dimension 113
Remarks 116
Exercises 117
Chapter 1 l. Projections of congruentes 119
1. Presentations of finitely generated cancellative monoids as projections of
affine semigroups 119
2. Lifting some projections 123
Remarks 127
Exercises 127
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
l. 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 W by a congruente is
cancellative
3. Elimination
4. Deciding whether a finitely generated monoid is separative
5. Deciding whether a finitely generated monoid is torsion free
6. N-semigroups
Remarks
Exercises
Appendix A. Graphs
Bibliography
Index of notation
Index of main results and algorithms
Index