Finitely Generated Commutative Monoids


J.C. Rosales and P.A. Garcia-Sanchez

The book is organized as follows. In Chapter 1 the editors introduce the concepts and basic results of the theory of monoids. Chapter 2 is devoted to the structure theorem of finitely generated Abelian groups and also shows how to compute a basis and the equations of a given subgroup of ℤn. In Chapter 3 they study finitely generated cancellative monoids and focus the attention on determining whether a given finitely generated monoid can be embedded in k or ℤk. Applications to monoids and resolution of systems of diophantine equations is given in Chapter 4.

Redei’s theorem ensures that every finitely generated monoid is finitely presented and is proven in Chapter 5. Associated to Redei’s theorem, arises the word problem for finitely generated monoids. The solution to this problem is presented in Chapter 6. The study of the systems of equations is performed in Chapter 7; full and normal affine semigroups come into scene.

In Chapter 8 the results obtained in Chapter 7 are used to give algorithms for computing a presentation of a finitely generated cancellative monoid and to decide from a presentation of a finitely generated monoid whether it is or is not cancellative. In Chapter 9 a characterization of minimal presentations of finitely generated reduced cancellative monoids are presented. Chapter 10 is devoted to numerical semigroups that are a subclass of affine semigroups. In Chapter 11 the editors show that presentations of finitely generated cancellative monoids are projections of presentations of affine semigroups. The last three chapters are devoted to the study of torsion of a finitely generated monoid.

This book is not only limited to people working in Semigroup Theory. The only knowledge required to follow and understand its contents is basic Linear Algebra. Thus, any student in their second year of Mathematics or Computer Science might find the book easy to understand. This monograph can also be used as a textbook for a course on finitely generated monoids.

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