Finitely generated commutative monoids and its applications

ABSTRACTMSc THESISFINITELY GENERATED COMMUTATIVE MONOIDS AND ITSAPPLICATIONSüOrhan SONMEZDEPARTMENT OF MATHEMATICSINSTITUTE OF NATURAL AND APPLIED SCIENCESUNIVERSITY OF CUKUROVAşü üSupervisor: Prof.Dr. Yusuf UNLUYear: 2005, Pages: 60ü üJury: Prof.Dr. Yusuf UNLUAssoc.Prof.Dr. Hayrullah AYIKAssist.Prof.Dr. Ersin KIRALIn this thesis we give an exposition of the fundamental results in abelian monoid theory.In addition to this we give a description of some classes in MS Visual Basic that can be usesto implement the algorithms which take place in this exposition. Following is a list of someimportant theorems and algorithms in this thesis.If M is a subgroup of Zn with invariant factors d1 , . . . , dr then it is shown thatZn /M Zd1 × . . . × Zdr × Zn−r .Furthermore, we give an algorithm to compute a basis of a subgroup of Zn from one of itssystems of generators or its defining equations.We discuss the properties of finitely generated monoids such as of being cancellative,torsion free, reduced and finite and give some results. We mention a result by Grillet onfinitely generated monoids.Minkowski-Farkas lemma and some of its consequences on finitely generated monoidsare given. Especially we apply this lemma to decide when a subspace of Qn has a non neg-ative or strongly positive elements, if any. We give algorithms to find out whether Nn / ∼Mis a group or an affine semigroup. We describe an algorithm to calculate U (Nn / ∼M ).An algorithmic method deciding whether a linear homogeneous system of equations has anontrivial nonnegative solution and another algorithmic method deciding whether two sub-monoids of Nn have nontrivial intersection and if so then finding an element in intersectionis given.Key Words: Finitely generated commutative monoids, being cancellative, being torsionfree, being reduced, strongly positive element.II