Abstract:

We discuss  the pregeometries of some finitely generated commutative semigroups.
In this article, the case of finitely generated commutative semigroups having a unique extension is considered, and their pregeometries are studied.  We prove that some such semigroups form a pregeometry with definable and algebraic closure operators. When the definable closure operator for such semigroups was studied, the degree of rigidity of these semigroups was evaluated. Moreover, it has been proven that a finitely generated, complete archimedean semigroup is a group, and its finite and infinite cases have been deterimined.