Extensions of Numerical Semigroups Generated by Triangular Numbers

Abstract

In this work, we study θ-gonal numerical semigroups, a natural generalization of semigroups generated by triangular numbers. A θ-gonal number represents the number of dots that can form a regular θ-sided polygon, with θ ≥ 3, and includes classical figurate numbers such as triangular, square, and pentagonal numbers. We investigate the algebraic structure of numerical semigroups generated by θ-gonal numbers, focusing on their minimal generating sets and fundamental properties. In particular, we characterize the systems of minimal generators, providing insights into the combinatorial and arithmetic structure of these semigroups. Our results generalize known properties of semigroups generated by triangular numbers and offer a framework for studying semigroups associated with other classes of polygonal numbers.