On unboundedness of some invariants of $\mathcal{C}$-semigroups

TL;DR

This paper investigates invariants of $\\mathcal{C}$-semigroups, demonstrating that certain invariants like type and reduced type can be unbounded relative to embedding dimension, and explores their structural properties and decompositions.

Contribution

It proves unboundedness of type and reduced type for $\\mathcal{C}$-semigroups and analyzes their irreducible decompositions, extending classical concepts to this broader setting.

Findings

Type equals the number of maximal elements of the Apéry set.

Type and almost symmetry are independent of term orders.

Existence of $\\mathcal{C}$-semigroups with arbitrarily large type and irreducible components.

Abstract

In this article, we first prove that the type of an affine semigroup ring is equal to the number of maximal elements of the Ap\'ery set with respect to the set of exponents of the monomials, which form a maximal regular sequence. Further, we consider $\mathcal{C}$-semigroups in $\mathbb{N}^d$ and prove that the notions of symmetric and almost symmetric $\mathcal{C}$-semigroups are independent of term orders. We further investigate the conductor and the Ap\'ery set of a $\mathcal{C}$-semigroup with respect to a minimal extremal ray. Building upon this, we extend the notion of reduced type to $\mathcal{C}$-semigroups and study its extremal behavior. For all $d$ and fixed $e \geq 2d$, we give a class of $\mathcal{C}$-semigroups of embedding dimension $e$ such that both the type and the reduced type do not have any upper bound in terms of the embedding dimension. We further explore irreducible decompositions of a $\mathcal{C}$-semigroup and give a lower bound on the irreducible components in an irreducible decomposition. Consequently, we deduce that for each positive integer $k$, there exists a $\mathcal{C}$-semigroup $S$ such that the number of irreducible components of $S$ is at least $k$.