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

TL;DR

This paper extends the theory of $\mathcal{C}$-semigroups by linking semigroup-ring invariants to combinatorial data from Apéry sets, and by establishing that symmetry notions are intrinsic to the semigroup rather than to term orders. It introduces the reduced type and analyzes its relationship with the conductor, providing criteria for minimal and maximal reduced type and demonstrating their behavior in classes like $T$-graded GNS and thickened semigroups. The authors prove strong unboundedness results: for any dimension, there exist $\mathcal{C}$-semigroups with unbounded type and reduced type at fixed embedding dimension, and a broad unboundedness result for the number of irreducible components in decompositions. These results highlight the rich asymptotic behavior of invariants in higher-dimensional semigroups and connect homological properties of semigroup rings to combinatorial gaps and Apéry data, with implications for the structure of irreducible decompositions.

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 Apéry 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 Apéry 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 $k$.

Figures (3)

• Figure 1: The elements belonging to the semigroups are all integer points inside the colored area, except for the marked points, which represent the gaps.
• Figure 2: Picture of gaps (black and blue), minimal generators (red) and pseudo-Frobenius elements (blue) of $S_T$, with $T=\langle 5,6,7 \rangle$
• Figure 3: Pictures of the semigroups $\textup{T}_{4}(S,2)$ and $\textup{T}_{5}(S',3)$ of Example \ref{['exa:kthick']}. The marked points are the gaps of the semigroups.

Theorems & Definitions (78)

• Proposition 2.1
• Theorem 3.1
• proof
• Definition 3.2
• Corollary 3.3
• proof
• Remark 3.4
• Remark 3.5
• Theorem 4.1
• proof