[Paper Review] The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences

Mahdi Omidali|arXiv (Cornell University)|Jan 11, 2010

Rings, Modules, and Algebras7 references3 citations

TL;DR

This paper computes the catenary degree and tame degree for numerical monoids generated by generalized arithmetic sequences, demonstrating that the difference between these invariants can be arbitrarily large even with a fixed number of minimal generators. The results resolve open questions about combinatorial invariants in non-unique factorizations within this class of monoids.

ABSTRACT

Studying ceratin combinatorial properties of non-unique factorizations have been a subject of recent literatures. Little is known about two combinatorial invariants, namely the catenary degree and the tame degree, even in the case of numerical monoids. In this paper we compute these invariants for a certain class of numerical monoids generated by generalized arithmetic sequences. We also show that the difference between the tame degree and the catenary degree can be arbitrary large even if the number of minimal generators is fixed.

Motivation & Objective

• To determine the catenary degree and tame degree for numerical monoids generated by generalized arithmetic sequences.
• To investigate the relationship between the catenary degree and tame degree in such monoids.
• To explore whether the gap between these invariants can grow without bound when the number of minimal generators is fixed.

Proposed method

• The authors analyze the structure of numerical monoids generated by generalized arithmetic sequences using combinatorial and algebraic techniques.
• They derive explicit formulas for the catenary degree based on the parameters of the sequence.
• They compute the tame degree by examining factorization properties and the behavior of irreducible elements.
• The analysis involves comparing factorization lengths and identifying critical elements that influence the invariants.
• The paper uses extremal constructions to demonstrate that the difference between the tame degree and catenary degree can be made arbitrarily large.

Experimental results

Research questions

• RQ1 What are the exact values of the catenary degree and tame degree for numerical monoids generated by generalized arithmetic sequences?
• RQ2 How do the catenary degree and tame degree relate to each other in this class of monoids?
• RQ3 Can the difference between the tame degree and catenary degree be arbitrarily large while keeping the number of minimal generators fixed?

Key findings

• The catenary degree and tame degree are explicitly computed for numerical monoids generated by generalized arithmetic sequences.
• The tame degree can exceed the catenary degree by an arbitrarily large margin, even when the number of minimal generators is fixed.
• The difference between the tame degree and catenary degree is not bounded, indicating a significant structural divergence in factorization behavior.
• The results show that the two invariants are not tightly correlated in this class of monoids.
• The constructions used to achieve large differences are based on specific parameter choices in the generalized arithmetic sequences.