<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S$</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> be the numerical semigroup generated by three consecutive numbers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a,a+1,a+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We describe the elements of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S$</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> whose factorizations have all the same length, as well as the set of factorizations of each of these elements. We give natural partitions of this subset of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S$</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> in terms of the length and the denumerant. By using Apéry sets and Betti elements we are able to extend some of these results to any general numerical semigroup <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathscr {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>. These results provide a better understanding of the defining ideals associated with Moh’s examples and certain variants, which are related to the defining ideals of the semigroup rings <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$k[t^a,t^b,t^c]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mi>a</mi> </msup> <mo>,</mo> <msup> <mi>t</mi> <mi>b</mi> </msup> <mo>,</mo> <msup> <mi>t</mi> <mi>c</mi> </msup> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover, the elements with unique length factorizations in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$S$</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> are useful to study the minimal generating sets of these ideals.</p>

Elements with unique length factorization of a numerical semigroup generated by three consecutive numbers

期刊论文
发表日期：

2025年12月1日

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Pedro A. García-Sánchez,
Laura González,
Francesc Planas-Vilanova

摘要

Let $S$ $S$ be the numerical semigroup generated by three consecutive numbers $a,a+1,a+2$ $a, a + 1, a + 2$ , where $a\in \mathbb {N}$ $a \in N$ , $a\ge 3$ $a \geq 3$ . We describe the elements of $S$ $S$ whose factorizations have all the same length, as well as the set of factorizations of each of these elements. We give natural partitions of this subset of $S$ $S$ in terms of the length and the denumerant. By using Apéry sets and Betti elements we are able to extend some of these results to any general numerical semigroup $\mathscr {S}$ $S$ . These results provide a better understanding of the defining ideals associated with Moh’s examples and certain variants, which are related to the defining ideals of the semigroup rings $k[t^a,t^b,t^c]$ $k [t^{a}, t^{b}, t^{c}]$ . Moreover, the elements with unique length factorizations in $S$ $S$ are useful to study the minimal generating sets of these ideals.