A numerical semigroup S is ordinary if its gap set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\text {G}}(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>G</mtext> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is either empty or of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{1,\ldots ,n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> for some positive integer n. For a positive integer d, by the quotient of S by d we mean the numerical semigroup <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{1}{d}S=\{n \in {\mathbb {Z}}_{\ge 0}: nd \in S\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> <mi>S</mi> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>n</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mi>d</mi> <mo>∈</mo> <mi>S</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For coprime positive integers a, b, we investigate values of d for which the numerical semigroup <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frac{1}{d}\langle a,b \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> <mrow> <mo stretchy="false">⟨</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is ordinary.

On quotients of two-dimensional numerical semigroups that are ordinary

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2026年7月29日

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Edgar Federico Elizeche,
Amitabha Tripathi

摘要

A numerical semigroup S is ordinary if its gap set ${\text {G}}(S)$ $G (S)$ is either empty or of the form $\{1,\ldots ,n\}$ ${1, \dots, n}$ for some positive integer n. For a positive integer d, by the quotient of S by d we mean the numerical semigroup $\frac{1}{d}S=\{n \in {\mathbb {Z}}_{\ge 0}: nd \in S\}$ $\frac{1}{d} S = {n \in Z_{\geq 0} : n d \in S}$ . For coprime positive integers a, b, we investigate values of d for which the numerical semigroup $\frac{1}{d}\langle a,b \rangle $ $\frac{1}{d} ⟨ a, b ⟩$ is ordinary.