Numerical Semigroups with a Fixed Fundamental Gap
A gap <i>a</i> of a numerical semigroup <i>S</i> is fundamental if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>2</mn><mi>a</mi>...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/1/95 |
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| Summary: | A gap <i>a</i> of a numerical semigroup <i>S</i> is fundamental if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>2</mn><mi>a</mi><mo>,</mo><mn>3</mn><mi>a</mi><mo>}</mo><mo>⊆</mo><mi>S</mi><mo>.</mo></mrow></semantics></math></inline-formula> In this work, we will study the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>=</mo><mfenced separators="" open="{" close="}"><mi>S</mi><mo>∣</mo><mi>S</mi><mspace width="4.pt"></mspace><mi>is</mi><mspace width="4.pt"></mspace><mi mathvariant="normal">a</mi><mspace width="4.pt"></mspace><mi>numerical</mi><mspace width="4.pt"></mspace><mi>semigroup</mi><mspace width="4.pt"></mspace><mi>and</mi><mspace width="4.pt"></mspace><mi>a</mi><mspace width="4.pt"></mspace><mrow><mi>is</mi><mspace width="4.pt"></mspace><mi mathvariant="normal">a</mi><mspace width="4.pt"></mspace><mi>fundamental</mi><mspace width="4.pt"></mspace><mi>gap</mi><mspace width="4.pt"></mspace><mi>of</mi><mspace width="4.pt"></mspace></mrow><mi>S</mi></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> In particular, we will give an algorithm to compute all the elements of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula> with a given genus. The intersection of two elements of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula> is again one element of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> A <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible numerical semigroup is an element of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula> that cannot be expressed as an intersection of two elements of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula> containing it properly. In this paper, we will study the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible numerical semigroups. In this sense we will give an algorithm to calculate all of them. Finally, we will study the submonoids of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="double-struck">N</mi><mo>,</mo><mo>+</mo><mo>)</mo></mrow></semantics></math></inline-formula> that can be expressed as an intersection (finite or infinite) of elements belonging to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> |
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| ISSN: | 2227-7390 |