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Numerical semigroups with maximal embedding dimension
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<div class="ChapSects"><a href="chap8.html#X7D2E70FC82D979D3">8 <span class="Heading">
Numerical semigroups with maximal embedding dimension
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X7D2E70FC82D979D3">8.1 <span class="Heading">
Numerical semigroups with maximal embedding dimension
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X867615F8846824EB">8.1-1 IsMEDNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X86C8358D8530106F">8.1-2 MEDNumericalSemigroupClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X848FD3FA7DB2DD4C">8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X82E40EFD83A4A186">8.2 <span class="Heading">
Numerical semigroups with the Arf property and Arf closures
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8255C5907F8968B9">8.2-1 IsArfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7FE10E2F85CB01A2">8.2-2 ArfNumericalSemigroupClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7C0D2F7986165DDE">8.2-3 MinimalArfGeneratingSystemOfArfNumericalSemigroup</a></span>
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</div>
<h3>8 <span class="Heading">
Numerical semigroups with maximal embedding dimension
</span></h3>
<p><a id="X7D2E70FC82D979D3" name="X7D2E70FC82D979D3"></a></p>
<h4>8.1 <span class="Heading">
Numerical semigroups with maximal embedding dimension
</span></h4>
<p>If S is a numerical semigroup and m is its multiplicity (the least positive integer belonging to it), then the embedding dimension e of S (the cardinality of the minimal system of generators of S) is less than or equal to m. We say that S has maximal embedding dimension (MED for short) when e=m. The intersection of two numerical semigroups with the same multiplicity and maximal embedding dimension is again of maximal embedding dimension. Thus we define the MED closure of a non-empty subset of positive integers M={m < m_1 < cdots < m_n <cdots} with gcd(M)=1 as the intersection of all MED numerical semigroups with multiplicity m.</p>
<p>Given a MED numerical semigroup S, we say that M={m_1 < cdots< m_k} is a MED system of generators if the MED closure of M is S. Moreover, M is a minimal MED generating system for S provided that every proper subset of M is not a MED system of generators of S. Minimal MED generating systems are unique, and in general are smaller that the classical minimal generating systems (see <a href="chapBib.html#biBRGGB03">[GB03]</a>).</p>
<p><a id="X867615F8846824EB" name="X867615F8846824EB"></a></p>
<h5>8.1-1 IsMEDNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsMEDNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>
<p>Returns true if <var class="Arg">S</var> is a MED numerical semigroup and false otherwise.</p>
<table class="example">
<tr><td><pre>
gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5,7));
true
gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5));
false
</pre></td></tr></table>
<p><a id="X86C8358D8530106F" name="X86C8358D8530106F"></a></p>
<h5>8.1-2 MEDNumericalSemigroupClosure</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MEDNumericalSemigroupClosure</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>
<p>Returns the MED closure of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> MEDNumericalSemigroupClosure(NumericalSemigroup(3,5));
<Numerical semigroup>
gap> MinimalGeneratingSystemOfNumericalSemigroup(last);
[ 3, 5, 7 ]
</pre></td></tr></table>
<p><a id="X848FD3FA7DB2DD4C" name="X848FD3FA7DB2DD4C"></a></p>
<h5>8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a MED numerical semigroup.</p>
<p>Returns the minimal MED generating system of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup(
> NumericalSemigroup(3,5,7));
[ 3, 5 ]
</pre></td></tr></table>
<p><a id="X82E40EFD83A4A186" name="X82E40EFD83A4A186"></a></p>
<h4>8.2 <span class="Heading">
Numerical semigroups with the Arf property and Arf closures
</span></h4>
<p>Numerical semigroups with the Arf property are a special kind of numerical semigroups with maximal embedding dimension. A numerical semigroup S is Arf if for every x,y,z in S with x>= y>= z, one has that x+y-zin S.</p>
<p>The intersection of two Arf numerical semigroups is again Arf, and thus we can consider the Arf closure of a set of nonnegative integers with greatest common divisor equal to one. Analogously as with MED numerical semigroups, we define Arf systems of generators and minimal Arf generating system for an Arf numerical semigroup. These are also unique(see <a href="chapBib.html#biBRGGB04">[GB04]</a>).</p>
<p><a id="X8255C5907F8968B9" name="X8255C5907F8968B9"></a></p>
<h5>8.2-1 IsArfNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsArfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>
<p>Returns true if <var class="Arg">S</var> is an Arf numerical semigroup and false otherwise.</p>
<table class="example">
<tr><td><pre>
gap> IsArfNumericalSemigroup(NumericalSemigroup(3,5,7));
true
gap> IsArfNumericalSemigroup(NumericalSemigroup(3,7,11));
false
gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,7,11));
true
</pre></td></tr></table>
<p><a id="X7FE10E2F85CB01A2" name="X7FE10E2F85CB01A2"></a></p>
<h5>8.2-2 ArfNumericalSemigroupClosure</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ArfNumericalSemigroupClosure</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>
<p>Returns the Arf closure of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> ArfNumericalSemigroupClosure(NumericalSemigroup(3,7,11));
<Numerical semigroup>
gap> MinimalGeneratingSystemOfNumericalSemigroup(last);
[ 3, 7, 8 ]
</pre></td></tr></table>
<p><a id="X7C0D2F7986165DDE" name="X7C0D2F7986165DDE"></a></p>
<h5>8.2-3 MinimalArfGeneratingSystemOfArfNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MinimalArfGeneratingSystemOfArfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is an Arf numerical semigroup.</p>
<p>Returns the minimal MED generating system of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> MinimalArfGeneratingSystemOfArfNumericalSemigroup(
> NumericalSemigroup(3,7,8));
[ 3, 7 ]
</pre></td></tr></table>
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