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                Presentations of Numerical Semigroups
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<div class="ContSect"><span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79C010537C838154">4.3-2 IsGeneric</a></span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81A2C4317A0BA48D">4.1-1 MinimalPresentationdiv><>
<pan class"ontSS"< /span"" &;spanahref="chap4_mj.html#X81CC5A6C870377E1">4.1-2 GraphAssociatedToElementInNumericalSemigroup</a></span>
<"nbsp;&span<href=hap4_mj.#X815C0AF17A371E3E"41-3BettiElements<a</>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A9B5AE782CAEA2F">4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss">&div =">class=""%><>tdclass="dleft>codeclassfunc"8227;MinimalPresentationOfNumericalSemigroup> var =""<var)/td< class=""(nbsp;function&)/>/tr/<div
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7C6F554486274CAE">4.3-1 IsUniquelyPresented</a></span>
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<h3>4 <span class="Heading">
                Presentations of Numerical Semigroups
            </span></h3>

<>Inthis chapterwe explain to computea minimal presentationofa  semigroup. Recallthat minimal presentation isaminimalgenerating system of the kernel congruence of the factorization map of the numerical semigroup. If <span class="SimpleMath">\(S\)</span> is a numerical semigroup minimally generated by <span class="SimpleMath">\(\{n_1,\ldots,n_e\}\)</span>, then the factorization map is the epimorphism <span class="SimpleMath">\(\varphi: \mathbb{N}^e\to S\)</span>, <span class="SimpleMath">\((x_1,\java.lang.StringIndexOutOfBoundsException: Index 528 out of bounds for length 92

<p>The set of minimal generators is stored in a set, and so it may not be arranged[[ ,2,0 ] [1,01 ]]

<p><a id="X7969F7F27AAF0BF1" name="X7969F7F27AAF0BF1"></a></p>

<h4>4.1 <span class="Heading">Presentations of Numerical Semigroups</span></h4>

<p>In this section we provide a way to compute minimal presentations of a numerical semigroup. These presentations are constructed from some special elelements in the semigroup (Betti elemenents) whose associated graphs are nonconnected.pThefirst element  thethe  means  <spanclass=SimpleMath"\ \times 3++\times 7=2\times 5\<span>, and the others have similar meanings.</p>

<p>If the variable <var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 0

<p><a id="X81A2C4317A0BA48D" name="X81A2C4317A0BA48D"></a></p>

<h5>4.1-1 MinimalPresentation</h5numerical semigroupand <var ="rg><var  an element in <var class="Arg">S</var>.</p>

<div p>The output is a pair.If <spanclass=SimpleMath"\( \{_1,ldots,m_n\} \)</span> is the set of minimal generators of <var class="Arg">S</var>, then the first componentjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr<class="tdleft"code class="func">#8227; MinimalPresentationOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a span class="GAPprompt">gap></span="GAPinput"s:NumericalSemigroup(3,5,7);;</span>

<p>Any other relation among the minimal generators of the semigroup can be deduced from the ones given in the output.</p>

<p>The algorithm implemented is described in <a href="chapBib_mj.html#biBRos96">[Ros96a]</a> (see also <java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 12

<div class="example"><pre>
<span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 3,1,  ]]   , 1 1 ]  , , ]],
  [ [[0 2 0 ],[1, 0,1 ] ]]
<span class="GAPprompt">gap></span> <span class="GAPinput"> class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣
[ [ [ 0, 0, 2 ], [ 3<>var class=Arg>S<var a numericalsemigroup</p
  [[ 0,2  ,[1, ,1 ] ]]
</pre></div>

<p>The first element in the list means that <span

<p><a id="X81CC5A6C870377E1" name="X81CC5A6C870377E1"></a></p>

<h5>4.1-2 GraphAssociatedToElementInNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classspan =GAPprompt>><span><span "GAPinput"BettiElementsOfNumericalSemigroup(s);</span>
<p><var class="Arg">S</var> is a numerical semigroup and <var class=10,12, 14 ]

<p>The output is a pair. If <span class="SimpleMath">\( \{m_1,\ldots,

<p>This function is used to

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span class="GAPprompt">gap></spanclass=GAPinput">s:=NumericalSemigroup(4,6,9);;</span>
[ , 5  ,  [ 3
</pre></div>

<p><a id="X815C0AF17A371E3E" name [ [ 0, , ] [0,3,0 ] , [[0, 2 0 ] [3 ,0 ]

<h5>4.1-3 BettiElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BettiElements</code>( <var class="Arg">S</var> )false
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BettiElementsOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<p>< class=Arg>S</var is a  semigroup./>

<p>The output is the set of elements in <var class="Arg">S</var> whose associated graph is nonconnected <a href="chapBib_mj.html#biBGS-O">[GO10]</a>.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p><var class="Arg">S</var> is a numerical semigroup. The output is the union of all minimal=Arg>S/var.Notice that if[,y isa minimalrelator,then [x,y]or y,]will be intheoutput,but not.</p>
[ 10, 12, 14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BettiElements(s);</span>
[ 10,,12,14 ]
</pre></div>

<p><a id="X7FC66A1B82E86FAF" name=" [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ]]

<h5>4.1-4 span class="GAPpromptgt;/pan><span class="GAPinput"AllMinimalRelationsOfNumericalSemigroup(s);</span>

<div class="func"><table class="func" width="100%"</re></iv>
<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
<spanclass"APprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[
<s"APprompt>gap&;/> spanGAPinput>([0,0,02,;</span>
false
<span class="GAPprompt">gap></span> <span class><var class=Arg></var>isa numerical semigroup./>
true
</pre></div>

<p><a id="X8750A6837EF75CA2" name="X8750A6837EF75CA2"></a></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>4.1-5 AllMinimalRelationsOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100="GAPinput">DegreesOfPrimitiveElementsOfNumericalSemigroup(s);</span>
<p<var =""S<var isa  semigroup.The output the union of allminimal presentations of <var class="Arg">S</var>. Notice that if [x,y] is a minimal relator, then either [x,y] or [y,x] will be in the output, but not both.</p>

<div class="example">div class"func"><table class="func" width="100%"<>td ="tdleft"code ="unc>&8227; ShadedSetOfElementInNumericalSemigroup</code>( <var class="Arg">n</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<panclass="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(4,6,9);;</span>
<spanis  is generalization of the  associated to var class=Arg"n/var>.</>
[ [ [ 0,
<span class=example">pre>
[[[0, 3 0 ,[0 0, 2 ]] [  3,0  ,[ 0,2 0  ]   3,1 0 ] [ , 0,2 ]]
</pre></div>[ [  ] [ 3 ] [ , 7 ], 5 ]  7]]

<p><a id="X7A9B5AE782CAEA2F" name="X7A9B5AE782CAEA2F"></a></p>

<h5>4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</h5>

<div class
<<varclass"Arg>S</var> is a numerical semigroup.</>

<p>The output is the set of elements <span class="SimpleMath">\(s\)</span> in <var class="Arg">S</var> such that there exists a minimal solution to <span class="SimpleMath">\(msg\cdot x-msg\cdot y = 0\)</span>, such that <span class="SimpleMath">\(x,y\)</span> are factorizations of <span class="SimpleMath">\(s\)</span>, and <spanp>< class=SimpleMath"\S)/spanbe a numerical semigroup, andlet span ="impleMath"\K)/pan bea .Let span =SimpleMath"\({n_1\dotsn_e\\<span> aset   generators spanclass=SimpleMath"\(\<span>,and let<spanclass=SimpleMath"\([x_1\ots,x_e])/span  the ring  polynomialin  span =java.lang.StringIndexOutOfBoundsException: Range [393, 392) out of bounds for length 659

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap&>a id"7E6BBAA7803DE7F3" =X7E6BBAA7803DE7F3"></a></p>
[ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
</pre>

<><a id"C42DEB68285F2B8" name="X7C42DEB68285F2B8"></a></p>

<h5>4.1-7 ShadedSetOfElementInNumericalSemigroupThe argument <class="Arg">K</var> is optional; when it is not supplied, the field of rational numbers is taken as java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><tdclass"tdleft"><ode class="func"&# style='color: green'>227; ShadedSetOfElementInNumericalSemigroup</code>( <var class="Arg">n</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> is an ^*x_2+x_3^,x_1+x_2x_3,x_1*+^ ]

<p>The output is a simplicial complex <span class="SimpleMath">\(C\)</span>. If <span class="SimpleMath">\( \{m_1,\ldots,m_n\} \)</span> is the set of minimal&ttwo-sidedideal  Rationalsx_1,,_] (3generators)gt

<p>[ -x_1^3*x_2+_32,-1^4+x_2*x_3, x_1x_3+x_2^2 ]

<div class="example"><pre>
<span">gap></span> <span class=GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap><[ [[0,0,2], [3 1, 0 ], [ 0 1,  ],[ 4,0,0]],java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
[ [  ], [ 3 ],
</pre></div>

<p>a ="X795E7F5682A6C8B3" name="X795E7F5682A6C8B3"></a></p>

<h4>4.2 <span class="Heading">Binomial ideals associated to numerical java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 0

spanclass>\\<spanbe  numericalsemigroup,andlet<spanclass="SimpleMath\(\<span>be a . Let <pan class=SimpleMath">\(\{n_1,\dots,n_e\}\)</span> be a set of minimal generators of <span class="SimpleMath">\(S\)</span>, and let <span class="SimpleMath">\(K[x_1,\dots,x_e]\)</span> be the ring of polynomial in the indeterminates <span class="SimpleMath">\(x_1,\dots,x_e\)</span> and div =func>table class=" width=100%<><java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 244

<p>Let <span class="SimpleMath">\(\varphi: K[x_1,\dots,x_e] \to K[t]\)</span> be the ring homomorphism determined by <span class="SimpleMath">\(\varphi(x_i)=t^{n_i}java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7E6BBAA7803DE7F3" name="X7E6BBAA7803DE7F3"></a></p>

<h5>4.2true

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 4
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
<spanclass"">gapgt;span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BinomialIdealOfNumericalSemigroup(GF(2),s);</span>
lt;wo- idealin GF(2[_1,_2x_3] (generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfTwoSidedIdeal(last);</span>
[ ^*x_2+^2, x_1^x_2*x_3,x_1x_3+x_2^2]
<span class="GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<two- ideal in Rationals[x_1,x_2,x_3], (3 generators)>
<span class="GAPprompt">gap></span> <span span class="GAPprompt">gap></span> <span class>IsGeneric(s);</span>
[ x_1^3*x_2+x_3^2,-x_14+x_2*x_3, -x_1*x_3+x_2^2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [0,, 0 ] [ 1, 0, 1] ]java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
</pre></div>

<p< id"X7D7EA20F818A5994"name"X7D7EA20F818A5994"><a>/p

<h4>4.3 <span class="r />

<p>A numerical semigroup <span class="SimpleMath">\(S\)</span> is uniquely presented if for any two minimal presentations /body>

<p><a id="X7C6F554486274CAE" name="X7C6F554486274CAE"></a></p>

<h5>4.3-1 IsUniquelyPresented</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsUniquelyPresented</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsUniquelyPresentedNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>

<p>The output is true if <var class="Arg">S</var> has uniquely presented. The implementation is based on <a href="chapBib_mj.html#biBGS-O">[GO10]</a>.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniquelyPresented(s);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniquelyPresentedNumericalSemigroup(s);</span>
true
</pre></div>

<p><a id="X79C010537C838154" name="X79C010537C838154"></a></p>

<h5>4.3-2 IsGeneric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneric</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGenericNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>

<p>The output is true if <var class="Arg">S</var> has a generic presentation, that is, in every minimal relation all generators occur. These semigroups are uniquely presented (see <a href="chapBib_mj.html#biBB-GS-G">[BGG11]</a>).</p>

<p>This filter implies <code class="func">IsUniquelyPresentedNumericalSemigroup</code> (<a href="chap4_mj.html#X7C6F554486274CAE"><span class="RefLink">4.3-1</span></a>).</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneric(s);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGenericNumericalSemigroup(s);</span>
true
</pre></div>

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