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                Presentations of Numerical Semigroups
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<p id="mathjaxlink" class="pcenter"><a href="chap4_mj.html">[MathJax on]</a></p>
<p><a id="X7969F7F27AAF0BF1" name="X7969F7F27AAF0BF1"></a></p>
<div class="ChapSects"><a href="chap4.html#X7969F7F27AAF0BF1">4 <span class="Heading">
                Presentations of Numerical Semigroups
            </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7969F7F27AAF0BF1">4.1 <span class="Heading">Presentations of Numerical Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81A2C4317A0BA48D">4.1-1 MinimalPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81CC5A6C870377E1">4.1-2 GraphAssociatedToElementInNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X815C0AF17A371E3E">4.1-3 BettiElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FC66A1B82E86FAF">4.1-4 IsMinimalRelationOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8750A6837EF75CA2">4.1-5 AllMinimalRelationsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A9B5AE782CAEA2F">4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C42DEB68285F2B8">4.1-7 ShadedSetOfElementInNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X795E7F5682A6C8B3">4.2 <span class="Heading">Binomial ideals associated to numerical semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E6BBAA7803DE7F3">4.2-1 BinomialIdealOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7D7EA20F818A5994">4.3 <span class="Heading">Uniquely Presented Numerical Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C6F554486274CAE">4.3-1 IsUniquelyPresented</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79C010537C838154">4.3-2 IsGeneric</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">
                Presentations of Numerical Semigroups
            </span></h3>

<p>In this chapter we explain how to compute a minimal presentation of a numerical semigroup. Recall that a minimal presentation is a minimal generating 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,...,n_e}</span>, then the factorization map is the epimorphism <span class="SimpleMath">φ: N^e-> S</span>, <span class="SimpleMath">(x_1,...,x_e)↦ x_1n_1+dots+ x_en_e</span>; its kernel is the congruence <span class="SimpleMath">{ (a,b) ∣ φ(a)=φ(b)}</span>.</p>

<p>The set of minimal generators is stored in a set, and so it may not be arranged as the user gave them. This may affect the arrangement of the coordinates of the pairs in a minimal presentation, since every coordinate is associated to a minimal generator.</p>

<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. A generalization of these graphs are the simplicial complexes called shaded sets of an element.</p>

<p>If the variable <var class="Arg">NumSgpsUseEliminationForMinimalPresentations</var> is set to true, then minimal presentations are computed via binomial ideals and elimination.</p>

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

<h5>4.1-1 MinimalPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MinimalPresentation</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>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 numerical semigroup. The output is a list of lists with two elements. Each list of two elements represents a relation between the minimal generators of the numerical semigroup. If <span class="SimpleMath">{ {x_1,y_1},...,{x_k,y_k}}</span> is the output and <span class="SimpleMath">{m_1,...,m_n}</span> is the minimal system of generators of the numerical semigroup, then <span class="SimpleMath">{x_i,y_i}={{a_i_1,...,a_i_n},{b_i_1,...,b_i_n}}</span> and <span class="SimpleMath">a_i_1m_1+⋯+a_i_nm_n= b_i_1m_1+ ⋯ +b_i_nm_n.</span></p>

<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.html#biBRos96">[Ros96a]</a> (see also <a href="chapBib.html#biBRGS99">[RG99b]</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">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0, 0 ] ], 
  [ [ 0, 2, 0 ], [ 1, 0, 1 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentationOfNumericalSemigroup(s);</span>
[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0, 0 ] ], 
  [ [ 0, 2, 0 ], [ 1, 0, 1 ] ] ]
</pre></div>

<p>The first element in the list means that <span class="SimpleMath">1× 3+1× 7=2× 5</span>, and the others have similar meanings.</p>

<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 class="func">&#an style='color: green'>8227; GraphAssociatedToElementInNumericalSemigroup</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 element in <var class="Arg">S</var>.</p>

<p>The output is a pair. If <span class="SimpleMath">{m_1,...,m_n}</span> is the set of minimal generators of <var class="Arg">S</var>, then the first component is the set of vertices of the graph associated to <var class="Arg">n</var> in <var class="Arg">S</var>, that is, the set <span class="SimpleMath">{ m_i | n-m_i∈ S}</span>, and the second component is the set of edges of this graph, that is, <span class="SimpleMath">{ {m_i,m_j} | n-(m_i+m_j)∈ S}.</span></p>

<p>This function is used to compute a minimal presentation of the numerical semigroup <var class="Arg">S</var>, as explained in <a href="chapBib.html#biBRos96">[Ros96a]</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">GraphAssociatedToElementInNumericalSemigroup(10,s);</span>
[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
</pre></div>

<p><a id="X815C0AF17A371E3E" name="X815C0AF17A371E3E"></a></p>

<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> )</td><td class="tdright">( operation )</td></tr></table></div>
<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="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>

<p>The output is the set of elements in <var class="Arg">S</var> whose associated graph is nonconnected <a href="chapBib.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">BettiElementsOfNumericalSemigroup(s);</span>
[ 10, 12, 14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BettiElements(s);</span>
[ 10, 12, 14 ]
</pre></div>

<p><a id="X7FC66A1B82E86FAF" name="X7FC66A1B82E86FAF"></a></p>

<h5>4.1-4 IsMinimalRelationOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsMinimalRelationOfNumericalSemigroup</code>( <var class="Arg">p</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">p</var> is a pair (a relation) of lists of integers. Determines if the pair <var class="Arg">p</var> is a minimal relation in a minimal presentation of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(4,6,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMinimalRelationOfNumericalSemigroup([[2,1,0],[0,0,2]],s);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMinimalRelationOfNumericalSemigroup([[3,1,0],[0,0,2]],s);</span>
true
</pre></div>

<p><a id="X8750A6837EF75CA2" name="X8750A6837EF75CA2"></a></p>

<h5>4.1-5 AllMinimalRelationsOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllMinimalRelationsOfNumericalSemigroup</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. The output is the union of all minimal 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"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(4,6,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllMinimalRelationsOfNumericalSemigroup(s);</span>
[ [ [ 0, 3, 0 ], [ 0, 0, 2 ] ], [ [ 3, 0, 0 ], [ 0, 2, 0 ] ], [ [ 3, 1, 0 ], [ 0, 0, 2 ] ] ]
</pre></div>

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

<h5>4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DegreesOfPrimitiveElementsOfNumericalSemigroup</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>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⋅ x-msg⋅ y = 0</span>, such that <span class="SimpleMath">x,y</span> are factorizations of <span class="SimpleMath">s</span>, and <span class="SimpleMath">msg</span> is the minimal generating system of <var class="Arg">S</var>. Betti elements are primitive, but not the way around in general.</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">DegreesOfPrimitiveElementsOfNumericalSemigroup(s);</span>
[ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
</pre></div>

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

<h5>4.1-7 ShadedSetOfElementInNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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 element in <var class="Arg">S</var>.</p>

<p>The output is a simplicial complex <span class="SimpleMath">C</span>. If <span class="SimpleMath">{m_1,...,m_n}</span> is the set of minimal generators of <var class="Arg">S</var>, then <span class="SimpleMath">L ∈ C</span> if <span class="SimpleMath">n-∑_i∈ L m_i∈ S</span> (<a href="chapBib.html#biBSzW">[SW86]</a>).</p>

<p>This function is a generalization of the graph associated to <var class="Arg">n</var>.</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">ShadedSetOfElementInNumericalSemigroup(10,s);</span>
[ [  ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]
</pre></div>

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

<h4>4.2 <span class="Heading">Binomial ideals associated to numerical semigroups</span></h4>

<p>Let <span class="SimpleMath">S</span> be a numerical semigroup, and let <span class="SimpleMath">K</span> be a field. Let <span 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 with coefficients in <span class="SimpleMath">K</span>. Let <span class="SimpleMath">K[t]</span> be the ring of polynomials in <span class="SimpleMath">t</span> with coefficients in <span class="SimpleMath">K</span>.</p>

<p>Let <span class="SimpleMath">φ: K[x_1,dots,x_e] -> K[t]</span> be the ring homomorphism determined by <span class="SimpleMath">φ(x_i)=t^n_i</span> for all <span class="SimpleMath">i</span>. The image of this morphism is usually known as the <em>semigroup ring associated</em> to <span class="SimpleMath">S</span>. The kernel is the <em>(binomial) ideal associated</em> to <span class="SimpleMath">S</span>. According to <a href="chapBib.html#biBMR0269762">[Her70]</a>, from the exponents of the binomials in this ideal we can recover a presentation of <span class="SimpleMath">S</span> and viceversa.</p>

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

<h5>4.2-1 BinomialIdealOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BinomialIdealOfNumericalSemigroup</code>( [<var class="Arg">K</var>, ]<var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The argument <var class="Arg">K</var> is optional; when it is not supplied, the field of rational numbers is taken as base field. <var class="Arg">S</var> is a numerical semigroup. The output is the binomial ideal associated to <var class="Arg">S</var>.</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">BinomialIdealOfNumericalSemigroup(GF(2),s);</span>
<two-sided ideal in GF(2)[x_1,x_2,x_3], (3 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfTwoSidedIdeal(last);</span>
[ x_1^3*x_2+x_3^2, x_1^4+x_2*x_3, x_1*x_3+x_2^2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BinomialIdealOfNumericalSemigroup(s);</span>
<two-sided ideal in Rationals[x_1,x_2,x_3], (3 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfTwoSidedIdeal(last);</span>
[ -x_1^3*x_2+x_3^2, -x_1^4+x_2*x_3, -x_1*x_3+x_2^2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0, 0 ] ], 
[ [ 0, 2, 0 ], [ 1, 0, 1 ] ] ]
</pre></div>

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

<h4>4.3 <span class="Heading">Uniquely Presented Numerical Semigroups</span></h4>

<p>A numerical semigroup <span class="SimpleMath">S</span> is uniquely presented if for any two minimal presentations <span class="SimpleMath">σ</span> and <span class="SimpleMath">τ</span> and any <span class="SimpleMath">(a,b)∈ σ</span>, either <span class="SimpleMath">(a,b)∈ τ</span> or <span class="SimpleMath">(b,a)∈ τ</span>, that is, there is essentially a unique minimal presentation (up to arrangement of the components of the pairs in it).</p>

<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.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.html#biBB-GS-G">[BGG11]</a>).</p>

<p>This filter implies <code class="func">IsUniquelyPresentedNumericalSemigroup</code> (<a href="chap4.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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