<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>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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, <spanclass="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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<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>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>
<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>
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