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  Nonunique invariants for factorizations in numerical semigroups
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<div class="ChapSects"><a href="chap9.html#X7B6F914879CD505F">9 <span class="Heading">
  Nonunique invariants for factorizations in numerical semigroups
 </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7FDB54217B15148F">9.1 <span class="Heading">
        Factorizations in Numerical Semigroups
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8429AECF78EE7EAB">9.1-1 FactorizationsIntegerWRTList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X80EF105B82447F30">9.1-2 Factorizations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87C9E03C818AE1AA">9.1-3 FactorizationsElementListWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X813D2A3A83916A36">9.1-4 RClassesOfSetOfFactorizations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7C5EED6D852C24DD">9.1-5 LShapes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86062FCA85A51870">9.1-6 RFMatrices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86D58E0084CFD425">9.1-7 DenumerantOfElementInNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X801DA4247A0BEBDA">9.1-8 DenumerantFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7D91A9377DAFAE35">9.1-9 DenumerantIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X846FEE457D4EC03D">9.2 <span class="Heading">
        Invariants based on lengths
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7D4CC092859AF81F">9.2-1 LengthsOfFactorizationsIntegerWRTList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7FDE4F94870951B1">9.2-2 LengthsOfFactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X860E461182B0C6F5">9.2-3 Elasticity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7A2B01BB87086283">9.2-4 Elasticity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X79C953B5846F7057">9.2-5 DeltaSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7DB8BA5B7D6F81CB">9.2-6 DeltaSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7A08CF05821DD2FC">9.2-7 DeltaSetPeriodicityBoundForNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8123FC0E83ADEE45">9.2-8 DeltaSetPeriodicityStartForNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X80B5DF908246BEB1">9.2-9 DeltaSetListUpToElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X85C6973E81583E8B">9.2-10 DeltaSetUnionUpToElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X83B06062784E0FD9">9.2-11 DeltaSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7AEFE27E87F51114">9.2-12 MaximumDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7F8B10C2870932B8">9.2-13 IsAdditiveNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X790308B07AB1A5C8">9.2-14 MaximalDenumerant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7DFC4ED0827761C1">9.2-15 MaximalDenumerantOfSetOfFactorizations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X811E5FFB83CCA4CE">9.2-16 MaximalDenumerant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87F633D98003DE52">9.2-17 Adjustment</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X84F5CA8D7B0F6C02">9.3 <span class="Heading">
        Invariants based on distances
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86F9D7868100F6F9">9.3-1 CatenaryDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7DDB40BB84FF0042">9.3-2 AdjacentCatenaryDegreeOfSetOfFactorizations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86E0CAD28655839C">9.3-3 EqualCatenaryDegreeOfSetOfFactorizations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X845D850F7812E176">9.3-4 MonotoneCatenaryDegreeOfSetOfFactorizations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X797147AA796D1AFE">9.3-5 CatenaryDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X80D478418403E7CB">9.3-6 TameDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X785B83F17BEEA894">9.3-7 CatenaryDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X863E3EF986764267">9.3-8 DegreesOffEqualPrimitiveElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X780E2C737FA8B2A9">9.3-9 EqualCatenaryDegreeOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E19683D7ADDE890">9.3-10 DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E0458187956C395">9.3-11 MonotoneCatenaryDegreeOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X809D97A179765EE6">9.3-12 TameDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7F7619BD79009B64">9.3-13 TameDegree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X78EBC6A57B8167E6">9.4 <span class="Heading">
        Primality
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X83075D7F837ACCB8">9.4-1 OmegaPrimality</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X85EB5E2581FFB8B2">9.4-2 OmegaPrimalityOfElementListInNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X80B48B7886A93FAC">9.4-3 OmegaPrimality</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X86735EEA780CECDA">9.5 <span class="Heading">
        Homogenization of Numerical Semigroups
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X856B689185C1F5D9">9.5-1 BelongsToHomogenizationOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X85D03DBB7BA3B1FB">9.5-2 FactorizationsInHomogenizationOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X857CC7FF85C05318">9.5-3 HomogeneousBettiElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7DFFCAC87B3B632B">9.5-4 HomogeneousCatenaryDegreeOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7A54E9FD7D4CB18F">9.6 <span class="Heading">
        Divisors, posets
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X853930E97F7F8A43">9.6-1 MoebiusFunctionAssociatedToNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7DF6825185C619AC">9.6-2 MoebiusFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8771F39A7C7E031E">9.6-3 DivisorsOfElementInNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X871CD69180783663">9.6-4 NumericalSemigroupByNuSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7F4CBFF17BBB37DE">9.6-5 NumericalSemigroupByTauSequence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X82D8A59083FCDF46">9.7 <span class="Heading">
        Feng-Rao distances and numbers
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7939BCE08655B62D">9.7-1 FengRaoDistance</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X83F9F4C67D4535EF">9.7-2 FengRaoNumber</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X79A8A15087CEE8C1">9.8 <span class="Heading">
        Numerical semigroups with Apéry sets having special factorization properties
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7B894ED27D38E4B5">9.8-1 IsPure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8400FB5D81EFB5FE">9.8-2 IsMpure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X80B707EE79990E1E">9.8-3 IsHomogeneousNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8630DEF77A350D76">9.8-4 IsSuperSymmetricNumericalSemigroup</a></span>
</div></div>
</div>

<h3>9 <span class="Heading">
  Nonunique invariants for factorizations in numerical semigroups
 </span></h3>

<p>Let <span class="SimpleMath">S</span> be a numerical semigroup minimally generated by <span class="SimpleMath">{m_1,...,m_n}</span>. A <em>factorization</em> of an element <span class="SimpleMath">s∈ S</span> is an n-tuple <span class="SimpleMath">a=(a_1,...,a_n)</span> of nonnegative integers such that <span class="SimpleMath">n=a_1 n_1+⋯+a_n m_n</span>. The <em>length</em> of <span class="SimpleMath">a</span> is <span class="SimpleMath">|a|=a_1+⋯+a_n</span>. Given two factorizations <span class="SimpleMath">a</span> and <span class="SimpleMath">b</span> of <span class="SimpleMath">n</span>, the <em>distance</em> between <span class="SimpleMath">a</span> and <span class="SimpleMath">b</span> is <span class="SimpleMath">d(a,b)=max { |a-gcd(a,b)|,|b-gcd(a,b)|}</span>, where <span class="SimpleMath">gcd((a_1,...,a_n),(b_1,...,b_n))=(min(a_1,b_1),...,min(a_n,b_n))</span>. In the literature, factorizations are sometimes called representations or expressions of the element in terms of the generators.</p>

<p>If <span class="SimpleMath">l_1>⋯ > l_k</span> are the lengths of all the factorizations of <span class="SimpleMath">s ∈ S</span>, the <em>delta set</em> associated to <span class="SimpleMath">s</span> is <span class="SimpleMath">∆(s)={l_1-l_2,...,l_k-l_k-1}</span>.</p>

<p>The <em>catenary degree</em> of an element in <span class="SimpleMath">S</span> is the least positive integer <span class="SimpleMath">c</span> such that for any two of its factorizations <span class="SimpleMath">a</span> and <span class="SimpleMath">b</span>, there exists a chain of factorizations starting in <span class="SimpleMath">a</span> and ending in <span class="SimpleMath">b</span> and so that the distance between two consecutive links is at most <span class="SimpleMath">c</span>. The <em>catenary degree</em> of <span class="SimpleMath">S</span> is the supremum of the catenary degrees of the elements in <span class="SimpleMath">S</span>.</p>

<p>The <em>tame degree</em> of <span class="SimpleMath">S</span> is the least positive integer <span class="SimpleMath">t</span> such that for any factorization <span class="SimpleMath">a</span> of an element <span class="SimpleMath">s</span> in <span class="SimpleMath">S</span>, and any <span class="SimpleMath">i</span> such that <span class="SimpleMath">s-m_i∈ S</span>, there exists another factorization <span class="SimpleMath">b</span> of <span class="SimpleMath">s</span> so that the distance to <span class="SimpleMath">a</span> is at most <span class="SimpleMath">t</span> and <span class="SimpleMath">b_inot = 0</span>.</p>

<p>The <em><span class="SimpleMath">ω</span>-primality</em> of an element <span class="SimpleMath">s</span> in <span class="SimpleMath">S</span> is the least positive integer <span class="SimpleMath">k</span> such that if <span class="SimpleMath">(∑_i∈ I s_i)-s∈ S, s_i∈ S</span>, then there exists <span class="SimpleMath">Ω⊆ I</span> with cardinality <span class="SimpleMath">k</span> such that <span class="SimpleMath">(∑_i∈ Ω s_i)-s∈ S</span>. The <em><span class="SimpleMath">ω</span>-primality</em> of <span class="SimpleMath">S</span> is the maximum of the <span class="SimpleMath">ω</span>-primality of its minimal generators.</p>

<p>The basic properties of these constants can be found in <a href="chapBib.html#biBGHKb">[GH06]</a>. The algorithm used to compute the catenary and tame degree is an adaptation of the algorithms appearing in <a href="chapBib.html#biBCGLPR">[CGL+06]</a> for numerical semigroups (see <a href="chapBib.html#biBCGL">[CGD07]</a>). The computation of the elasticity of a numerical semigroup reduces to <span class="SimpleMath">m/n</span> with <span class="SimpleMath">m</span> the multiplicity of the semigroup and <span class="SimpleMath">n</span> its largest minimal generator (see <a href="chapBib.html#biBCHM06">[CHM06]</a> or <a href="chapBib.html#biBGHKb">[GH06]</a>).</p>

<p><a id="X7FDB54217B15148F" name="X7FDB54217B15148F"></a></p>

<h4>9.1 <span class="Heading">
        Factorizations in Numerical Semigroups
    </span></h4>

<p>Denumerants, sets of factorizations, R-classes, and L-shapes are described in this section.</p>

<p><a id="X8429AECF78EE7EAB" name="X8429AECF78EE7EAB"></a></p>

<h5>9.1-1 FactorizationsIntegerWRTList</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorizationsIntegerWRTList</code>( <var class="Arg">n</var>, <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a list of integers and <var class="Arg">n</var> an integer. The output is the set of factorizations of <var class="Arg">n</var> in terms of the elements in the list <var class="Arg">ls</var>. This function uses <code class="func">RestrictedPartitions</code> (<a href="../../../doc/ref/chap16.html#X7A70D4F3809494E7"><span class="RefLink">Reference: RestrictedPartitions</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ],
  [ 5, 2, 0, 1 ], [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
</pre></div>

<p><a id="X80EF105B82447F30" name="X80EF105B82447F30"></a></p>

<h5>9.1-2 Factorizations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Factorizations</code>( <var class="Arg">n</var>, <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">‣ Factorizations</code>( <var class="Arg">S</var>, <var class="Arg">n</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">‣ FactorizationsElementWRTNumericalSemigroup</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> an element of <var class="Arg">S</var>. The output is the set of factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Factorizations(1100,s);</span>
[ [ 0, 8, 1, 0, 0, 0 ], [ 0, 0, 0, 2, 2, 0 ], [ 5, 1, 1, 0, 0, 1 ],
  [ 0, 2, 3, 0, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Factorizations(s,1100)=Factorizations(1100,s);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsElementWRTNumericalSemigroup(1100,s)=Factorizations(1100,s);</span>
true
</pre></div>

<p><a id="X87C9E03C818AE1AA" name="X87C9E03C818AE1AA"></a></p>

<h5>9.1-3 FactorizationsElementListWRTNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorizationsElementListWRTNumericalSemigroup</code>( <var class="Arg">l</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">l</var> a list of elements of <var class="Arg">S</var>.</p>

<p>Computes the factorizations of all the elements in <var class="Arg">l</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(10,11,13);</span>
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsElementListWRTNumericalSemigroup([100,101,103],s);</span>
[ [ [ 0, 2, 6 ], [ 1, 7, 1 ], [ 3, 4, 2 ], [ 5, 1, 3 ], [ 10, 0, 0 ] ],
  [ [ 0, 8, 1 ], [ 1, 0, 7 ], [ 2, 5, 2 ], [ 4, 2, 3 ], [ 9, 1, 0 ] ],
  [ [ 0, 7, 2 ], [ 2, 4, 3 ], [ 4, 1, 4 ], [ 7, 3, 0 ], [ 9, 0, 1 ] ] ]
    </pre></div>

<p><a id="X813D2A3A83916A36" name="X813D2A3A83916A36"></a></p>

<h5>9.1-4 RClassesOfSetOfFactorizations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RClassesOfSetOfFactorizations</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a set of factorizations (a list of lists of nonnegative integers with the same length). The output is the set of <span class="SimpleMath">mathcal R</span>-classes of this set of factorizations as defined in Chapter 7 of <a href="chapBib.html#biBRGbook">[RG09]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(10,11,19,23);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BettiElements(s);</span>
[ 30, 33, 42, 57, 69 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Factorizations(69,s);</span>
[ [ 5, 0, 1, 0 ], [ 2, 1, 2, 0 ], [ 0, 0, 0, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RClassesOfSetOfFactorizations(last);</span>
[ [ [ 2, 1, 2, 0 ], [ 5, 0, 1, 0 ] ], [ [ 0, 0, 0, 3 ] ] ]
</pre></div>

<p><a id="X7C5EED6D852C24DD" name="X7C5EED6D852C24DD"></a></p>

<h5>9.1-5 LShapes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LShapes</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">‣ LShapesOfNumericalSemigroup</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 number of LShapes associated to <var class="Arg">S</var>. These are ways of arranging the set of factorizations of the elements in the Apéry set of the largest generator, so that if one factorization <span class="SimpleMath">x</span> is chosen for <span class="SimpleMath">w</span> and <span class="SimpleMath">w-w'∈ S, then only the factorization of x'</span> of <span class="SimpleMath">w' with x'≤ x</span> can be in the LShape (and if there is no such a factorization, then we have no LShape with <span class="SimpleMath">x</span> in it), see <a href="chapBib.html#biBAG-GS">[AG10]</a>.</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">LShapes(s);</span>
[ [ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ], [ 2, 0 ], [ 1, 1 ], [ 0, 2 ], [ 2, 1 ],
      [ 1, 2 ], [ 2, 2 ] ],
  [ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ], [ 2, 0 ], [ 1, 1 ], [ 3, 0 ], [ 2, 1 ],
      [ 4, 0 ], [ 5, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LShapesOfNumericalSemigroup(s) = LShapes(s);</span>
true     
</pre></div>

<p><a id="X86062FCA85A51870" name="X86062FCA85A51870"></a></p>

<h5>9.1-6 RFMatrices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RFMatrices</code>( <var class="Arg">f</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">f</var> is a pseudo-Frobenius number of <var class="Arg">S</var>.</p>

<p>The output is the list of RF-matrices associated to <var class="Arg">f</var>. The ith row of each matrix contains the coefficients of a combination of <var class="Arg">f</var> in terms of the minimal generators of the semigroup, obtained by substraction the ith generator to the factorizations of <var class="Arg">f</var> plus the ith generator of <var class="Arg">S</var>, see <a href="chapBib.html#biBRFMatrix">[Mos16]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(6, 7, 9, 10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RFMatrices(8,s);</span>
[ [ [ -1, 2, 0, 0 ], [ 1, -1, 1, 0 ], [ 0, 1, -1, 1 ], [ 3, 0, 0, -1 ] ],
  [ [ -1, 2, 0, 0 ], [ 1, -1, 1, 0 ], [ 0, 1, -1, 1 ], [ 0, 0, 2, -1 ] ] ]
</pre></div>

<p><a id="X86D58E0084CFD425" name="X86D58E0084CFD425"></a></p>

<h5>9.1-7 DenumerantOfElementInNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DenumerantOfElementInNumericalSemigroup</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> a positive integer. The output is the number of factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,195,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DenumerantOfElementInNumericalSemigroup(1311,s);</span>
6
</pre></div>

<p><a id="X801DA4247A0BEBDA" name="X801DA4247A0BEBDA"></a></p>

<h5>9.1-8 DenumerantFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DenumerantFunction</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup. The output is a function that for a given <span class="SimpleMath">n</span> computes the number of factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,195,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DenumerantFunction(s)(1311);</span>
6
</pre></div>

<p><a id="X7D91A9377DAFAE35" name="X7D91A9377DAFAE35"></a></p>

<h5>9.1-9 DenumerantIdeal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DenumerantIdeal</code>( <var class="Arg">n</var>, <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">‣ DenumerantIdeal</code>( <var class="Arg">S</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> a nonnegative integer. The output is the ideal of elements in <var class="Arg">S</var> with more than <var class="Arg">n</var> factorizations. If we add zero to this set, we obtain what is is called in <a href="chapBib.html#biBkomatsu">[Kom24]</a> an <var class="Arg">n</var>-semigroup.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,195,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">1311 in DenumerantIdeal(6,s);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">1311 in DenumerantIdeal(5,s);</span>
true
</pre></div>

<p><a id="X846FEE457D4EC03D" name="X846FEE457D4EC03D"></a></p>

<h4>9.2 <span class="Heading">
        Invariants based on lengths
    </span></h4>

<p>This section is devoted to nonunique factorization invariants based on lengths of factorizations. There are some families of numerical semigroups related to maximal denumerantes; membership tests for these families are provede here.</p>

<p><a id="X7D4CC092859AF81F" name="X7D4CC092859AF81F"></a></p>

<h5>9.2-1 LengthsOfFactorizationsIntegerWRTList</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LengthsOfFactorizationsIntegerWRTList</code>( <var class="Arg">n</var>, <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a list of integers and <var class="Arg">n</var> an integer. The output is the set of lengths of the factorizations of <var class="Arg">n</var> in terms of the elements in <var class="Arg">ls</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthsOfFactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ 6, 8 ]
</pre></div>

<p><a id="X7FDE4F94870951B1" name="X7FDE4F94870951B1"></a></p>

<h5>9.2-2 LengthsOfFactorizationsElementWRTNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LengthsOfFactorizationsElementWRTNumericalSemigroup</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> an element of <var class="Arg">S</var>. The output is the set of lengths of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);</span>
<Numerical semigroup with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthsOfFactorizationsElementWRTNumericalSemigroup(1100,s);</span>
[ 4, 6, 8, 9 ]
</pre></div>

<p><a id="X860E461182B0C6F5" name="X860E461182B0C6F5"></a></p>

<h5>9.2-3 Elasticity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Elasticity</code>( <var class="Arg">n</var>, <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">‣ Elasticity</code>( <var class="Arg">S</var>, <var class="Arg">n</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">‣ ElasticityOfFactorizationsElementWRTNumericalSemigroup</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> an element of <var class="Arg">S</var>. The output is the maximum length divided by the minimum length of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e := Elasticity(1100,s);</span>
9/4
<span class="GAPprompt">gap></span> <span class="GAPinput">Elasticity(1100,s) = Elasticity(s,1100);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ElasticityOfFactorizationsElementWRTNumericalSemigroup(1100,s)= e;</span>
true
</pre></div>

<p><a id="X7A2B01BB87086283" name="X7A2B01BB87086283"></a></p>

<h5>9.2-4 Elasticity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Elasticity</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">‣ ElasticityOfNumericalSemigroup</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 elasticity of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Elasticity(s);</span>
286/101
<span class="GAPprompt">gap></span> <span class="GAPinput">ElasticityOfNumericalSemigroup(s);</span>
286/101
</pre></div>

<p><a id="X79C953B5846F7057" name="X79C953B5846F7057"></a></p>

<h5>9.2-5 DeltaSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSet</code>( <var class="Arg">ls</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">‣ DeltaSetOfSetOfIntegers</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is list of integers. The output is the Delta set of the elements in <var class="Arg">ls</var>, that is, the set of differences of consecutive elements in the list.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthsOfFactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ 6, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSet(last);</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSetOfSetOfIntegers(last2);</span>
[ 2 ]
</pre></div>

<p><a id="X7DB8BA5B7D6F81CB" name="X7DB8BA5B7D6F81CB"></a></p>

<h5>9.2-6 DeltaSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSet</code>( <var class="Arg">n</var>, <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">‣ DeltaSet</code>( <var class="Arg">S</var>, <var class="Arg">n</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">‣ DeltaSetOfFactorizationsElementWRTNumericalSemigroup</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> an element of <var class="Arg">S</var>. The output is the Delta set of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d := DeltaSet(1100,s);</span>
[ 1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSet(s,1100) = d;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSetOfFactorizationsElementWRTNumericalSemigroup(1100,s) = d;</span>
true
</pre></div>

<p><a id="X7A08CF05821DD2FC" name="X7A08CF05821DD2FC"></a></p>

<h5>9.2-7 DeltaSetPeriodicityBoundForNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSetPeriodicityBoundForNumericalSemigroup</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. Computes the bound were the periodicity starts for Delta sets of the elements in <var class="Arg">S</var>; see <a href="chapBib.html#biBGG-MF-VT">[GMV15]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSetPeriodicityBoundForNumericalSemigroup(s);</span>
60
    </pre></div>

<p><a id="X8123FC0E83ADEE45" name="X8123FC0E83ADEE45"></a></p>

<h5>9.2-8 DeltaSetPeriodicityStartForNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSetPeriodicityStartForNumericalSemigroup</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>Computes the element were the periodicity starts for Delta sets of the elements in <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSetPeriodicityStartForNumericalSemigroup(s);</span>
21
    </pre></div>

<p><a id="X80B5DF908246BEB1" name="X80B5DF908246BEB1"></a></p>

<h5>9.2-9 DeltaSetListUpToElementWRTNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSetListUpToElementWRTNumericalSemigroup</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, <var class="Arg">n</var> an integer.</p>

<p>Computes the Delta sets of the integers up to (and including) <var class="Arg">n</var>, if an integer is not in <var class="Arg">S</var>, the corresponding Delta set is empty.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSetListUpToElementWRTNumericalSemigroup(31,s);</span>
[ [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ],
  [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [ 2 ], [  ], [  ], [ 2 ], [  ],
  [ 2 ], [  ], [ 2 ], [ 2 ], [  ] ]
    </pre></div>

<p><a id="X85C6973E81583E8B" name="X85C6973E81583E8B"></a></p>

<h5>9.2-10 DeltaSetUnionUpToElementWRTNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSetUnionUpToElementWRTNumericalSemigroup</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, <var class="Arg">n</var> a nonnegative integer.</p>

<p>Computes the union of the delta sets of the elements of <var class="Arg">S</var> up to and including <var class="Arg">n</var>, using a ring buffer to conserve memory.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSetUnionUpToElementWRTNumericalSemigroup(60,s);</span>
[ 2 ]
    </pre></div>

<p><a id="X83B06062784E0FD9" name="X83B06062784E0FD9"></a></p>

<h5>9.2-11 DeltaSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSet</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">‣ DeltaSetOfNumericalSemigroup</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>Computes the Delta set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSet(s);</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSetOfNumericalSemigroup(s);</span>
[ 2 ]
    </pre></div>

<p><a id="X7AEFE27E87F51114" name="X7AEFE27E87F51114"></a></p>

<h5>9.2-12 MaximumDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximumDegree</code>( <var class="Arg">S</var>, <var class="Arg">n</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">‣ MaximumDegreeOfElementWRTNumericalSemigroup</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> a nonnegative integer. The output is the maximum length of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);</span>
<Numerical semigroup with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximumDegree(1100,s);</span>
9
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximumDegreeOfElementWRTNumericalSemigroup(1100,s);</span>
9
</pre></div>

<p><a id="X7F8B10C2870932B8" name="X7F8B10C2870932B8"></a></p>

<h5>9.2-13 IsAdditiveNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAdditiveNumericalSemigroup</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. Detects if <var class="Arg">S</var> is additive, that is, <span class="SimpleMath">ord(m+x)=ord(x)+1</span> for all <span class="SimpleMath">x</span> in <var class="Arg">S</var>, where <span class="SimpleMath">m</span> is the multiplicity of <var class="Arg">S</var> (<span class="SimpleMath">ord</span> corresponds to <code class="func">MaximumDegreeOfElementWRTNumericalSemigroup</code> (<a href="chap9.html#X7AEFE27E87F51114"><span class="RefLink">9.2-12</span></a>); see Section <a href="chap9.html#X79A8A15087CEE8C1"><span class="RefLink">9.8</span></a> for an alternate definition). For these semigroups <span class="SimpleMath">gr_mathfrakm(K[[S]])</span> is Cohen-Macaulay (see <a href="chapBib.html#biBBH">[BH13]</a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=IrreducibleNumericalSemigroupsWithFrobeniusNumber(31);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(l);</span>
109
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(Filtered(l,IsAdditiveNumericalSemigroup));</span>
20
</pre></div>

<p><a id="X790308B07AB1A5C8" name="X790308B07AB1A5C8"></a></p>

<h5>9.2-14 MaximalDenumerant</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalDenumerant</code>( <var class="Arg">n</var>, <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">‣ MaximalDenumerant</code>( <var class="Arg">S</var>, <var class="Arg">n</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">‣ MaximalDenumerantOfElementInNumericalSemigroup</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> an element of <var class="Arg">S</var>. The output is the number of factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var> with maximal length.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDenumerant(1100,s);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDenumerant(s,1311);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDenumerantOfElementInNumericalSemigroup(1311,s);</span>
2
</pre></div>

<p><a id="X7DFC4ED0827761C1" name="X7DFC4ED0827761C1"></a></p>

<h5>9.2-15 MaximalDenumerantOfSetOfFactorizations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalDenumerantOfSetOfFactorizations</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is list of factorizations (a list of lists of nonnegative integers with the same length). The output is number of elements in <var class="Arg">ls</var> with maximal length.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
  [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDenumerantOfSetOfFactorizations(last);</span>
6
</pre></div>

<p><a id="X811E5FFB83CCA4CE" name="X811E5FFB83CCA4CE"></a></p>

<h5>9.2-16 MaximalDenumerant</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalDenumerant</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">‣ MaximalDenumerantOfNumericalSemigroup</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 maximal denumerant of <var class="Arg">S</var>, that is, the maximum of the maximal denumerants of the elements in <var class="Arg">S</var> (see <a href="chapBib.html#biBBH">[BH13]</a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDenumerant(s);</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalDenumerantOfNumericalSemigroup(s);</span>
4
</pre></div>

<p><a id="X87F633D98003DE52" name="X87F633D98003DE52"></a></p>

<h5>9.2-17 Adjustment</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Adjustment</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">‣ AdjustmentOfNumericalSemigroup</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 adjustment of <var class="Arg">S</var> as defined in <a href="chapBib.html#biBBH">[BH13]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := Adjustment(s);</span>
[ 0, 12, 24, 36, 48, 60, 72, 84, 95, 96, 107, 108, 119, 120, 131, 132, 143,
  144, 155, 156, 167, 168, 171, 177, 179, 180, 183, 185, 189, 190, 191, 192,
  195, 197, 201, 203, 204, 207, 209, 213, 215, 216, 219, 221, 225, 227, 228,
  231, 233, 237, 239, 240, 243, 245, 249, 251, 252, 255, 257, 261, 263, 264,
  266, 267, 269, 273, 275, 276, 279, 280, 281, 285, 287, 288, 292, 293, 299,
  300, 304, 305, 311, 312, 316, 317, 323, 324, 328, 329, 335, 336, 340, 341,
  342, 347, 348, 352, 353, 354, 356, 359, 360, 361, 362, 364, 365, 366, 368,
  370, 371, 372, 374, 376, 377, 378, 380, 382, 383, 384, 388, 389, 390, 394,
  395, 396, 400, 401, 402, 406, 407, 408, 412, 413, 414, 418, 419, 420, 424,
  425, 426, 430, 431, 432, 436, 437, 438, 442, 444, 448, 450, 451, 454, 456,
  460, 465, 466, 472, 477, 478, 484, 489, 490, 496, 501, 502, 508, 513, 514,
  519, 520, 525, 526, 527, 531, 532, 533, 537, 539, 543, 545, 549, 551, 555,
  561, 567, 573, 579, 585, 591, 597, 603, 609, 615, 621, 622, 627, 698, 704,
  710, 716, 722 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AdjustmentOfNumericalSemigroup(s) = a;</span>
true
</pre></div>

<p><a id="X84F5CA8D7B0F6C02" name="X84F5CA8D7B0F6C02"></a></p>

<h4>9.3 <span class="Heading">
        Invariants based on distances
    </span></h4>

<p>This section is devoted to invariants that rely on the concept of distance between two factorizations.</p>

<p><a id="X86F9D7868100F6F9" name="X86F9D7868100F6F9"></a></p>

<h5>9.3-1 CatenaryDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CatenaryDegree</code>( <var class="Arg">ls</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">‣ CatenaryDegreeOfSetOfFactorizations</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a set of factorizations (a list of lists of nonnegative integers with the same length). The output is the catenary degree of this set of factorizations.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ],
  [ 5, 2, 0, 1 ], [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegree(last);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegreeOfSetOfFactorizations(last2);</span>
5
</pre></div>

<p><a id="X7DDB40BB84FF0042" name="X7DDB40BB84FF0042"></a></p>

<h5>9.3-2 AdjacentCatenaryDegreeOfSetOfFactorizations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AdjacentCatenaryDegreeOfSetOfFactorizations</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a set of factorizations. The output is the adjacent catenary degree of this set of factorizations, that is, the supremum of the distance between to sets of factorizations with adjacent lengths. More precisely, if <span class="SimpleMath">l_1,...,l_t</span> are the lengths of the factorizations of the elements in <var class="Arg">ls</var>, and <span class="SimpleMath">Z_l_i</span> is the set of factorizations in <var class="Arg">ls</var> with length <span class="SimpleMath">l_i</span>, then the adjacent catenary degree is the maximum of the distances <span class="SimpleMath">mathrm d (Z_l_i,Z_l_i+1})</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
  [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AdjacentCatenaryDegreeOfSetOfFactorizations(last);</span>
5
</pre></div>

<p><a id="X86E0CAD28655839C" name="X86E0CAD28655839C"></a></p>

<h5>9.3-3 EqualCatenaryDegreeOfSetOfFactorizations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EqualCatenaryDegreeOfSetOfFactorizations</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a set of factorizations. The same as CatenaryDegreeOfSetOfFactorizations, but now the factorizations joined by the chain must have the same length, and the elements in the chain also. Equivalently, if <span class="SimpleMath">l_1,...,l_t</span> are the lengths of the factorizations of the elements in <var class="Arg">ls</var>, and <span class="SimpleMath">Z_l_i</span> is the set of factorizations in <var class="Arg">ls</var> with length <span class="SimpleMath">l_i</span>, then the equal catenary degree is the maximum of the CatenaryDegreeOfSetOfFactorizations of <span class="SimpleMath">mathrm d (Z_l_i,Z_l_i+1})</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
  [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EqualCatenaryDegreeOfSetOfFactorizations(last);</span>
2
</pre></div>

<p><a id="X845D850F7812E176" name="X845D850F7812E176"></a></p>

<h5>9.3-4 MonotoneCatenaryDegreeOfSetOfFactorizations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MonotoneCatenaryDegreeOfSetOfFactorizations</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a set of factorizations. The same as CatenaryDegreeOfSetOfFactorizations, but now the factorizations are joined by a chain with nondecreasing lengths. Equivalently, it is the maximum of the AdjacentCatenaryDegreeOfSetOfFactorizations and the EqualCatenaryDegreeOfSetOfFactorizations.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ], [ 5, 2, 0, 1 ],
  [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MonotoneCatenaryDegreeOfSetOfFactorizations(last);</span>
5
</pre></div>

<p><a id="X797147AA796D1AFE" name="X797147AA796D1AFE"></a></p>

<h5>9.3-5 CatenaryDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CatenaryDegree</code>( <var class="Arg">n</var>, <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">‣ CatenaryDegree</code>( <var class="Arg">S</var>, <var class="Arg">n</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">‣ CatenaryDegreeOfElementInNumericalSemigroup</code>( <var class="Arg">n</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">n</var> is a nonnegative integer and <var class="Arg">S</var> is a numerical semigroup. The output is the catenary degree of <var class="Arg">n</var> relative to <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegree(157,NumericalSemigroup(13,18));</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegree(NumericalSemigroup(13,18),1157);</span>
18
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegreeOfElementInNumericalSemigroup(1157,NumericalSemigroup(13,18));</span>
18
</pre></div>

<p><a id="X80D478418403E7CB" name="X80D478418403E7CB"></a></p>

<h5>9.3-6 TameDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TameDegree</code>( <var class="Arg">ls</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">‣ TameDegreeOfSetOfFactorizations</code>( <var class="Arg">ls</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">ls</var> is a set of factorizations (a list of lists of nonnegative integers with the same length). The output is the tame degree of this set of factorizations.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorizationsIntegerWRTList(100,[11,13,15,19]);</span>
[ [ 2, 6, 0, 0 ], [ 3, 4, 1, 0 ], [ 4, 2, 2, 0 ], [ 5, 0, 3, 0 ],
  [ 5, 2, 0, 1 ], [ 6, 0, 1, 1 ], [ 0, 1, 2, 3 ], [ 1, 1, 0, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TameDegree(last);</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">TameDegreeOfSetOfFactorizations(last2);</span>
4
</pre></div>

<p><a id="X785B83F17BEEA894" name="X785B83F17BEEA894"></a></p>

<h5>9.3-7 CatenaryDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CatenaryDegree</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">‣ CatenaryDegreeOfNumericalSemigroup</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 catenary degree of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);</span>
<Numerical semigroup with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegree(s);</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegreeOfNumericalSemigroup(s);</span>
8
</pre></div>

<p><a id="X863E3EF986764267" name="X863E3EF986764267"></a></p>

<h5>9.3-8 DegreesOffEqualPrimitiveElementsOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreesOffEqualPrimitiveElementsOfNumericalSemigroup</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 with the same length of <span class="SimpleMath">s</span>, and <span class="SimpleMath">msg</span> is the minimal generating system of <var class="Arg">S</var>. These elements are used to compute the equal catenary degree of <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">DegreesOfEqualPrimitiveElementsOfNumericalSemigroup(s);</span>
[ 3, 5, 7, 10 ]
</pre></div>

<p><a id="X780E2C737FA8B2A9" name="X780E2C737FA8B2A9"></a></p>

<h5>9.3-9 EqualCatenaryDegreeOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EqualCatenaryDegreeOfNumericalSemigroup</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 equal catenary degree of <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">EqualCatenaryDegreeOfNumericalSemigroup(s);</span>
2
</pre></div>

<p><a id="X7E19683D7ADDE890" name="X7E19683D7ADDE890"></a></p>

<h5>9.3-10 DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup</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>, with <span class="SimpleMath">|x|≤ |y|</span>; <span class="SimpleMath">msg</span> stands the minimal generating system of <var class="Arg">S</var>. These elements are used to compute the monotone catenary degree of <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">DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup(s);</span>
[ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
</pre></div>

<p><a id="X7E0458187956C395" name="X7E0458187956C395"></a></p>

<h5>9.3-11 MonotoneCatenaryDegreeOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MonotoneCatenaryDegreeOfNumericalSemigroup</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 monotone catenary degree of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(10,23,31,44);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CatenaryDegreeOfNumericalSemigroup(s);</span>
9
<span class="GAPprompt">gap></span> <span class="GAPinput">MonotoneCatenaryDegreeOfNumericalSemigroup(s);</span>
21
</pre></div>

<p><a id="X809D97A179765EE6" name="X809D97A179765EE6"></a></p>

<h5>9.3-12 TameDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TameDegree</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">‣ TameDegreeOfNumericalSemigroup</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 tame degree of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(101,113,196,272,278,286);</span>
<Numerical semigroup with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">TameDegree(s);</span>
14
<span class="GAPprompt">gap></span> <span class="GAPinput">TameDegreeOfNumericalSemigroup(s);</span>
14
</pre></div>

<p><a id="X7F7619BD79009B64" name="X7F7619BD79009B64"></a></p>

<h5>9.3-13 TameDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TameDegree</code>( <var class="Arg">n</var>, <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">‣ TameDegree</code>( <var class="Arg">S</var>, <var class="Arg">n</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">‣ TameDegreeOfElementInNumericalSemigroup</code>( <var class="Arg">n</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">n</var> is an element of the numerical semigroup <var class="Arg">S</var>. The output is the tame degree of <var class="Arg">n</var> in <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(10,11,13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TameDegree(100,s);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">TameDegree(s,100);</span>
5
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