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                Constructing numerical semigroups from others
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<p><a id="X8148F05A830EE2D5" name="X8148F05A830EE2D5"></a></p>
<div class="ChapSects"><a href="chap5_mj.html#X8148F05A830EE2D5">5 <span class="Heading">
                Constructing numerical semigroups from others
            </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X782F3AB97ACF84B8">5.1 <span class="Heading">
                    Adding and removing elements of a numerical semigroup
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C94611F7DD9E742">5.1-1 RemoveMinimalGeneratorFromNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X865EA8377D632F53">5.1-2 AddSpecialGapOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7DC65D547FB274D8">5.2 <span class="Heading">Intersections, sums, quotients, dilatations, numerical duplications and multiples by integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X875A8D2679153D4B">5.2-1 Intersection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F308BCE7A0E9D91"><code>5.2-2 \+</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83CCE63C82F34C25">5.2-3 QuotientOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7BE8DD6884DE693F">5.2-4 MultipleOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F395079839BBE9D">5.2-5 NumericalDuplication</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8176CEB4829084B4">5.2-6 AsNumericalDuplication</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7DCEC67A82130CD8">5.2-7 InductiveNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X81632C597E3E3DFE">5.2-8 DilatationOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X867D9A9A87CEB869">5.3 <span class="Heading">
                    Constructing the set of all numerical semigroups containing a given numerical semigroup
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7FBA34637ADAFEDA">5.3-1 OverSemigroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8634CFB1848430DC">5.4 <span class="Heading"> Constructing the set of numerical semigroups with given Frobenius number</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X81759C3482B104D6">5.4-1 NumericalSemigroupsWithFrobeniusNumberFG</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7DB3994B872C4940">5.4-2 NumericalSemigroupsWithFrobeniusNumberAndMultiplicity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87369D567AA6DBA0">5.4-3 NumericalSemigroupsWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X80CACB287B4609E1">5.4-4 NumericalSemigroupsWithFrobeniusNumberPC</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8021419483185FE3">5.5 <span class="Heading"> Constructing the set of numerical semigroups with given maximum primitive</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C17AB04877559B6">5.5-1 NumericalSemigroupsWithMaxPrimitiveAndMultiplicity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X875A8B337DFA01F0">5.5-2 NumericalSemigroupsWithMaxPrimitive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7DA1FA7780684019">5.5-3 NumericalSemigroupsWithMaxPrimitivePC</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7D6635CB7D041A54">5.6 <span class="Heading">
                    Constructing the set of numerical semigroups with genus g
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86970F6A868DEA95">5.6-1 NumericalSemigroupsWithGenus</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B4F3B5E841E3853">5.6-2 NumericalSemigroupsWithGenusPC</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8265233586477CC7">5.7 <span class="Heading">
                    Constructing the set of numerical semigroups with a given set of pseudo-Frobenius numbers
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X874B252180BD7EB4">5.7-1 ForcedIntegersForPseudoFrobenius</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87AAFFF9814E9BD2">5.7-2 SimpleForcedIntegersForPseudoFrobenius</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D6775A57B800892">5.7-3 NumericalSemigroupsWithPseudoFrobeniusNumbers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X862DBFA379D52E2C">5.7-4 ANumericalSemigroupWithPseudoFrobeniusNumbers</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">
                Constructing numerical semigroups from others
            </span></h3>

<p>This chapter provides several functions to construct numerical semigroups from others (via intersections, quotients by an integer, removing or adding integers, etc.).</p>

<p><a id="X782F3AB97ACF84B8" name="X782F3AB97ACF84B8"></a></p>

<h4>5.1 <span class="Heading">
                    Adding and removing elements of a numerical semigroup
                </span></h4>

<p>In this section we show how to construct new numerical semigroups from a given numerical semigroup. Two dual operations are presented. The first one removes a minimal generator from a numerical semigroup. The second adds a special gap to a semigroup (see <a href="chapBib_mj.html#biBRGGJ03">[RGGJ03]</a>).</p>

<p><a id="X7C94611F7DD9E742" name="X7C94611F7DD9E742"></a></p>

<h5>5.1-1 RemoveMinimalGeneratorFromNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RemoveMinimalGeneratorFromNumericalSemigroup</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 one if its minimal generators.</p>

<p>The output is the numerical semigroup <span class="SimpleMath">\( \textit{S} \setminus\{\textit{n}\} \)</span> (see <a href="chapBib_mj.html#biBRGGJ03">[RGGJ03]</a>; <span class="SimpleMath">\(S\setminus\{n\}\)</span> is a numerical semigroup if and only if <span class="SimpleMath">\(n\)</span> is a minimal generator of <span class="SimpleMath">\(S\)</span>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);</span>
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">RemoveMinimalGeneratorFromNumericalSemigroup(7,s);</span>
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGeneratingSystemOfNumericalSemigroup(last);</span>
[ 3, 5 ]
</pre></div>

<p><a id="X865EA8377D632F53" name="X865EA8377D632F53"></a></p>

<h5>5.1-2 AddSpecialGapOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddSpecialGapOfNumericalSemigroup</code>( <var class="Arg">g</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">g</var> is a special gap of <var class="Arg">S</var>.</p>

<p>The output is the numerical semigroup <span class="SimpleMath">\( \textit{S} \cup\{\textit{g}\} \)</span> (see <a href="chapBib_mj.html#biBRGGJ03">[RGGJ03]</a>, where it is explained why this set is a numerical semigroup).</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">s2:=RemoveMinimalGeneratorFromNumericalSemigroup(5,s);</span>
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3:=AddSpecialGapOfNumericalSemigroup(5,s2);</span>
<Numerical semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElementsOfNumericalSemigroup(s) =</span>
<span class="GAPprompt">></span> <span class="GAPinput">SmallElementsOfNumericalSemigroup(s3);</span>
true                
<span class="GAPprompt">gap></span> <span class="GAPinput">s=s3;</span>
true
</pre></div>

<p><a id="X7DC65D547FB274D8" name="X7DC65D547FB274D8"></a></p>

<h4>5.2 <span class="Heading">Intersections, sums, quotients, dilatations, numerical duplications and multiples by integers</span></h4>

<p>We provide functions to build numerical semigroups from others by means of intersections, quotients, multiples and related constructions.</p>

<p><a id="X875A8D2679153D4B" name="X875A8D2679153D4B"></a></p>

<h5>5.2-1 Intersection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Intersection</code>( <var class="Arg">S</var>, <var class="Arg">T</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">‣ IntersectionOfNumericalSemigroups</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> and <var class="Arg">T</var> are numerical semigroups. Computes the intersection of <var class="Arg">S</var> and <var class="Arg">T</var> (which is a numerical semigroup).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup("modular", 5,53);</span>
<Modular numerical semigroup satisfying 5x mod 53 <= x >
<span class="GAPprompt">gap></span> <span class="GAPinput">T := NumericalSemigroup(2,17);</span>
<Numerical semigroup with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(S);</span>
[ 0, 11, 12, 13, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(T);</span>
[ 0, 2, 4, 6, 8, 10, 12, 14, 16 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection(S,T);</span>
<Numerical semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(last);</span>
[ 0, 12, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IntersectionOfNumericalSemigroups(S,T) = Intersection(S,T);</span>
true
</pre></div>

<p><a id="X7F308BCE7A0E9D91" name="X7F308BCE7A0E9D91"></a></p>

<h5><code>5.2-2 \+</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \+</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">S</var> and <var class="Arg">T</var> are numerical semigroups. Computes the sum of <var class="Arg">S</var> and <var class="Arg">T</var> (which is a numerical semigroup).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(4,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:=NumericalSemigroup(6,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGenerators(s+t);</span>
[ 4, 6, 7, 9 ]
</pre></div>

<p><a id="X83CCE63C82F34C25" name="X83CCE63C82F34C25"></a></p>

<h5>5.2-3 QuotientOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuotientOfNumericalSemigroup</code>( <var class="Arg">S</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \/</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> is an integer. Computes the quotient of <var class="Arg">S</var> by <var class="Arg">n</var>, that is, the set <span class="SimpleMath">\(\{ x\in {\mathbb N}\ |\ nx \in S\}\)</span>, which is again a numerical semigroup. <code class="code">S / n</code> may be used as a short for <code class="code">QuotientOfNumericalSemigroup(S, n)</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,29);</span>
<Numerical semigroup with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(s);</span>
[ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 29, 30, 32, 33, 35, 36, 38,
39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">t:=QuotientOfNumericalSemigroup(s,7);</span>
<Numerical semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(t);</span>
[ 0, 3, 5, 6, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u := s / 7;</span>
<Numerical semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(u);</span>
[ 0, 3, 5, 6, 8 ]
</pre></div>

<p><a id="X7BE8DD6884DE693F" name="X7BE8DD6884DE693F"></a></p>

<h5>5.2-4 MultipleOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MultipleOfNumericalSemigroup</code>( <var class="Arg">S</var>, <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup, and <var class="Arg">a</var> and <var class="Arg">b</var> are positive integers. Computes <span class="SimpleMath">\(a S\cup \{b,b+1,\to\}\)</span>. If <var class="Arg">b</var> is smaller than <span class="SimpleMath">\(a c\)</span>, with <span class="SimpleMath">\(c\)</span> the conductor of <span class="SimpleMath">\(S\)</span>, then a warning is displayed.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">N:=NumericalSemigroup(1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=MultipleOfNumericalSemigroup(N,4,20);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(s);</span>
[ 0, 4, 8, 12, 16, 20 ]
</pre></div>

<p><a id="X7F395079839BBE9D" name="X7F395079839BBE9D"></a></p>

<h5>5.2-5 NumericalDuplication</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalDuplication</code>( <var class="Arg">S</var>, <var class="Arg">E</var>, <var class="Arg">b</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup, and <var class="Arg">E</var> and ideal of <var class="Arg">S</var>, and <var class="Arg">b</var> is a positive odd integer, so that <span class="SimpleMath">\(2S\cup (2E+b)\)</span> is a numerical semigroup (this extends slightly the original definition where <var class="Arg">b</var> was imposed to be in <var class="Arg">S</var>, <a href="chapBib_mj.html#biBduplication">[DS13]</a>; now the condition imposed is <span class="SimpleMath">\(E+E+b\subseteq S\)</span>). Computes <span class="SimpleMath">\(2S\cup (2E+b)\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);</span>
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=6+s;</span>
<Ideal of numerical semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">ndup:=NumericalDuplication(s,e,3);</span>
<Numerical semigroup with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(ndup);</span>
[ 0, 6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24 ]
</pre></div>

<p><a id="X8176CEB4829084B4" name="X8176CEB4829084B4"></a></p>

<h5>5.2-6 AsNumericalDuplication</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsNumericalDuplication</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">T</var> is a numerical semigroup. Detects whether or not <var class="Arg">T</var> can be expressed as <code class="code">NumericalDuplication</code>(S,E,b), with E a proper ideal of S. Returns the list [S,E,b] if this is possible, and <code class="code">fail</code> otherwise.</p>

<p>Notice that a numerical semigroup can be represented in different ways as a numerical duplication.</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">ndup:=NumericalDuplication(s,6+s,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">asdup:=AsNumericalDuplication(ndup);</span>
[ <Numerical semigroup with 3 generators>, <Ideal of numerical semigroup>, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ndup = CallFuncList(NumericalDuplication,asdup);</span>
true
</pre></div>

<p><a id="X7DCEC67A82130CD8" name="X7DCEC67A82130CD8"></a></p>

<h5>5.2-7 InductiveNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InductiveNumericalSemigroup</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">a</var> and <var class="Arg">b</var> are lists of positive integers, with <span class="SimpleMath">\(k\)</span> the length of <var class="Arg">a</var> and <var class="Arg">b</var>, and such that <span class="SimpleMath">\(b[i+1]\ge a[i]b[i]\)</span> (<span class="SimpleMath">\(0\le i\le k-1\)</span>). Computes inductively <span class="SimpleMath">\(S_0=\mathbb N\)</span> and <span class="SimpleMath">\(S_{i+1}=a[i]S_i\cup \{a[i]b[i],a[i]b[i]+1,\to\}\)</span>, and returns <span class="SimpleMath">\(S_{k}\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=InductiveNumericalSemigroup([4,2],[5,23]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(s);</span>
[ 0, 8, 16, 24, 32, 40, 42, 44, 46 ]
</pre></div>

<p><a id="X81632C597E3E3DFE" name="X81632C597E3E3DFE"></a></p>

<h5>5.2-8 DilatationOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DilatationOfNumericalSemigroup</code>( <var class="Arg">S</var>, <var class="Arg">a</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup, and <var class="Arg">a</var> is a positive integer. If <span class="SimpleMath">\(M\)</span> is the maximal ideal of <var class="Arg">S</var>, then <var class="Arg">a</var> must be in <span class="SimpleMath">\(M-2M\)</span>. Computes the numerical semigroup <span class="SimpleMath">\(\{0\} \cup \{a+s \mid s\in M\}\)</span>, see <a href="chapBib_mj.html#biBdilatation">[BS19]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,4,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=MaximalIdeal(s);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(m-2*m);</span>
[ -3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d:=DilatationOfNumericalSemigroup(s,3);</span>
<Numerical semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(d);</span>
[ 0, 6 ]
</pre></div>

<p><a id="X867D9A9A87CEB869" name="X867D9A9A87CEB869"></a></p>

<h4>5.3 <span class="Heading">
                    Constructing the set of all numerical semigroups containing a given numerical semigroup
                </span></h4>

<p>In order to construct the set of numerical semigroups containing a fixed numerical semigroup <span class="SimpleMath">\(S\)</span>, one first constructs its unitary extensions, that is to say, the sets <span class="SimpleMath">\(S\cup\{g\}\)</span> that are numerical semigroups with <span class="SimpleMath">\(g\)</span> a positive integer. This is achieved by constructing the special gaps of the semigroup, and then adding each of them to the numerical semigroup. Then we repeat the process for each of these new numerical semigroups until we reach <span class="SimpleMath">\( {\mathbb N} \)</span>.</p>

<p>These procedures are described in <a href="chapBib_mj.html#biBRGGJ03">[RGGJ03]</a>.</p>

<p><a id="X7FBA34637ADAFEDA" name="X7FBA34637ADAFEDA"></a></p>

<h5>5.3-1 OverSemigroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OverSemigroups</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">‣ OverSemigroupsNumericalSemigroup</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 set of numerical semigroups containing it.</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">OverSemigroups(s);</span>
[ <The numerical semigroup N>, <Numerical semigroup with 2 generators>, 
  <Numerical semigroup with 3 generators>, 
  <Numerical semigroup with 3 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(last,s->MinimalGenerators(s));</span>
[ [ 1 ], [ 2, 3 ], [ 3 .. 5 ], [ 3, 5, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OverSemigroupsNumericalSemigroup(s) = OverSemigroups(s);</span>
true
</pre></div>

<p><a id="X8634CFB1848430DC" name="X8634CFB1848430DC"></a></p>

<h4>5.4 <span class="Heading"> Constructing the set of numerical semigroups with given Frobenius number</span></h4>

<p>Finding the set of all numerical semigroups with a given Frobenius number is not accomplished via over semigroups. In order to achieve this, we use fundamental gaps. If the multiplicity is fixed, then the construction relies on the calculation of irreducible numerical semigroups with that Frobenius number and multiplicity.</p>

<p><a id="X81759C3482B104D6" name="X81759C3482B104D6"></a></p>

<h5>5.4-1 NumericalSemigroupsWithFrobeniusNumberFG</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithFrobeniusNumberFG</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. The output is the set of numerical semigroups with Frobenius number <var class="Arg">f</var>. The algorithm implemented is given in <a href="chapBib_mj.html#biBRGSGGJM">[RGGJM04]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithFrobeniusNumberFG(15));</span>
200
</pre></div>

<p><a id="X7DB3994B872C4940" name="X7DB3994B872C4940"></a></p>

<h5>5.4-2 NumericalSemigroupsWithFrobeniusNumberAndMultiplicity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithFrobeniusNumberAndMultiplicity</code>( <var class="Arg">f</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> and <var class="Arg">m</var> are integers. The output is the set of numerical semigroups with Frobenius number <var class="Arg">f</var> and multiplicity <var class="Arg">m</var>. The algorithm implemented is given in <a href="chapBib_mj.html#biBBOR19">[BOR21]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithFrobeniusNumberAndMultiplicity(15,6));</span>
28
</pre></div>

<p><a id="X87369D567AA6DBA0" name="X87369D567AA6DBA0"></a></p>

<h5>5.4-3 NumericalSemigroupsWithFrobeniusNumber</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. As happens with the function <code class="func">NumericalSemigroupsWithFrobeniusNumberFG</code> (<a href="chap5_mj.html#X81759C3482B104D6"><span class="RefLink">5.4-1</span></a>), the output is the set of numerical semigroups with Frobenius number <var class="Arg">f</var>. It makes use of <code class="func">NumericalSemigroupsWithFrobeniusNumberAndMultiplicity</code> (<a href="chap5_mj.html#X7DB3994B872C4940"><span class="RefLink">5.4-2</span></a>) to compute the semigroups with the Frobenius number given for all the possible multiplicities.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithFrobeniusNumber(15));</span>
200
</pre></div>

<p><a id="X80CACB287B4609E1" name="X80CACB287B4609E1"></a></p>

<h5>5.4-4 NumericalSemigroupsWithFrobeniusNumberPC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithFrobeniusNumberPC</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. The output is the set of numerical semigroups with Frobenius number <var class="Arg">f</var>. It relies on pre-computed data, which is available for small values of <var class="Arg">f</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithFrobeniusNumberPC(15));</span>
200
</pre></div>

<p><a id="X8021419483185FE3" name="X8021419483185FE3"></a></p>

<h4>5.5 <span class="Heading"> Constructing the set of numerical semigroups with given maximum primitive</span></h4>

<p>Finding the set of all numerical semigroups with a given maximum primitive.</p>

<p><a id="X7C17AB04877559B6" name="X7C17AB04877559B6"></a></p>

<h5>5.5-1 NumericalSemigroupsWithMaxPrimitiveAndMultiplicity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithMaxPrimitiveAndMultiplicity</code>( <var class="Arg">M</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">M</var> and <var class="Arg">m</var> are integers. The output is the set of numerical semigroups with maximum primitive <var class="Arg">M</var> and multiplicity <var class="Arg">m</var>. The algorithm implemented is based on work by M. Delgado and Neeraj Kumar.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithMaxPrimitiveAndMultiplicity(15,6));</span>
27
</pre></div>

<p><a id="X875A8B337DFA01F0" name="X875A8B337DFA01F0"></a></p>

<h5>5.5-2 NumericalSemigroupsWithMaxPrimitive</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithMaxPrimitive</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">M</var> is an integer. The output is the set of numerical semigroups with maximum primitive <var class="Arg">M</var>. It makes use of <code class="func">NumericalSemigroupsWithMaxPrimitiveAndMultiplicity</code> (<a href="chap5_mj.html#X7C17AB04877559B6"><span class="RefLink">5.5-1</span></a>) to compute the semigroups with the given maximum primitive for all the possible multiplicities.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumericalSemigroupsWithMaxPrimitive(5);</span>
[ <Numerical semigroup with 2 generators>, 
  <Numerical semigroup with 2 generators>, 
  <Numerical semigroup with 3 generators>, 
  <Numerical semigroup with 2 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithMaxPrimitive(15));</span>
194
</pre></div>

<p><a id="X7DA1FA7780684019" name="X7DA1FA7780684019"></a></p>

<h5>5.5-3 NumericalSemigroupsWithMaxPrimitivePC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithMaxPrimitivePC</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">M</var> is an integer. The output is the set of numerical semigroups with maximum primitive <var class="Arg">M</var>. It relies on pre-computed data, which is available for small values of <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithMaxPrimitivePC(15));</span>
194
</pre></div>

<p><a id="X7D6635CB7D041A54" name="X7D6635CB7D041A54"></a></p>

<h4>5.6 <span class="Heading">
                    Constructing the set of numerical semigroups with genus g
                </span></h4>

<p>Given a numerical semigroup of genus g (that is, with exactly g gaps), removing minimal generators, one obtains numerical semigroups of genus g+1. In order to avoid repetitions, we only remove minimal generators greater than the Frobenius number of the numerical semigroup (this is accomplished with the local function sons).</p>

<p>These procedures are described in <a href="chapBib_mj.html#biBRGGB03">[RGGB03]</a> and <a href="chapBib_mj.html#biBB-A08">[Bra08]</a>.</p>

<p><a id="X86970F6A868DEA95" name="X86970F6A868DEA95"></a></p>

<h5>5.6-1 NumericalSemigroupsWithGenus</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithGenus</code>( <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">g</var> is a nonnegative integer. The output is the set of numerical semigroups with genus <var class="Arg">g</var>. If the user just wants to use some numerical semigroup with a given genus pseudo-randomly choosen, he is probably looking for the function <code class="func">RandomNumericalSemigroupWithGenus</code> (<a href="chapB_mj.html#X78A2A0107CCBBB79"><span class="RefLink">B.1-7</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumericalSemigroupsWithGenus(5);</span>
[ <Numerical semigroup with 6 generators>, 
  <Numerical semigroup with 5 generators>, 
  <Numerical semigroup with 5 generators>, 
  <Numerical semigroup with 5 generators>, 
  <Numerical semigroup with 5 generators>, 
  <Numerical semigroup with 4 generators>, 
  <Numerical semigroup with 4 generators>, 
  <Numerical semigroup with 4 generators>, 
  <Numerical semigroup with 4 generators>, 
  <Numerical semigroup with 3 generators>, 
  <Numerical semigroup with 3 generators>, 
  <Numerical semigroup with 2 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(last,MinimalGenerators);</span>
[ [ 6 .. 11 ], [ 5, 7, 8, 9, 11 ], [ 5, 6, 8, 9 ], [ 5, 6, 7, 9 ], 
  [ 5, 6, 7, 8 ], [ 4, 6, 7 ], [ 4, 7, 9, 10 ], [ 4, 6, 9, 11 ], 
  [ 4, 5, 11 ], [ 3, 8, 10 ], [ 3, 7, 11 ], [ 2, 11 ] ]
</pre></div>

<p><a id="X7B4F3B5E841E3853" name="X7B4F3B5E841E3853"></a></p>

<h5>5.6-2 NumericalSemigroupsWithGenusPC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithGenusPC</code>( <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">g</var> is a nonnegative integer. The output is the set of numerical semigroups with genus <var class="Arg">g</var>. It relies on pre-computed data, which is available for small values of <var class="Arg">g</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithGenusPC(15));</span>
2857
</pre></div>

<p><a id="X8265233586477CC7" name="X8265233586477CC7"></a></p>

<h4>5.7 <span class="Heading">
                    Constructing the set of numerical semigroups with a given set of pseudo-Frobenius numbers
                </span></h4>

<p>Refer to <code class="func">PseudoFrobeniusOfNumericalSemigroup</code> (<a href="chap3_mj.html#X861DED207A2B5419"><span class="RefLink">3.1-24</span></a>).</p>

<p>These procedures are described in <a href="chapBib_mj.html#biBDGSRP15">[DGR16]</a>, and are used to find the set of numerical semigroups with a prescribed set of pseudo-Frobenius numbers.</p>

<p><a id="X874B252180BD7EB4" name="X874B252180BD7EB4"></a></p>

<h5>5.7-1 ForcedIntegersForPseudoFrobenius</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ForcedIntegersForPseudoFrobenius</code>( <var class="Arg">PF</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">PF</var> is a list of positive integers (given as a list or individual elements). The output is:</p>


<ul>
<li><p>in case there exists a numerical semigroup <span class="SimpleMath">\(S\)</span> such that <span class="SimpleMath">\(PF(S)=PF\)</span>:</p>


<ul>
<li><p>a list <span class="SimpleMath">\([forced\_gaps,forced\_elts]\)</span> such that:</p>


<ul>
<li><p><span class="SimpleMath">\(forced\_gaps\)</span> is contained in <span class="SimpleMath">\({\mathbb N} - S\)</span> for any numerical semigroup S such that <span class="SimpleMath">\(PF(S)=\{g\_1,\ldots,g\_n\}\)</span></p>

</li>
<li><p>forced_elts is contained in <span class="SimpleMath">\(S\)</span> for any numerical semigroup <span class="SimpleMath">\(S\)</span> such that <span class="SimpleMath">\(PF(S)=\{g\_1,\ldots,g\_n\}\)</span></p>

</li>
</ul>
</li>
</ul>
</li>
<li><p>"fail" in case it is found some condition that fails.</p>

</li>
</ul>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pf := [ 58, 64, 75 ];</span>
[ 58, 64, 75 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ForcedIntegersForPseudoFrobenius(pf);                              </span>
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 11, 15, 16, 17, 25, 29, 32, 58, 64, 75 ], 
  [ 0, 59, 60, 67, 68, 69, 70, 71, 72, 73, 74, 76 ] ]
</pre></div>

<p><a id="X87AAFFF9814E9BD2" name="X87AAFFF9814E9BD2"></a></p>

<h5>5.7-2 SimpleForcedIntegersForPseudoFrobenius</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleForcedIntegersForPseudoFrobenius</code>( <var class="Arg">fg</var>, <var class="Arg">fe</var>, <var class="Arg">PF</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Is just a quicker version of <code class="func">ForcedIntegersForPseudoFrobenius</code> (<a href="chap5_mj.html#X874B252180BD7EB4"><span class="RefLink">5.7-1</span></a>)</p>

<p><var class="Arg">fg</var> is a list of integers that we require to be gaps of the semigroup; <var class="Arg">fe</var> is a list of integers that we require to be elements of the semigroup; <var class="Arg">PF</var> is a list of positive integers. The output is:</p>


<ul>
<li><p>in case there exists a numerical semigroup <span class="SimpleMath">\(S\)</span> such that <span class="SimpleMath">\(PF(S)=PF\)</span>:</p>


<ul>
<li><p>a list <span class="SimpleMath">\([forced\_gaps,forced\_elts]\)</span> such that:</p>


<ul>
<li><p><span class="SimpleMath">\(forced\_gaps\)</span> is contained in <span class="SimpleMath">\({\mathbb N} - S\)</span> for any numerical semigroup S such that <span class="SimpleMath">\(PF(S)=\{g\_1,\ldots,g\_n\}\)</span></p>

</li>
<li><p>forced_elts is contained in <span class="SimpleMath">\(S\)</span> for any numerical semigroup <span class="SimpleMath">\(S\)</span> such that <span class="SimpleMath">\(PF(S)=\{g\_1,\ldots,g\_n\}\)</span></p>

</li>
</ul>
</li>
</ul>
</li>
<li><p>"fail" in case it is found some condition that fails.</p>

</li>
</ul>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pf := [ 15, 20, 27, 35 ];;                                               </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fint := ForcedIntegersForPseudoFrobenius(pf); </span>
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 20, 27, 35 ], 
  [ 0, 19, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">free := Difference([1..Maximum(pf)],Union(fint));</span>
[ 11, 13, 14, 17, 18, 21, 22, 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SimpleForcedIntegersForPseudoFrobenius(fint[1],Union(fint[2],[free[1]]),pf);</span>
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 20, 24, 27, 35 ], 
  [ 0, 11, 19, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36 ] ]
</pre></div>

<p><a id="X7D6775A57B800892" name="X7D6775A57B800892"></a></p>

<h5>5.7-3 NumericalSemigroupsWithPseudoFrobeniusNumbers</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsWithPseudoFrobeniusNumbers</code>( <var class="Arg">PF</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">PF</var> is a list of positive integers (given as a list or individual elements). The output is: a list of numerical semigroups S such that PF(S)=PF. When Length(PF)=1, it makes use of the function <code class="func">NumericalSemigroupsWithFrobeniusNumber</code> (<a href="chap5_mj.html#X87369D567AA6DBA0"><span class="RefLink">5.4-3</span></a>)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pf := [ 58, 64, 75 ];</span>
[ 58, 64, 75 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(NumericalSemigroupsWithPseudoFrobeniusNumbers(pf));</span>
561
<span class="GAPprompt">gap></span> <span class="GAPinput">pf := [11,19,22];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumericalSemigroupsWithPseudoFrobeniusNumbers(pf);</span>
[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>, 
  <Numerical semigroup>, <Numerical semigroup> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(last,MinimalGenerators);   </span>
[ [ 7, 9, 17, 20 ], [ 7, 10, 13, 16, 18 ], [ 9, 12, 14, 15, 16, 17, 20 ], 
  [ 10, 13, 14, 15, 16, 17, 18, 21 ], 
  [ 12, 13, 14, 15, 16, 17, 18, 20, 21, 23 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Set(last2,PseudoFrobeniusOfNumericalSemigroup);    </span>
[ [ 11, 19, 22 ] ]
g</pre></div>

<p><a id="X862DBFA379D52E2C" name="X862DBFA379D52E2C"></a></p>

<h5>5.7-4 ANumericalSemigroupWithPseudoFrobeniusNumbers</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ANumericalSemigroupWithPseudoFrobeniusNumbers</code>( <var class="Arg">PF</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">PF</var> is a list of positive integers (given as a list or individual elements). Alternatively, a record with fields "pseudo_frobenius" and "max_attempts" may be given. The output is: A numerical semigroup S such that <span class="SimpleMath">\(PF(S)=PF\)</span>. Returns fail if it concludes that it does not exist and suggests to use NumericalSemigroupsWithPseudoFrobeniusNumbers if it is not able to conclude...</p>

<p>When <span class="SimpleMath">\(Length(PF)=1\)</span> or <span class="SimpleMath">\(Length(PF)=2\)</span> and <span class="SimpleMath">\(2*PF[1] = PF[2]\)</span>, it makes use of the function <code class="func">AnIrreducibleNumericalSemigroupWithFrobeniusNumber</code> (<a href="chap6_mj.html#X7C8AB03F7E0B71F0"><span class="RefLink">6.1-4</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pf := [ 83, 169, 173, 214, 259 ];;                     </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ANumericalSemigroupWithPseudoFrobeniusNumbers(pf);</span>
<Numerical semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">gen := MinimalGeneratingSystem(last);</span>
[ 38, 57, 64, 72, 79, 98, 99, 106, 118, 120, 124, 132, 134, 146, 147, 154, 
  165, 168, 179 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ns := NumericalSemigroup(gen);       </span>
<Numerical semigroup with 19 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">PseudoFrobeniusOfNumericalSemigroup(ns);</span>
[ 83, 169, 173, 214, 259 ]
</pre></div>


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